Talk:Conglomerate (mathematics)

2070:

does this mean?" does not appear, and after that they construct models (again with a honest explanation of how the objects in the new theory are interpreted in the old theory, so that misunderstandings don't appear). But here the situation is opposite: people pretend that they extend the modern set theories, NBG and MK, by "adding new objects", but when you look closer at what they suggest as "axiomatic theories" it becomes clear that those theories are absurd, because these people simply don't understand the difference between naive set theory and modern axiomatic set theories, trying to construct modern set theory by methods of 19 century. As if they never heard about paradoxes. And moreover, the only reasonable explanation of what they do, as it turns out, is that they don't add something, but on the contrary cut some pieces off NBG and MK. And what looks absolutely scandalous, they use for this a trick that normal people can't expect: they just rename the objects. What becomes visible only after a long study in an appendix, in which no one had guessed to look. And the most puzzling questions like what we discussed, -- "how can a proper class belong to something?" --- disappear just because proper classes are thrown away from their "great theory". This is called "breach of trust".

3265:"there is no universe in MK"? Not quite so. Note that ZFC does not prove existence of Grothendieck universes (which can be proved, assuming that ZFC is consistent), but it also does not disprove their existence (according to a common belief that ZFC+large cardinal is still consistent). Thus, the number of Grothendieck universes in ZFC is ill defined (and, informally, it means for me that the "real" place of the border between sets and proper classes is left uncertain). And again, "universe" in general (not quite "Grothendieck") is a vague idea; but surely it is reasonable to treat the class of all sets (be it in NBG or MK) as a universe. And in addition, any number of Grothendieck universes may exist (or not) in MK just as in ZFC. 3339:. And anyway, "the larger universe" is replaced soon by "the class of all sets". On the other hand, if it exists among sets (which could happen within ZFC), then it can be used as well. And "on the third hand", the underlying theory can be taken to be NBG, and then the class of all sets is a legitimate universe even in your approach, isn't it? I did not write whether I mean ZFC or NBG there. And, after all, my text is intended to be the lead; detailed explanations and clarifications may follow in sections; but the big picture should be reasonably understandable, – not only what conglomerate is (basically), but also, what is it good for (basically). 281: 479:, with a series of formulas "that define them" (and a new term, "conglomerate"). This is an extension by definition. The new formulas don't cancel the old theorems of GB, everything what was true in GB, is true in the new theory as well. In particular, proper classes still can't belong to anything. If he would define a new relation "X belongs to Y", we could understand this phrase — "proper class belongs to this object" — differently. But he does not do this (and he uses the same symbol for this relation 271: 250: 2179:. I also did not see that the definition of class can be "shifted" (and I don't understand why this can be useful). Despite all this, as I told before, I would not be against keeping this article, provided it will contain an honest explanation of the essense of the matter so that these traps would disappear. And the essense in my opinion is that this is a trick that cuts off some parts of NBG and MK and renames objects. I am not sure that this will not be an original research. 3170:

is a virtual machine REALLY a finite sequence of bits, or not? For me, this is a wrong way to ask. A real number is a member of a complete ordered field. Nowadays all this is usually interpreted within ZFC, and therefore every number is a set anyway. But this is rather an implementation. Likewise, a file containing an image of a virtual machine is stored on a hard disk and therefore is a finite sequence of bits anyway. But again, this is rather an implementation. See also

3174:. A mathematical object has no onthology; its implementation has (more or less...), but implementations may differ. Closer to our business: for me, the phrase '"conglomerate" is a synonym of "set"' (even with light reservations) sounds strange and a bit misleading, just like 'a "real number" is a special case of a "set"' or 'a "virtual machine" is a special case of a "finite sequence of bits"'. My feeling is that you (rather than me) introduce some strange diplomacy. 2234:" is me? :-) Not at all. I know about him not more than is published on his page (available to both of us). Why before? Since the consensus to "keep" is already reached; and now everyone (including you) is free to edit this article (as well us revert other's edit, when appropriate). "Closed as keep" does not mean "no more edits to the article", nor "no more edits to the AFD page"; it means only "no more edits to the closed discussion there". 219: 5755:. This seems to be the most reasonable alternative, with most of the initial supporters of the proposed move commenting that it is acceptable. Unless there is a competing use of "conglomerate" used as prominently in some other field of mathematics, the rarity of the descriptor is no barrier to the use of "mathematics" as a disambiguation term, compared to things like 3237:"), thus it is not forbidden to write "the larger universe is usually the class of all sets". A Grothendieck universe must be a set, but the general idea of universe is more permissive. Still, if something can be said more clear, I'll be glad. But for me, your wording is unclear for another reason. Also, the lead must be short; clarifications may follow in sections. 190: 949: 2175:

for me, and I still don't understand why it is necessary. I also was working in category theory and faced some problems in its "foundations". In particular, I needed a proposition that each category has a skeleton. But I resolved this problem inside MK, without this strange trick of cutting off something, see Property 1 at page 5

3333:"In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC." 5282:

should follow the same path. First give (probably) informal requirements and then give definition (that we already have). I like Eozhik's text on this page (04:44, 4 May 2019). I will try to combine this text, my old text and other comments, and make a suggestion, unless someone else propose something before me.

4341:

Boris, in my understanding, this is not a lead, but almost all what can (and should) be written here. I would just add references and mention that this construction describes a model of ZFC/NBG/MK in the "extended ZFC/NBG/MK". This model, actually, makes this theory reasonable. Here is the extended

3169:

About "straitforward, non-diplomatic", I do not understand you. It seems, for you the question "is a conglomerate REALLY a set or not?" is somewhat political or religious; or, at the very least, ontological. Let me ask you then: is a number (no matter, natural, or real, etc) REALLY a set, or not? Or:

2069:

I have another explanation: the point is that I am a punctual person, attentive to details. The difference between your geometric examples and what we see here is that in non-Euclidean geometries people honestly describe the axiomatic theory, with all the necessary details, so that the question "what

1613:

Zhang's paper is absent in mathscinet. I am not against if in the article it would be written that this is "folklore", not a rigor theory. The times changed: when Cantor lived there was no necessity to specify this, but now this necessity appeared, since there is a sharp border between "folklore" and

751:

No-no! Mathematicians are obliged to be accurate, otherwise what they do is not called mathematics. Moreover, this part of mathematics is called mathematical logic. It is built directly for studying proofs in mathematics, so mathematicians must be especially scrupulous about their constructions here.

5907:

All this is original research on my part, as I cannot find the axioms of ACG, or of any system, in any of the references I have looked at. Category people might not care which implementation is made, but any that we can find in reliable sources should be included if the article is to remain as (set

5281:

for your work. I like the new sections "Definition", "Corollaries" and "Terminology". The lead section, however, is too narrow now. I like the approach by Herrlich & Strecker: they first describe the properties of conglomerates that are required and then in Appendix give a definition. I think we

1300:

build a first order theory and does prove what he claims and what I used in the article. Finally, you cannot just say that all sources are unreliable. If a source is published in reliable peer reviewed journal or by well-known scientific publisher you have to provide a good argument why do you think

823:

Well, tastes differ. Yes, mathematicians are obliged to be accurate. But I really do not understand why do you write "absurd idea" (!) when I observe rather understandable lapsus (in the formalization, not in the idea). Clearly, the author, writing "the first part is just GB", did not think that you

516:

But, as far as I understand, this is not an extension by definition. Yes, axioms of GB still hold, as well as all theorems of GB. But new objects are added, of new types "conglomerate" and "second class"; these are neither sets nor classes (and therefore never appear in axioms, nor theorems, of GB).

5988:

As an aside, my late mother, and one of her students, were working on topology without AC. I don't know of further progress; her death was about 17 years ago. My father and I had been working on adapting a 1939 Czech paper to work without AC. Topology is significantly different without AC, but I

1724:

must be supplied with correct explanations of what this exactly means. Because when a reader understands this literally, this becomes lies: not all classes from the point of view of the commonly used set theories, where this term is used, NBG or MK, form a conglomerate in this "terminology". To say

1675:

Yes, I see. A pity that the book does not say explicitly (and prove) that consistency of this theory follows from consistency of ZFC plus large cardinal(s). Many set theorists tend to believe in the latter consistency. On the other hand, the book is very close to this point, and probably, an expert

1061:

are only applied to ACG which is described in Jinwen. So you should not expect to find this term in other references. About Grothendieck universe: I've added a link to a scan of Herrlich & Strecker. You can check that they talk about Grothendieck universe. Nel and Laan name this just "universe"

780:

its intended interpretation. First mathematicians generate a new idea, try it informally, get some intuition, then the idea matures, and is formalized (sometimes with some trials and errors; of course, I also prefer everyone to be always correct, but...). And, I believe, the relativization outlined

5730:

Yes the main application of conglomerates is category theory, but the notion itself is from the set theory. This is set-theoretic foundation of category theory. Category theory folk do not use this term unless they talk about set theoretic foundation. There is no known contradiction with the right

3188:

Boris, I don't see anything religious in my understanding of sets (and I think, it actually does not matter). Look: there is a necessity to explain the differences between "conglomerates" and "sets" for people who perceive sets as objects of axiomatic set theories like ZFC, NBG and MK. Your phrase

2400:

This approach has been used and developed by Isbell, Mac Lane, and Feferman and is the foundational approach taken for this text. It consists 'essentially' of the addition of one extra stage of flexibility to the Gödel-Bernays-von Neumann approach, as opposed to the addition of a plethora of extra

2174:

Boris, I understood what can be done for constructing first order theory after seeing how they really define these objects in that Appendix in Herrlich&Strecker. But we can't write this here because this will be an original research. This trick with cutting off parts of NBG or MK is indeed new

647:(see J. Shoenfield, Mathematical logic, 1967, 4.4.7) of the first order theory in another first order theory (or a model what is the same up to terminology), so that this would look like some kind of extension, but Zhang Jinwen does not do this. And the same with the other autors. This is absurd. 802:

Boris, I would not protest if the article contained an explanation like "conglomerates are just an idea which did not find a rigor definition yet..." and something like this. But the article does not contain such explanations. And this is a cheating, as I wrote above. Creating a parallel reality

4901:

Yes, applications are desirable. I don't know them. If there will appear people who know, that would be good, although as I told before, I don't believe that this notion is really useful. The facts from category theory that I used all are proved without conglomerates (and without difficulties).

1456:

After a search on MathSciNet I feel astonished. Conglomerates are used in more than 10 publications reviewed (and reviews). No one of these defines this notion, nor cites a source where it is defined (or did I miss something?); all just accept it to be well-known. I guess, J.Adamek, H.Herrlich,

4930:

OK with me. A pity that Alexei Kopylov is not with us now; but nothing is lost here, everything can be restored from the history anyway. Being not at all a category theorist, I have no opinion, what is really useful and what is not. However, I see sections "categories of categories" in and ;

3289:-- does not exist in MK. It can be defined, I think, by introducing some supplementary symbols and axioms, but people don't do this. And that is why the reasoning about two universes, big and small, sounds confusing. (As far as I understand, in NBG and even in ZFC the situation is the same.) 4034:

Yes, the latter sounds better. But still, it may seem а nonsense ("replace of a term does not change the meaning") until reading the second paragraph (and thinking). Also I bother a bit about "extensions ... like ZF, NBG or MK"; the reader that knows already that NBG and MK extend ZF could

5479:

the move, as the sources I found were all from the categorical or closely related literature. I can understand the desire for no redirect, but I would only support that option if links to this article are all updated to point to the new title; there is no good reason to break wiki links.

1652:

they said that there are several attempts to build foundation for category theory, each of them with some disadvantages. Then they list the features that they requiry of such foundation (including conglomerates). And they say that these features can be realized. They show how to do it in

2349:

The first phrase "In mathematics, a conglomerate is a collection of classes, just as a class is a collection of sets" is incorrect (and I did not find it in the sources). A member of a conglomerate is generally another conglomerate, and in particular, it may be a class or even a set.

3142:

By the way, this picture seems to give an explanation of how the notion of cardinality for all "conglomerates", including "proper classes in new sense" (and this means, in old sense as well, since they are presented in this model), can be defined? An interesting theory...

1547:. 528 means that people cite different facts from category theory, not connected to "conglomerates". 10 illustrates the fact that there is some percent of scientists who are very narrow specialists. They don't check the facts even if they lay near (but not in their field). 3279:

Boris, the problem is not that Grothendieck universes can exist or not in MK (and because of this we need a supplementary axiom that it exists). The problem is that there is no universe in MK as an object that contains all other objects as elements. What is described in

5937:

I think, the question "What IS the theory" applies to category theory no more than to (say) topology, or algebra. Generally, they are insensitive to reasonable variations of the underlying set theory. In rare cases of such sensitivity, it is stated explicitly. Right?

1743:

Whoever intermixes words taken from different contexts, faces the consequences. (By the way, this is a well-known problem of math on wikipedia; editor "fixes an error" and does not bother to check that his terminology/notation agrees with that in the article; see

1457:

G.Strecker (The Joy of Cats...) is the implicit source. The review of this source is linked 528 times from refs and 10 times from reviews. In the end of the review the source is criticized for esoteric terminology (but not conglomerates). What does it all mean??

2113:(Unindent) You really do not understand that the boundary between sets and classes is not ontological, it shifts whenever we accept new large cardinal (and its first shift was the axiom of infinity). But, well, you want a first order theory anyway; and this is 3088:) this does not lead to a loss of information about objects of MK since the representation of MK as a model in "MK+Grothedieck universe" means that what can be proved in MK about its usual objects can be proved in "MK+Grothedieck universe" about subsets of 2883:

Do you think it is correct to say that when we add to NBG or to MK the axiom of existence of a Grothendieck universe we obtain two universes? Initially there were no big universes in NBG and in MK, when we add one universe we obtain one universe, not two.

517:

Accordingly, there cannot be a definition of a conglomerate as "a class satisfying ...", as it should be in any extension by definition. And in fact, the extension is non-conservative (as noted in the article). Yes, he adds to GB new predicate symbols

3489:

of all sets can't be a universe in NBG or in MK. There are also proper classes which are not sets. This is confusing. If on the other hand you are speaking about ZFC, then its "universe" is not an object of ZFC. This is also confusing. (By the way,

2142:

Second group of axioms: "just" the axioms of ZFC, but relativized to U, which is automatic if you say them by words (the word "set" has already the new meaning); but technically, use bounded (by U) quantifiers. Or instead, one axiom: κ is a worldly

1762:

About conservative extensions: now, to your liking, it IS an extention by definitions of "ZFC + worldly cardinal" (up to a pedagogical inconvenience that some words are REdefined); and "ZFC + worldly cardinal" is a nonconservative extension of ZFC.

362:

The term "conglomerate" does not have rigorous definition in mathematics. The authors don't define it as an object in a known axiomatic first order theory, and don't construct a special first order theory for it. 7 years ago we discussed this at

5119:

I think, we can treat this as a reference for our claim that this construction gives a model of ZFC/NBG in "ZFC/NBG+Grothendieck universe". As far as I understand, he does not write this directly, I see only an indirect mention (again at page

4800:

I must say, I don't see what can be added (not counting references). The section about "axiomatic theory of conglomerates" does not have reliable sources. And what is discussed "beyond conglomerates" is logical to discuss beyond this article.

4945:

For being with us it is desirable to understand the details. OK, I've made the edits. I removed only the sources where conglomerates are not mentioned or are mentioned in passing. Perhaps, some other sources can be removed as well. Or added.

3379:

of all sets is not an object of ZFC, it is defined in MK (and in NBG), not in ZFC. NBG (and MK) do not have "big universe" because there is no an object in NBG (and in MK) which contains all objects (i.e. all classes) as elements. The class

3067:

As far as I understand, in this philosophy they refuse to consider objects which are not sets in MK, i.e. which are "proper classes in the old sense", but (not counting the possible problems caused by the supplementary axiom of existence of

3883:, what is desirable in category theory (so that each class can be considered as an element of a "more general collection", a conglomerate). Technicaly this is organized by the following changes in terminology: it is considered convenient 2963:, what is desirable in category theory (so that each class can be considered as an element of a "more general collection", a conglomerate). Technicaly this is organized by the following changes in terminology: it is considered convenient 1355:

5. We came to the understanding that Herrlich and Strecker is the only source that can be discussed (in particularly, Adamek et al refer to them). The problem with this text is that the notion of conglomerate is introduced there just by

1343: 4960:

I added a partition into sections. What is written in "Corollaries" is, in my opinion, an original research. (And that is why I initiated the procedure of deletion.) I forsee accusations if the necessary sources will not be found.

3446:

It need not contain everything (including classes). I wrote clearly "class of all sets"; and not "big universe" but just "the larger universe". Once again, it seems to be usual enough, to say "the class of all sets is a universe".

838:

Absurd because those people don't see absurdities in their constructions. And I dare to say that this grows from their poor understanding of what logic is. If they would be better educated there would not be problems with this:

1028:

From Google books I conclude also that Zhang Jinwen mentiones only the term "non-conservative extension" from this list, but it is seen only that he applies it to abstract systems. It is not seen that he claims what you write

3128:) since this is another cardinality, inside the new theory. I would add explanations like this into the text, if customs allow this (what seems doubtful for me, since what we discuss here I perceive as an original research). 153: 4457:, what is desirable in category theory (so that each class can be considered as an element of a "more general collection", a conglomerate). Technicaly this is organized by the following changes in terminology: when a 4176:, what is desirable in category theory (so that each class can be considered as an element of a "more general collection", a conglomerate). Technicaly this is organized by the following changes in terminology: when a 2259:

is absolutely correct, "keep" does not mean that we keep the article as it is, it just means that the topic is not deleted, and editors may (and of course are roundly encouraged to) improve upon it. Happy editing!

3527: 3702: 2851:, these models are guests, supported by the host (the set theory underlying all that). Sure, one may extend the host by its classes (be it NBG or MK), but this is probably irrelevant to the virtual "machines". 991:

Done badly. The question was not what all these terms mean, but how they are related to conglomerates. Among authors who mention conglomerates in their texts, the following people do not speak at all about

824:

will interpret it this way. He meant that the classes satisfy GB, but are only a part of his wider universe. And I do not think that the privilege to introduce new objects must be reserved for logicians.

873:

Well, now our (different) opinions are voiced clearly; we agree that some authors are not so educated in some branches of mathematics, but we treat this fact differently. I have nothing to add here.

5968:

I would think that KP+vNBG should be adequate, but category theorists generally don't care. (KP is essentially the minimum required for transfinite recursion over non-set ordinals to be defined.) I

2148:

Did I forgot something? Maybe. Anyway, this is of course my "original research" (until someone will find it somewhere, which will happen inevitably, but I do not know, when). Anyway, it shows that

3626: 424:

In this article we develope the axiom system of conglomerates denoted as ACG. The system ACG falls into two parts, the first part is just GB, the second part is composed of five groups A,B,C,D,E.

1349:

3. There is no requirement to build a formal theory for people who are not related to mathematics. Mathematicians are obliged to give formal definitions and formal proofs. That is the difference.

1690:

And, really, who needs a special first order theory for conglomerates? Let it be just a convenient (for category theory) special terminology over the theory "ZFC + worldly cardinal". The set

3199:-- seems confusing for me, because if a person tries to understand this applying his knowledge about, say, MK, that would sound strange for him, since there is no universe in MK. When we add 1120:. It will be honest with respect to the reader to rewrite the article in such a way that this becomes clear (and to remove everything that is not supported by the reliable sources, including 4775:). At the same time, the passage to the new theory does not give an equivalent theory in this situation, since there can appear some extra propositions about classes which can be proved in " 337: 2205: 1173: 409: 200: 733:; a human often is able to catch author's idea and interpret the text accordingly. Later, when, and if, the matter becomes well-known and enters textbooks, author's lapsus will be fixed. 1997:. Do you find it ugly, that "In this model, lines and segments are straight Euclidean segments, whereas in the Poincaré disk model, lines are arcs that meet the boundary orthogonally"? 147: 2799:, only the first two or three levels are used in "ordinary analysis and probability theory", and nevertheless, the prevailing notion is σ-algebra (including the whole hierarchy). 1382:

with these properties. What was called "sets" before we now call "conglomerates". By "classes" we now mean (not arbitrary classes, or arbitrary sets in NBG, but) only subsets of

3726: 3650: 3592: 3568: 2562:, one translates the three-level terminology of sets, classes and conglomerates into the one-level terminology of the set theory as follows: "set" is translated as "member of 2464:, one translates the three-level terminology of sets, classes and conglomerates into the one-level terminology of the set theory as follows: "set" is translated as "member of 4883:

About "what can be added". Maybe, an example of a notion or/and result of category theory that involves conglomerates. Probably it will also show that the power-powerset P(P(

3728:, i.e. again sets. Every time we pass from sets to sets. All numbers can be treated as sets. Or you mean something else? Are there theories where this scheme does not work? 1815: 6026: 44: 3171: 5679: 5655: 4788: 4776: 4768: 4716: 4708: 4696: 4688: 4656: 4587: 4575: 4485: 4404: 4278: 4123: 4008: 3962: 3834: 3038: 2914: 2708: 2655: 1841: 1221: 1091: 2715:, they don't consider a "big universe" that "contains all objects (classes) as elements". Maybe this means that the first paragraph also needs some more corrections. 5094: 4831:. This is bad. (By model I mean the construction from Takeuti's "Proof theory", I.8, but the definition must be presented in different books on mathematical logic.) 3961:

So, in particular, if the new theory was constructed as a modification of the theories where the term "class" was used (as a contraposition to the term "set"), like

3037:

So, in particular, if the new theory was constructed as a modification of the theories where the term "class" was used (as a contraposition to the term "set"), like

1881: 497: 4649: 4277:(So, in particular, if the new theory was constructed as a modification of the theories where the term "class" was used as a contraposition to the term "set", like 3430: 3404: 3377: 694: 641: 615: 567: 541: 477: 451: 5161: 5141: 5114: 4765: 4745: 4685: 4616: 4561: 4538: 4515: 4478: 4436: 4394: 4370: 4268: 4245: 4222: 4197: 4155: 4113: 4089: 3953: 3930: 3907: 3862: 3217: 3126: 3106: 3086: 3030: 3007: 2984: 2942: 1915: 1861: 1777:

And the redefinition of "sets" and "classes" is not my "original research", it is taken from the source mentioned. I only added some relation to large cardinals.

1579:

Yes, I excluded intentionally such authors as Adamek and Zhang from this list. And probably all this shows again that (like it or not) a formal theory is created

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I made that decision. It was "past due" for closure by four days, mostly because I don't think any administrator wanted to read through it, per my close note.

845:

And in each field there are some rules. If you introduce new objects in foundations of mathematics you must do this by the rules of foundations of mathematics.

1509:

Summary: "...Galois equivalence between the conglomerate of idempotent and weakly hereditary closure operators of X and the conglomerate of subclasses of M..."

3331:

I do not think this is a problem, as long as the word "universe" refers to a vague idea rather than a well-defined math notion. My usage conforms to the quote

662:

On the other hand, probably, this is an inadvertence of Zhang Jinwen. He should clarify that the first part is not "just GB"; rather, here each axiom of GB is

2577:(In MK functions between classes are considered without problem. And I think there is no difference between "family of classes" and "collection of classes".) 734: 2541:(whose existence is postulated); the larger universe is usually the class of all sets. A single universe is sufficient for most branches of mathematics, but 2447:(whose existence is postulated); the larger universe is usually the class of all sets. A single universe is sufficient for most branches of mathematics, but 5015:

Boris, I can't find the whole text of MacLane's paper, I see only two pages. What is written there, because of what you refer to this text as the source?

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its intended interpretation. And this: for category theory, the details of the definition are not important. Note also that most of these works appeared

1032:

ACG is a non-conservative extension of Quine-Morse set theory and ZF# (Zermelo–Fraenkel set theory with an axiom of existence of inaccessible cardinals).

327: 79: 1157:, you have added "Unreliable source" after two sources. Could you provide reasons why do you believe this sources are unreliable in subsections bellow? 6056: 2159:. But I know, you are dissatisfied. Well, I did my best. And fortunately, category theorists do not need your permission (nor wikipedia's permisson). 5096:) themselves satisfy the ZFC axioms for set theory. For that matter, if we take "set" to be small set and "class" to be any subset of the universe 1544: 804: 3406:

of all sets is defined in NBG and in MK, but it does not contain all objects of NBG and MK as elements, only sets (proper classes don't belong to

2558:

Universes are instrumental when formalizing concepts used in category theory inside set-theoretical foundations. Denoting the smaller universe by

2460:

Universes are instrumental when formalizing concepts used in category theory inside set-theoretical foundations. Denoting the smaller universe by

1748:.) But, of course, it is wise to warn the reader whenever a misunderstanding is expected. This is a pedagogical matter, rather than mathematical. 1288:

First of all, there is no requirement to build a first order theory. It can be high order theory of anything else. Second, this is not a test for

1301:

it is unreliable. I suggest for each source that you believe is unreliable you create a separate subsection bellow and put your argument there.

4916:

If there are no objections, I will put this text into the article instead of what is written now, add references, and forget about this story.

2689:

About "conglomerate conversion": you are right; I followed the current article, but in fact, the term in is rather "conglomerate convention".

2376:"Thus we can form the conglomerate of all classes as well as such entities as functions between conglomerates and families of conglomerates." 1494:

Review (by Herrlich): "...illegitimate collection (i.e. one which can be mapped onto the conglomerate of all subclasses of a proper class)..."

303: 5789:

to avoid needing to determine which subfield the term is part of, since there seem to be no other senses of "conglomerate" in mathematics. --

5602:

This term is too rare. It will be an exaggeration to say that it is used in mathematics. Several authors in category theory, and that is all.

1539:

Boris, you could notice also that the paper by Jinwen Zhang is not mentioned there. Moreover, this publishing house, World Scientific, has a

85: 3233:

OK, to this end my formulation includes "(whose existence is postulated)". Also, "In set theory, universes are often classes" (quoted from "

2545:

may need a hierarchy of two or more universes. Conglomerates may be used when two universes are sufficient; the infinite hierarchy given by

2451:

may need a hierarchy of two or more universes. Conglomerates may be used when two universes are sufficient; the infinite hierarchy given by

5803: 5317:, what you were writing so far was impossible to understand, I am not happy of thinking about this extra work for me and for other people. 5350:

Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a

3655: 1937:

Since not every class in the sense of NBG is a conglomerate in the sense of this theory (but only those which are called "sets" in NBG).

3303:

There is no "big universe" in first order theories. When we introduce "small universe", it appears, but "big universe" does not appear.

4443: 4162: 3869: 2949: 2737: 2546: 2452: 168: 1517:Šlapal 1995 "A conglomerate of exponential supercategories of the category of finitely generated topological spaces" Aequationes Math 135: 2421:"Universes are of critical importance to formalizing concepts in category theory inside set-theoretical foundations." (Quoted from " 294: 255: 2804:

That is, mathematicians often prefer foundations with great margin, since "economical" foundations tend to make unwanted troubles.

2155:

About "working mathematician": a working category theorist just uses conglomerates long ago, since he understands intuitively that

643:), new letters to alphabet, new formulas to the list of axioms, but not "new objects". There is also a possibility to construct an 591:

Boris, there is no such an option in first order theories, "adding new objects". One can add new symbols to signature (like those

1745: 1561:

I wonder, if there is a possibility to let these people know about what happens here. It would be interesting to listen to them.

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First group of axioms: "just" the axioms of ZFC (only with the word "set" replaced with "conglomerate", if you say it in words).

711:

Again a hint. Zhang Jinwen should have constructed an accurate first order theory, but he did not do this. That is his problem.

2086:

Your examples from set theory in my opinion are too technical, I don't see how they can be useful for "working mathematician".

99: 30: 3108:. Even the cardinality of sets is not restricted (what one could expect since sets in this theory are elements of a given set 5624: 4816: 2654:

So, in particular, if the new theory was constructed as a modification of the theories where the term "class" was used, like

1270:

In fact, this applies to all the sources in this article: all of them are unreliable, since nowhere these claims are proved.

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And, I guess, ZFC+ZFC looks the most natural choice, and is equivalent to ZFC+MK, which confirms the insensitivity. Right?

752:

All the formal requirements must be met. Furthermore, since we are here, "catching someone's idea" is called in Knowledge

5683: 1125: 1058: 1001: 928: 129: 74: 5931:"Notice that this Axiom of Choice implies an Axiom of Choice for Classes and also the familiar Axiom of Choice for Sets." 5875: 1432:, since they don't construct a formal theory for which this construction could be considered as model. This is renaming. 5687: 3597: 1994: 756:. It is not allowed. And since we are humans, presenting somebody's absurd ideas as reliable theories is a violation of 368: 230: 5889: 5850: 5224:

Strangely, I do not see the paper in the bibliography of , nor . But i see in the former. Maybe this book embraces ?

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The second edition of Mac Lane's "Categories for the Working Mathematician", of 1998, mentiones in the bibliography.

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BTW, I never claimed that Grothendieck used the term "conglomerate". I change the name of the section as suggested by

65: 2208:

and why it was made before the article was edited so that this would satisfy all the participants of the discussion?

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MK is described in Kelley's "General topology" (appendix), NBG in Mendelson's "Introduction to mathematical logic".

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Boris, neither ZFC, nor NBG, nor MK have universe in this sense. ZFC doesn't have "big universe" because the class

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conglomerates are needed there; and I'll try to write a bit on this in the article if neither you nor Kopylov do.

4845:"A model of a theory is a structure (e.g. an interpretation) that satisfies the sentences of that theory." (From " 4446:, where not only existence of a single Grothendieck universe is postulated, but of an infinite hierarchy of them.) 4165:, where not only existence of a single Grothendieck universe, but of an infinite hierarchy of them is postulated.) 2443:, of functions between classes, of families of classes etc. The smaller universe is usually assumed to be a given 1985:

No, I think, you are just not used to models. Look at the classical Euclidean models of non-Euclidean geometries:

5819: 5628: 5552: 5420: 5394: 5341: 196: 5497:

Agree that reasonable uses should be re-pointed to the new name, regardless of whether the redirect is kept. —

2736:

I did not see this in their considerations. As far as I understand, they use only one universe. (But mentioning

1986: 175: 5798: 5740: 5287: 3872:, where not only existence of a single Grothendiech universe, but of an infinite hierarchy of them is assumed. 1662: 1482:

Hušek 1976 "Lattices of reflections and coreflections in continuous structures" Lecture Notes in Math. Vol. 540

1309: 1162: 1071: 978: 3251:

I don't insist, we can find another formulation. But I don't understand, what can be called "universe" in MK?

2952:, where not only existence of a single Grothendiech universe, but of an infinite hierarchy of them is assumed. 1487:Činčura 1979/80 "Reflective and coreflective subcategories of some category of generalized topological spaces" 218: 5699: 2781:), and nevertheless, the prevailing theory is ZFC (rather than Z), and the prevailing universe is the whole 1504:

Review (by Pumplün): "...Galois correspondence between the conglomerate of all subclasses of a category..."

387: 109: 5076:

From the definition of universe it follows readily that the small sets (with the given membership relation

5043:

I'm not a category theorist, but I would also like to see it, if possible. I have borrowing privileges at

2645:

to use the term "conglomerate" for all sets (i.e. elements of the bigger universe, not necessary elements

5993: 5980: 5912: 5760: 5756: 5670: 5635: 5619:. I see no references other than in and/or referring to category theory. I would accept a redirect from 5536: 5501: 5427: 5184: 5051: 4993: 4850: 3491: 3476: 3281: 3234: 3193: 2570:"; and "conglomerate" is translated as just "set". This translation is known as "conglomerate conversion". 2513: 2472:"; and "conglomerate" is translated as just "set". This translation is known as "conglomerate conversion". 2436: 2422: 2012: 1990: 1425: 1198: 394: 5831: 5827: 5715: 5589: 5487: 5351: 4784: 4724: 4704: 4664: 4595: 4458: 4416: 4350: 4177: 4135: 4069: 3842: 2922: 2770: 2538: 2444: 1202: 1063: 993: 916: 236: 280: 3526:

And by the way, in SOME implementations, a natural number is a set, but a real number is a class, see

2439:

that contains another universe as a member. This concept has been created to deal with collections of

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Review (by Vorster): "...where L denotes the conglomerate of all coreflective subcategories of C..."

3482:

universe is a collection that contains all the entities one wishes to consider in a given situation

2759:(a) the Tarski-Grothendieck theory provides infinite (moreover, transfinite) hierarchy of universes; 2733:

About "really use an hierarchy of more than two universes": a good question; I do not know; do you?

2314:; his opinions and participation is welcome, preferably, before substantial changes in the article. 189: 5793: 5790: 5736: 5544: 5354:

after discussing it on the closer's talk page. No further edits should be made to this discussion.

5314: 5283: 4667:"). However, (not counting the possible problems caused by the supplementary axiom of existence of 3707: 3631: 3573: 3549: 3286:

universe is a collection that contains all the entities one wishes to consider in a given situation

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Boris, people who don't like cheating, they need specification. For example, the phrases like this

1658: 1305: 1292:. For example, Adamek et. al. is a secondary source, it may not include all proofs. Third, Osborne 1158: 1067: 974: 915:

can you provide the references to the places in the sources you cite where the authors speak about

910: 371:. As far as I can see nothing changed since that time. I would nominate this article for deletion. 364: 161: 55: 302:

on Knowledge. If you would like to participate, please visit the project page, where you can join

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are redirected, the formal definition of model in mathematical logic is not given. The same with

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is applied to arbitrary sets as a contraposition to the distinguished sets that are elements of

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is applied to arbitrary sets as a contraposition to the distinguished sets that are elements of

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is postulated (in addition to the other axioms). An example of the non-conservative extension of

3219:

we obtain only one (small) universe, not two (small and big). There must be another explanation.

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is postulated (in addition to the other axioms). An example of the non-conservative extension of

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Well, never mind. (Just to show that "to be a proper class" may depend on the implementation.)

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to Conglomerate (mathematics), supported by most people (on both sides of the debate) — Martin

5990: 5977: 5909: 5667: 5632: 5540: 5498: 5424: 5181: 5048: 4990: 1704:κ is the "universe" discussed in the Appendix (linked above)... and everyone should be happy. 391: 51: 4687:) this in some sense does not lead to a loss of information about objects of the old theory ( 1591:

of formalization. And surely you know that Cantor worked in the set theory a lot before ZFC.

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about its usual objects called classes (including proper classes) can be proved as well in "

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as elements). The notion of "big universe" is confusing, it must be eliminated in the text.

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Adamek (and Zhang). Conglomerates existed during decades as "folklore", and are now in the

1007:

1. J.Adamek, H.Herrlich, G.Strecker in "Abstract and Concrete Categories: The Joy of Cats",

5411:, as there is no proposed axiom system, and a naive approach leads to contradictions. It 4346: 4065: 3996: 3822: 2902: 2796: 2542: 2502: 2448: 2270: 730: 5079: 2824:

About relations to NBG and MK. According to the conglomerate convention, only members of

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Holgate et al 2016 "Topogenous and Nearness Structures on Categories" Appl Categor Struct

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Formulated in terms of set theory, but used only in (some branches of) category theory.

4867:

This definition is too vague. I don't know, maybe later I will find some time for this.

4628: 3409: 3383: 3356: 2367:"The concept of 'conglomerate' has been created to deal with 'collections of classes'." 1951:

An ugly terminology, provoking a variety of misunderstandings on very different levels.

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J.Adamek, H.Herrlich, G.Strecker in "Abstract and Concrete Categories: The Joy of Cats",

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above (by me) is the right interpretation of the (formally wrong) text of Zhang Jinwen.

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Yes, sounds good. A bit long for a lead, but this is probably inevitable in this case.

4302: 4286: 4044: 4028: 3986: 3970: 3815: 3779: 3765: 3751: 3737: 3704:, i.e. again a set. Then real numbers can be defined as Dedekind cuts, i.e. subsets in 3539: 3503: 3470: 3456: 3441: 3348: 3326: 3312: 3298: 3274: 3260: 3246: 3228: 3183: 3152: 3137: 3062: 3046: 2893: 2860: 2845: 2813: 2749: 2724: 2698: 2680: 2663: 2616: 2602: 2586: 2485: 2410: 2385: 2359: 2323: 2301: 2279: 2243: 2217: 2188: 2168: 2095: 2079: 2040: 2028: 2006: 1960: 1946: 1926: 1892: 1786: 1772: 1757: 1734: 1713: 1685: 1666: 1654: 1637: 1623: 1600: 1570: 1556: 1533: 1484:

Review (by Kannan): "...the order structure of the conglomerate of all reflections..."

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Boris, but you are at least a mathematician. I, on the conrary, don't like this idea.

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Hint: "The source is unreliable because I don't agree with it" is not a good argument.

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6. G.Preuß in "Theory of Topological Structures: An Approach to Categorical Topology".

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Thanks for your help. With updated links, a move without redirect is fine with me. --

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arising after adding to the system of axioms a supplementary axiom of existence of a

4134:

arising after adding to the system of axioms a supplementary axiom of existence of a

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Yes, I mean proper class. And I gave you a link (which you did not follow, I guess).

1649: 970: 5892:(probably adequate for category theory; but I don't think vNBG implies MK within Z) 5694:

to point to the proposed new title, which I had also made into a redirect for now.

4846: 4828: 4622: 2292:) should start this, since it was he who first suggested a reasonable formulation. 2016: 1098:), in contraposition to the sets that are subsets of a given Grothendieck universe 4853:". And Knowledge is not a textbook. Still, if you feel able to do better, try it. 1931:

And this means that this phrase should also be understood in a very cunning sense:

4289:

as objects of the old theory are discarded from consideration in the new theory.)

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as objects of the old theory are discarded from consideration in the new theory.

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Intro: "...diagrams for which the 'conglomerate' of natural transformations..."

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as objects of the old theory are discarded from consideration in the new theory.

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as objects of the old theory are discarded from consideration in the new theory.

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can be defined as the smallest infinite ordinal number, i.e. a set. After that

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Korostenski et al 1986 "On left-cancellable classes of morphisms" Comm. Algebra

4449:

The concept of conglomerate has been created for the possibility to deal with

4168:

The concept of conglomerate has been created for the possibility to deal with

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Greve 1980 "How many monoidal closed structures are there in TOP?" Arch. Math.

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Sets are not all entities that are considered in NBG and in MK, so the class

5547:

in the title as such details are best explained in the body of the article.

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Börger et al 1981 "Compact and hypercomplete categories" J Pure Appl Algebra

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As we understood, the only reasonable meaning of the term "conglomerate" is

1019:

As far as I can see from the references to Google books the same is true for

1066:. If you don't think it is obvious, I can rewrite it closer to the sources. 4625:(i.e. an object that can't be element of any other object, like the class 1522:

Page 449: "...the ordered conglomerate of all neighbourhood operators..."

1402:. And by "sets" we mean (not arbitrary sets in NBG, but) only elements of 4727:" about its objects called classes in this theory (i.e. about subsets of 3461:

But, well, your proposal is also OK with me (may be up to some remarks).

757: 727:

there is no such an option in first order theories, "adding new objects".

390:. The theory, itself, might be notable, but not individual elements. — 5872:(probably still implies the former, but with class being a defined term) 4543:

to apply the term "conglomerate" to all sets (not necessary elements of

4199:

is added to the chosen axiomatic set theory it is considered convenient

725:

Yes, somewhat a problem, but surmountable. Yes, you are right, formally

421:

Zhang Jinwen constructs his theory as an extension by definition of GB:

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where extensions of axiomatic set theories assuming the existence of a

4250:

to use the term "conglomerate" for all sets (not necessary elements of

4068:

where extensions of axiomatic set theories assuming the existence of a

3977:

We can try to express what we need better. How would you correct this?

3935:

to use the term "conglomerate" for all sets (not necessary elements of

3806:

Well, let me repeat your text here, as the basis for later discussion.

3012:

to use the term "conglomerate" for all sets (not necessary elements of

737:. You are not writing the referee report to Zhang Jinwen's manuscript. 1479:

Review (by Bergman): "A/z(A) is a conglomerate of members of Σ(S)..."

1628:

And details of definitions are important everywhere in mathematics.

1352:

4. Osborne does not construct a formal theory. As well as the others.

1325:

you have to provide a good argument why do you think it is unreliable

1062:

or "universal conglomerate", but is is obvious that this is equal to

1022:

5. H.Herrlich, G.Strecker, in "Category theory: an introduction", and

5047:, but they don't seem to have a copy of MacLane's paper, either. — 4035:

misinterpret this phrase (even if grammatically it is unambiguous).

3825:, the term "conglomerate" is used as a synonim of the term "set" in 3697:{\displaystyle {\mathbb {Z} }\times ({\mathbb {N} }\setminus \{0\})} 2905:, the term "conglomerate" is used as a synonim of the term "set" in 1201:

and ZF# (Zermelo–Fraenkel set theory with an axiom of existence of

499:). This means that he understands this relation in the same sense. 5976:+ZF. Still, at this point, it's original research on my part. — 5658:, although I believe the statement to be misleading, as vNBG is a 5073:Спасибо, добрый человек! Received. I see the phrase (at page 195) 2176: 967:(personally, I don't like this style, but alternatives are worse). 5297:

Nice. I got tired, pretending to be a category theorist. :-)

5116:, these sets and classes satisfy the usual Gödel-Bernays axioms. 4570:

Formally this gives a model of the initial axiomatic set theory (

2779:

Von Neumann universe#Applications of V as models for set theories

2628:

entails some changes in terminology: it is considered convenient

1180:, for claiming what is written in this article, in particularly, 5973: 5879: 5859: 4851:

Interpretation (logic)#Interpretations of a first-order language

2607:

And do they really use an hierarchy of more than two universes?

1191:

to allow classes, there are extensions that allow conglomerates.

3875:

This concept has been created for the possibility to deal with

2955:

This concept has been created for the possibility to deal with

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Review (by Bentley): "...M a conglomerate of mono-sources..."

1360:, not by constructing a formal theory. The authors just say: 1217: 1188: 1087: 212: 184: 15: 2593:

Are you sure that there is a term "conglomerate conversion"?

5826:

Subsequent comments should be made in a new section on this

803:

like what people do in Russian Knowledge, see, for example,

4655:) are discarded from the consideration in the new theory (" 2898:

What about this (a straitforward, non-diplomatic) variant?

4695:) since its representation as a model in the new theory (" 2535:, of functions between classes, of families of classes etc 1507:

Dikranjan et al 1987 "Closure operators. I" Topology Appl.

1424:. In contrast to what you write here this is not called a 410:

Knowledge:Articles for deletion/Conglomerate (set theory)

2537:. The smaller universe is usually assumed to be a given 1648:

I think the best source is Herrlich&Strecke. In the

1339:

build a first order theory and does prove what he claims

1252:

do this. We discussed the details with Boris Tsirelson

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If the initial axiomatic set theory admits the idea of

2865:

This is what we see after those changes in terminology:

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I can support "conglomerate (mathematics)" as well. --

2145:

Definition: "class" is a conglomerate contained in U.

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The above discussion is preserved as an archive of a

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Also "conglomerate (mathematics)" looks good for me.

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I agree, "synonim" is a bad word here. We can write

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Also I would change the second paragraph as follows:

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Richter 1990/91 "Algebra ⊂ topology?!" Res. Exp. Math

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You forgot also Adamek-Herrlich-Strecker. As I wrote

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is a good alternative. Anyway redirect is required.

5666:

of conglomerates has been pointed to, as of yet. —

5001:

The corollaries are now sourced (to Mac Lane 1969).

1228:), where all these propositions are proved. Neither 298:, a collaborative effort to improve the coverage of 6001:Oh yes, without AC it should be different. Indeed, 4651:

of all sets in NBG and in MK), then these objects (

3621:{\displaystyle {\mathbb {N} }\times {\mathbb {N} }} 1725:nothing about "non-conservative extensions", etc. 6020: 5834:. No further edits should be made to this section. 5155: 5135: 5108: 5088: 4759: 4739: 4679: 4643: 4610: 4555: 4532: 4509: 4472: 4430: 4388: 4364: 4262: 4239: 4216: 4191: 4149: 4107: 4083: 3947: 3924: 3901: 3856: 3756:I did not understand why this can be interesting. 3720: 3696: 3644: 3620: 3586: 3562: 3475:When you write about universe you give this link: 3424: 3398: 3371: 3211: 3120: 3100: 3080: 3024: 3001: 2978: 2936: 1909: 1875: 1855: 1835: 1809: 1459:(The same was just written by me on the AFD page.) 1414: 1394: 1374: 1296:prove what he claims (Proposition 6.13) and Zhang 1110: 688: 635: 609: 561: 535: 491: 471: 445: 3494:is written badly, it must be corrected as well.) 3172:Equivalent definitions of mathematical structures 1746:WT:WPM#"Where triangle's area is triangle's area" 1676:in set theory can infer this consequence easily. 842:did not think that you will interpret it this way 6028:may be a countable union of countable ordinals! 5731:definition, so current title is not misleading. 3546:I don't understand. Do you mean proper classes? 33:for general discussion of the article's subject. 4975:Nice. And, I'll try to source the corollaries. 3192:In mathematics, conglomerates are members of a 2435:In mathematics, conglomerates are members of a 2029:Large cardinal#Motivations and epistemic status 1477:Skornjakov 1973 "Radicals of Ω-rings" (Russian) 1016:4. V.Laan in "Introduction to category theory" 1010:2. M.S.Osborne in "Basic Homological Algebra", 5989:don't know who is studying it, if anyone. — 4747:, including subsets that are not elements of 4520:to apply the term "class" only to subsets of 4480:is added to the chosen axiomatic set theory ( 4227:to apply the term "class" only to subsets of 3912:to apply the term "class" only to subsets of 2989:to apply the term "class" only to subsets of 2844:) is a model of NBG and MK. In the spirit of 2773:of "ordinary mathematics", and is a model of 2638:to apply the term "class" only to subsets of 2624:The introduction of a new (smaller) universe 2523:. This concept has been created to deal with 2306:Hmmm... The current version is mostly due to 174: 8: 4497:to apply the term "set" only to elements of 4204:to apply the term "set" only to elements of 4053:Yes, you are right. What about this variant? 3889:to apply the term "set" only to elements of 3688: 3682: 2966:to apply the term "set" only to elements of 2631:to apply the term "set" only to elements of 1239:Scott Osborne in "Basic Homological Algebra" 729:On the other hand, a mathematician is not a 4767:, which are analogs of proper classes from 3196:that contains another universe as a member. 859:And these are not tastes. These are rules. 776:After all, every formal theory was created 735:Don't throw the baby out with the bathwater 216: 5415:be used in category theory. Request move 5340:The following is a closed discussion of a 369:specialists in logic explained the details 244: 6012: 6006: 5654:as inappropriate, but left the link from 5148: 5128: 5101: 5081: 5059:I just sent email to you, too. Received? 4752: 4732: 4672: 4630: 4603: 4548: 4525: 4502: 4465: 4423: 4399:The most popular axiomatic set theories, 4381: 4357: 4255: 4232: 4209: 4184: 4142: 4118:The most popular axiomatic set theories, 4100: 4076: 3940: 3917: 3894: 3849: 3713: 3712: 3711: 3709: 3674: 3673: 3672: 3661: 3660: 3659: 3657: 3637: 3636: 3635: 3633: 3613: 3612: 3611: 3603: 3602: 3601: 3599: 3579: 3578: 3577: 3575: 3555: 3554: 3553: 3551: 3411: 3385: 3358: 3204: 3113: 3093: 3073: 3017: 2994: 2971: 2929: 2401:stages as with Grothendieck Universes." 2284:OK, we'll edit the text. I think, Boris ( 1902: 1868: 1848: 1822: 1796: 1407: 1387: 1367: 1103: 675: 622: 596: 548: 522: 484: 458: 432: 2491:Boris, I would make several corrections: 4438:. (An example of such an extension of 4157:. (An example of such an extension of 3679: 2566:"; "class" is translated as "subset of 2468:"; "class" is translated as "subset of 1197:ACG is a non-conservative extension of 386:Concur. Appears to be an element of a 246: 2671:It seems to me that would be clearer. 1247:Jinwen Zhang in "The axiom system ACG" 569:; but they apply not only to classes. 2740:is useful in this context, I think.) 2140:Definition: "set" is an element of U. 7: 5896:In any case, "set" is an element of 5680:Von Neumann–Bernays–Gödel set theory 5656:von_Neumann–Bernays–Gödel set theory 5359:The result of the move request was: 5254:Yes, it does, in Sections I.6, I.7. 4707:") means that what can be proved in 4582:) in the extension of this theory (" 4405:Von Neumann–Bernays–Gödel set theory 4279:Von Neumann–Bernays–Gödel set theory 4124:Von Neumann–Bernays–Gödel set theory 4009:Von Neumann–Bernays–Gödel set theory 3963:Von Neumann–Bernays–Gödel set theory 3835:Von Neumann–Bernays–Gödel set theory 3652:can be defined as a quotient set of 3594:can be defined as a quotient set of 3039:Von Neumann–Bernays–Gödel set theory 2915:Von Neumann–Bernays–Gödel set theory 2872:are sets; thus, only members of the 2828:are sets; thus, only members of the 2709:Von Neumann–Bernays–Gödel set theory 2656:Von Neumann–Bernays–Gödel set theory 1216:to built a first order theory (like 292:This article is within the scope of 2027:etc. Still closer to the business: 1791:Boris, in which sense the relation 1049:The answer to the last question is 235:It is of interest to the following 23:for discussing improvements to the 5375:13:38, 20 May 2019 (UTC) — Martin 2516:that contains another universe as 14: 6062:Mid-priority mathematics articles 5029:I just sent you email. Received? 2549:serves more demanding situations. 2455:serves more demanding situations. 1657:. Hope it answer your questions. 312:Knowledge:WikiProject Mathematics 6057:Start-Class mathematics articles 5163:, is not itself a model of ZFC. 4003:of axiomatic set theories, like 3829:of axiomatic set theories, like 2909:of axiomatic set theories, like 1721:all classes form a conglomerate 1013:3. L.Nel in "Continuity theory". 947: 417:Boris, I already told this: in 315:Template:WikiProject Mathematics 279: 269: 248: 217: 199:on 15 April 2019. The result of 188: 45:Click here to start a new topic. 2230:Do you suspect that the admin " 1053:: Theorem 21 (+definition 10). 956:: added pages using a template 921:strongly inaccessible cardinals 332:This article has been rated as 195:This article was nominated for 5678:I have updated the links from 5625:conglomerate (category theory) 5531:The target would be better as 5400:Conglomerate (category theory) 4817:Structure (mathematical logic) 4492:) it is considered convenient 4444:Tarski–Grothendieck set theory 4163:Tarski–Grothendieck set theory 3870:Tarski–Grothendieck set theory 3721:{\displaystyle {\mathbb {Q} }} 3691: 3669: 3645:{\displaystyle {\mathbb {Q} }} 3587:{\displaystyle {\mathbb {Z} }} 3563:{\displaystyle {\mathbb {N} }} 2950:Tarski–Grothendieck set theory 2795:(c) in the (uncountably high) 2738:Tarski–Grothendieck set theory 2547:Tarski–Grothendieck set theory 2453:Tarski–Grothendieck set theory 2340:Changes proposed and discussed 2011:Or, closer to set theory, try 1934:Every class is a conglomerate. 1428:. This even can't be called a 1205:). Moreover, it can prove the 408:Here is a message copied from 358:This article should be deleted 1: 5862:(probably implies the former) 2204:Boris, who made the decision 1126:many-sorted first-order logic 1059:many-sorted first-order logic 1002:many-sorted first-order logic 929:many-sorted first-order logic 670:by another predicate symbols 306:and see a list of open tasks. 42:Put new text under old text. 5972:doubt that we need to go to 5750:Support alternative move to 5688:Category of small categories 3492:the article that you mention 1810:{\displaystyle A\subseteq U} 6021:{\displaystyle \omega _{1}} 5900:and "class" is a subset of 5692:Glossary of category theory 5650:. I removed the link from 5166:But maybe, this is enough. 4413:non-conservative extensions 4401:Zermelo-Fraenkel set theory 4132:non-conservative extensions 4120:Zermelo-Fraenkel set theory 4005:Zermelo-Fraenkel set theory 4001:non-conservative extensions 3841:, where the existence of a 3831:Zermelo-Fraenkel set theory 3827:non-conservative extensions 3479:. It is written there that 2921:, where the existence of a 2911:Zermelo-Fraenkel set theory 2907:non-conservative extensions 1122:non-conservative extensions 1055:Non-conservative extensions 998:non-conservative extensions 925:non-conservative extensions 50:New to Knowledge? Welcome! 6078: 5786:Conglomerate (mathematics) 5752:Conglomerate (mathematics) 5733:Conglomerate (mathematics) 5621:conglomerate (mathematics) 5533:conglomerate (mathematics) 4825:model (mathematical logic) 4815:By the way in the article 2431:I try possible beginning. 2189:21:44, 28 April 2019 (UTC) 2169:21:08, 28 April 2019 (UTC) 2096:20:16, 28 April 2019 (UTC) 2080:20:12, 28 April 2019 (UTC) 2041:18:34, 28 April 2019 (UTC) 2021:Minimal model (set theory) 2017:Absoluteness#In set theory 2007:18:27, 28 April 2019 (UTC) 1961:11:37, 28 April 2019 (UTC) 1947:11:26, 28 April 2019 (UTC) 1927:11:01, 28 April 2019 (UTC) 1893:10:38, 28 April 2019 (UTC) 1787:10:29, 28 April 2019 (UTC) 1773:10:22, 28 April 2019 (UTC) 1758:10:16, 28 April 2019 (UTC) 1735:09:23, 28 April 2019 (UTC) 1714:09:10, 28 April 2019 (UTC) 1686:04:34, 28 April 2019 (UTC) 1667:21:42, 27 April 2019 (UTC) 1638:10:15, 27 April 2019 (UTC) 1624:09:59, 27 April 2019 (UTC) 1601:09:34, 27 April 2019 (UTC) 1571:09:16, 27 April 2019 (UTC) 1557:09:05, 27 April 2019 (UTC) 1534:08:46, 27 April 2019 (UTC) 1470:07:59, 27 April 2019 (UTC) 1442:08:12, 28 April 2019 (UTC) 1426:non-conservative extension 1314:21:27, 27 April 2019 (UTC) 1280:05:06, 27 April 2019 (UTC) 1266:04:56, 27 April 2019 (UTC) 1209:of both of these theories. 1167:20:37, 26 April 2019 (UTC) 1138:08:33, 28 April 2019 (UTC) 1076:20:57, 27 April 2019 (UTC) 1045:04:37, 27 April 2019 (UTC) 983:20:31, 26 April 2019 (UTC) 941:06:16, 26 April 2019 (UTC) 883:07:45, 26 April 2019 (UTC) 869:07:42, 26 April 2019 (UTC) 855:07:33, 26 April 2019 (UTC) 834:06:24, 26 April 2019 (UTC) 819:06:08, 26 April 2019 (UTC) 791:05:50, 26 April 2019 (UTC) 770:06:08, 26 April 2019 (UTC) 747:05:31, 26 April 2019 (UTC) 721:04:41, 26 April 2019 (UTC) 706:20:57, 25 April 2019 (UTC) 657:04:41, 26 April 2019 (UTC) 579:20:18, 25 April 2019 (UTC) 509:07:04, 25 April 2019 (UTC) 398:18:36, 22 April 2019 (UTC) 381:15:11, 28 March 2019 (UTC) 25:Conglomerate (mathematics) 5629:conglomerate (set theory) 5421:Conglomerate (set theory) 5395:Conglomerate (set theory) 5333:Requested move 3 May 2019 4372:are considered, the term 4091:are considered, the term 2840:is a model of ZFC, and P( 1995:Poincaré half-plane model 331: 264: 243: 80:Be welcoming to newcomers 5824:Please do not modify it. 5810:20:58, 15 May 2019 (UTC) 5779:17:30, 11 May 2019 (UTC) 5453:17:26, 11 May 2019 (UTC) 5430:17:19, 3 May 2019 (UTC) 5388:13:38, 20 May 2019 (UTC) 5347:Please do not modify it. 1843:for some "conglomerate" 1185:Gödel-Bernays set theory 338:project's priority scale 6038:06:29, 7 May 2019 (UTC) 5997:05:54, 7 May 2019 (UTC) 5984:06:17, 6 May 2019 (UTC) 5964:16:11, 5 May 2019 (UTC) 5948:04:23, 5 May 2019 (UTC) 5916:20:59, 4 May 2019 (UTC) 5745:08:29, 9 May 2019 (UTC) 5721:19:39, 8 May 2019 (UTC) 5704:19:12, 8 May 2019 (UTC) 5674:20:21, 3 May 2019 (UTC) 5639:20:01, 4 May 2019 (UTC) 5612:08:18, 4 May 2019 (UTC) 5595:18:16, 3 May 2019 (UTC) 5578:18:12, 3 May 2019 (UTC) 5557:18:00, 3 May 2019 (UTC) 5524:17:37, 3 May 2019 (UTC) 5505:20:24, 3 May 2019 (UTC) 5493:17:32, 3 May 2019 (UTC) 5472:17:28, 3 May 2019 (UTC) 5327:13:05, 9 May 2019 (UTC) 5307:09:48, 9 May 2019 (UTC) 5292:08:17, 9 May 2019 (UTC) 5264:07:34, 7 May 2019 (UTC) 5249:07:35, 7 May 2019 (UTC) 5234:07:21, 7 May 2019 (UTC) 5188:08:33, 7 May 2019 (UTC) 5176:06:18, 7 May 2019 (UTC) 5069:06:06, 7 May 2019 (UTC) 5055:05:57, 7 May 2019 (UTC) 5039:05:42, 7 May 2019 (UTC) 5025:05:19, 7 May 2019 (UTC) 5011:05:11, 7 May 2019 (UTC) 4997:06:21, 6 May 2019 (UTC) 4989:Looks good, so far. — 4985:05:41, 6 May 2019 (UTC) 4971:05:11, 6 May 2019 (UTC) 4956:04:56, 6 May 2019 (UTC) 4941:19:23, 5 May 2019 (UTC) 4926:17:01, 5 May 2019 (UTC) 4912:16:58, 5 May 2019 (UTC) 4897:16:03, 5 May 2019 (UTC) 4877:16:58, 5 May 2019 (UTC) 4863:15:47, 5 May 2019 (UTC) 4841:12:04, 5 May 2019 (UTC) 4811:11:27, 5 May 2019 (UTC) 4409:Morse–Kelley set theory 4325:04:32, 5 May 2019 (UTC) 4303:04:12, 5 May 2019 (UTC) 4283:Morse–Kelley set theory 4128:Morse–Kelley set theory 4045:20:30, 4 May 2019 (UTC) 4029:19:19, 4 May 2019 (UTC) 4013:Morse–Kelley set theory 3987:19:13, 4 May 2019 (UTC) 3967:Morse–Kelley set theory 3839:Morse–Kelley set theory 3816:19:06, 4 May 2019 (UTC) 3780:10:36, 4 May 2019 (UTC) 3766:10:22, 4 May 2019 (UTC) 3752:10:16, 4 May 2019 (UTC) 3738:10:09, 4 May 2019 (UTC) 3540:09:50, 4 May 2019 (UTC) 3504:19:07, 4 May 2019 (UTC) 3471:19:01, 4 May 2019 (UTC) 3457:18:51, 4 May 2019 (UTC) 3442:18:42, 4 May 2019 (UTC) 3349:17:50, 4 May 2019 (UTC) 3327:11:29, 4 May 2019 (UTC) 3313:11:03, 4 May 2019 (UTC) 3299:10:58, 4 May 2019 (UTC) 3275:10:36, 4 May 2019 (UTC) 3261:10:20, 4 May 2019 (UTC) 3247:10:14, 4 May 2019 (UTC) 3229:09:57, 4 May 2019 (UTC) 3184:09:39, 4 May 2019 (UTC) 3153:07:55, 4 May 2019 (UTC) 3138:07:45, 4 May 2019 (UTC) 3063:06:11, 4 May 2019 (UTC) 3043:Morse–Kelley set theory 2919:Morse–Kelley set theory 2894:06:21, 4 May 2019 (UTC) 2861:06:04, 4 May 2019 (UTC) 2836:) are classes; indeed, 2814:05:38, 4 May 2019 (UTC) 2750:06:50, 4 May 2019 (UTC) 2725:05:30, 4 May 2019 (UTC) 2713:Morse–Kelley set theory 2699:05:14, 4 May 2019 (UTC) 2681:05:07, 4 May 2019 (UTC) 2660:Morse–Kelley set theory 2617:04:19, 4 May 2019 (UTC) 2603:04:15, 4 May 2019 (UTC) 2587:04:44, 4 May 2019 (UTC) 2486:20:02, 3 May 2019 (UTC) 2411:18:35, 3 May 2019 (UTC) 2386:18:24, 3 May 2019 (UTC) 2360:18:17, 3 May 2019 (UTC) 2324:18:07, 3 May 2019 (UTC) 2302:17:42, 3 May 2019 (UTC) 2280:17:31, 3 May 2019 (UTC) 2244:17:22, 3 May 2019 (UTC) 2218:16:52, 3 May 2019 (UTC) 1543:since its publications 295:WikiProject Mathematics 6022: 5761:Conglomerate (geology) 5757:Conglomerate (company) 5543:. We don't need more 5537:function (mathematics) 5419:redirect, as the term 5165: 5157: 5137: 5118: 5110: 5090: 4819:to which the quieries 4761: 4741: 4681: 4645: 4612: 4557: 4534: 4511: 4474: 4432: 4390: 4366: 4264: 4241: 4218: 4193: 4151: 4109: 4085: 3949: 3926: 3903: 3858: 3722: 3698: 3646: 3622: 3588: 3564: 3484: 3477:Universe (mathematics) 3426: 3400: 3373: 3288: 3235:Universe (mathematics) 3213: 3198: 3122: 3102: 3082: 3026: 3003: 2980: 2938: 2882: 2423:Universe (mathematics) 2013:Constructible universe 1936: 1911: 1877: 1857: 1837: 1836:{\displaystyle A\in C} 1811: 1723: 1416: 1396: 1376: 1341: 1327: 1211: 1203:inaccessible cardinals 1199:Quine-Morse set theory 1193: 1112: 1034: 903:By the way, about the 844: 809: 690: 637: 611: 563: 537: 493: 473: 447: 426: 225:This article is rated 75:avoid personal attacks 6023: 5514:I agree with Arthur. 5158: 5138: 5121: 5111: 5091: 5074: 4785:Grothendieck universe 4762: 4742: 4725:Grothendieck universe 4705:Grothendieck universe 4682: 4665:Grothendieck universe 4646: 4613: 4596:Grothendieck universe 4558: 4535: 4512: 4475: 4459:Grothendieck universe 4433: 4417:Grothendieck universe 4391: 4367: 4351:Grothendieck universe 4265: 4242: 4219: 4194: 4178:Grothendieck universe 4152: 4136:Grothendieck universe 4110: 4086: 4070:Grothendieck universe 3950: 3927: 3904: 3859: 3843:Grothendieck universe 3723: 3699: 3647: 3623: 3589: 3565: 3480: 3427: 3401: 3374: 3284: 3214: 3190: 3123: 3103: 3083: 3027: 3004: 2981: 2939: 2923:Grothendieck universe 2866: 2539:Grothendieck universe 2445:Grothendieck universe 2157:this is not a problem 2150:this is not a problem 1932: 1912: 1878: 1858: 1838: 1817:implies the relation 1812: 1719: 1417: 1397: 1377: 1333: 1323: 1195: 1181: 1178:here at the talk page 1113: 1064:Grothendieck universe 1035:Where is it written? 1030: 994:Grothendieck universe 917:Grothendieck universe 840: 800: 691: 638: 612: 564: 538: 494: 474: 448: 422: 100:Neutral point of view 6005: 5147: 5127: 5100: 5089:{\displaystyle \in } 5080: 4751: 4731: 4671: 4629: 4602: 4547: 4524: 4501: 4464: 4422: 4380: 4356: 4254: 4231: 4208: 4183: 4141: 4099: 4075: 3939: 3916: 3893: 3848: 3708: 3656: 3632: 3598: 3574: 3550: 3410: 3384: 3357: 3337:Von Neumann universe 3203: 3112: 3092: 3072: 3016: 2993: 2970: 2928: 2783:Von Neumann universe 2505:, conglomerates are 1987:Beltrami–Klein model 1901: 1876:{\displaystyle \in } 1867: 1847: 1821: 1795: 1406: 1386: 1366: 1102: 674: 621: 595: 547: 521: 492:{\displaystyle \in } 483: 457: 431: 318:mathematics articles 105:No original research 5841:What IS the theory? 4644:{\displaystyle Set} 4019:What do you think? 3628:, i.e. a set. Then 3425:{\displaystyle Set} 3399:{\displaystyle Set} 3372:{\displaystyle Set} 2707:OK. Boris, also in 2476:Remarks, opinions? 1991:Poincaré disk model 1863:? What is meant by 1362:"We consider a set 754:"original research" 696:("to be a class"). 689:{\displaystyle cls} 668:quantifiers bounded 636:{\displaystyle cog} 610:{\displaystyle scl} 562:{\displaystyle cog} 536:{\displaystyle scl} 472:{\displaystyle cog} 446:{\displaystyle scl} 6018: 5652:class (set theory) 5631:is just wrong. — 5153: 5133: 5106: 5086: 4887:)) is not enough. 4757: 4737: 4677: 4641: 4608: 4553: 4530: 4507: 4470: 4428: 4386: 4362: 4342:text as I see it: 4260: 4237: 4214: 4189: 4147: 4105: 4081: 3945: 3922: 3899: 3854: 3718: 3694: 3642: 3618: 3584: 3560: 3422: 3396: 3369: 3209: 3118: 3098: 3078: 3022: 2999: 2976: 2934: 2775:Zermelo set theory 1917:is a set. Pardon. 1907: 1873: 1853: 1833: 1807: 1541:dubious reputation 1412: 1392: 1372: 1150:Unreliable source? 1108: 1086:"arbitrary set in 795:I told about this 686: 633: 607: 559: 533: 489: 469: 443: 287:Mathematics portal 231:content assessment 86:dispute resolution 47: 5932: 5924: 5541:set (mathematics) 5455: 5423:is misleading. — 5386: 5374: 5156:{\displaystyle U} 5136:{\displaystyle s} 5109:{\displaystyle U} 4849:".) Detailed in " 4760:{\displaystyle U} 4740:{\displaystyle U} 4680:{\displaystyle U} 4611:{\displaystyle U} 4556:{\displaystyle U} 4533:{\displaystyle U} 4510:{\displaystyle U} 4473:{\displaystyle U} 4431:{\displaystyle U} 4389:{\displaystyle U} 4365:{\displaystyle U} 4263:{\displaystyle U} 4240:{\displaystyle U} 4217:{\displaystyle U} 4192:{\displaystyle U} 4150:{\displaystyle U} 4108:{\displaystyle U} 4084:{\displaystyle U} 3948:{\displaystyle U} 3925:{\displaystyle U} 3902:{\displaystyle U} 3857:{\displaystyle U} 3334: 3212:{\displaystyle U} 3121:{\displaystyle U} 3101:{\displaystyle U} 3081:{\displaystyle U} 3025:{\displaystyle U} 3002:{\displaystyle U} 2979:{\displaystyle U} 2937:{\displaystyle U} 2756:I only know that 1910:{\displaystyle U} 1856:{\displaystyle C} 1460: 1415:{\displaystyle U} 1395:{\displaystyle U} 1375:{\displaystyle U} 1304: 1111:{\displaystyle U} 968: 905:original research 899:Original research 728: 511: 352: 351: 348: 347: 344: 343: 211: 210: 183: 182: 66:Assume good faith 43: 6069: 6027: 6025: 6024: 6019: 6017: 6016: 5930: 5922: 5806: 5801: 5796: 5771: 5719: 5593: 5491: 5445: 5431: 5402: 5376: 5364: 5349: 5162: 5160: 5159: 5154: 5142: 5140: 5139: 5134: 5115: 5113: 5112: 5107: 5095: 5093: 5092: 5087: 4766: 4764: 4763: 4758: 4746: 4744: 4743: 4738: 4686: 4684: 4683: 4678: 4650: 4648: 4647: 4642: 4618:as the universe. 4617: 4615: 4614: 4609: 4562: 4560: 4559: 4554: 4539: 4537: 4536: 4531: 4516: 4514: 4513: 4508: 4479: 4477: 4476: 4471: 4437: 4435: 4434: 4429: 4395: 4393: 4392: 4387: 4371: 4369: 4368: 4363: 4269: 4267: 4266: 4261: 4246: 4244: 4243: 4238: 4223: 4221: 4220: 4215: 4198: 4196: 4195: 4190: 4156: 4154: 4153: 4148: 4114: 4112: 4111: 4106: 4090: 4088: 4087: 4082: 3954: 3952: 3951: 3946: 3931: 3929: 3928: 3923: 3908: 3906: 3905: 3900: 3863: 3861: 3860: 3855: 3727: 3725: 3724: 3719: 3717: 3716: 3703: 3701: 3700: 3695: 3678: 3677: 3665: 3664: 3651: 3649: 3648: 3643: 3641: 3640: 3627: 3625: 3624: 3619: 3617: 3616: 3607: 3606: 3593: 3591: 3590: 3585: 3583: 3582: 3569: 3567: 3566: 3561: 3559: 3558: 3431: 3429: 3428: 3423: 3405: 3403: 3402: 3397: 3378: 3376: 3375: 3370: 3332: 3218: 3216: 3215: 3210: 3127: 3125: 3124: 3119: 3107: 3105: 3104: 3099: 3087: 3085: 3084: 3079: 3031: 3029: 3028: 3023: 3008: 3006: 3005: 3000: 2985: 2983: 2982: 2977: 2943: 2941: 2940: 2935: 2868:only members of 2313: 2291: 2266: 2254: 2203: 2121:A new constant: 2025:Transitive model 1916: 1914: 1913: 1908: 1882: 1880: 1879: 1874: 1862: 1860: 1859: 1854: 1842: 1840: 1839: 1834: 1816: 1814: 1813: 1808: 1702:worldly cardinal 1614:"rigor theory". 1545:are not refereed 1458: 1421: 1419: 1418: 1413: 1401: 1399: 1398: 1393: 1381: 1379: 1378: 1373: 1302: 1117: 1115: 1114: 1109: 966: 965: 959: 955: 951: 950: 914: 726: 695: 693: 692: 687: 642: 640: 639: 634: 616: 614: 613: 608: 590: 568: 566: 565: 560: 542: 540: 539: 534: 498: 496: 495: 490: 478: 476: 475: 470: 452: 450: 449: 444: 416: 388:WP:fringe theory 320: 319: 316: 313: 310: 289: 284: 283: 273: 266: 265: 260: 252: 245: 228: 222: 221: 213: 192: 185: 179: 178: 164: 95:Article policies 16: 6077: 6076: 6072: 6071: 6070: 6068: 6067: 6066: 6047: 6046: 6030:Boris Tsirelson 6008: 6003: 6002: 5956:Boris Tsirelson 5940:Boris Tsirelson 5843: 5838: 5804: 5799: 5794: 5765: 5709: 5583: 5570:Boris Tsirelson 5481: 5464:Boris Tsirelson 5439: 5398: 5345: 5335: 5299:Boris Tsirelson 5271: 5256:Boris Tsirelson 5226:Boris Tsirelson 5222: 5145: 5144: 5125: 5124: 5098: 5097: 5078: 5077: 5061:Boris Tsirelson 5031:Boris Tsirelson 5003:Boris Tsirelson 4977:Boris Tsirelson 4933:Boris Tsirelson 4889:Boris Tsirelson 4855:Boris Tsirelson 4749: 4748: 4729: 4728: 4669: 4668: 4627: 4626: 4600: 4599: 4545: 4544: 4522: 4521: 4499: 4498: 4462: 4461: 4420: 4419: 4378: 4377: 4354: 4353: 4347:category theory 4345:In the part of 4317:Boris Tsirelson 4252: 4251: 4229: 4228: 4206: 4205: 4181: 4180: 4139: 4138: 4097: 4096: 4073: 4072: 4066:category theory 4064:In the part of 4037:Boris Tsirelson 3997:category theory 3937: 3936: 3914: 3913: 3891: 3890: 3846: 3845: 3823:category theory 3808:Boris Tsirelson 3804: 3772:Boris Tsirelson 3744:Boris Tsirelson 3706: 3705: 3654: 3653: 3630: 3629: 3596: 3595: 3572: 3571: 3548: 3547: 3532:Boris Tsirelson 3463:Boris Tsirelson 3449:Boris Tsirelson 3408: 3407: 3382: 3381: 3355: 3354: 3341:Boris Tsirelson 3267:Boris Tsirelson 3239:Boris Tsirelson 3201: 3200: 3176:Boris Tsirelson 3110: 3109: 3090: 3089: 3070: 3069: 3014: 3013: 2991: 2990: 2968: 2967: 2926: 2925: 2903:category theory 2853:Boris Tsirelson 2806:Boris Tsirelson 2797:Borel hierarchy 2791: 2768: 2691:Boris Tsirelson 2543:category theory 2503:category theory 2478:Boris Tsirelson 2449:category theory 2403:Boris Tsirelson 2378:Boris Tsirelson 2352:Boris Tsirelson 2347: 2342: 2316:Boris Tsirelson 2307: 2285: 2278: 2262: 2257:Boris Tsirelson 2248: 2236:Boris Tsirelson 2197: 2161:Boris Tsirelson 2137: 2033:Boris Tsirelson 1999:Boris Tsirelson 1899: 1898: 1865: 1864: 1845: 1844: 1819: 1818: 1793: 1792: 1779:Boris Tsirelson 1765:Boris Tsirelson 1750:Boris Tsirelson 1706:Boris Tsirelson 1696: 1678:Boris Tsirelson 1593:Boris Tsirelson 1526:Boris Tsirelson 1462:Boris Tsirelson 1454: 1404: 1403: 1384: 1383: 1364: 1363: 1322:1. I did this: 1152: 1100: 1099: 963: 957: 948: 946: 908: 901: 875:Boris Tsirelson 826:Boris Tsirelson 783:Boris Tsirelson 739:Boris Tsirelson 731:proof assistant 698:Boris Tsirelson 672: 671: 666:to classes via 619: 618: 593: 592: 584: 571:Boris Tsirelson 545: 544: 519: 518: 481: 480: 455: 454: 429: 428: 406: 360: 317: 314: 311: 308: 307: 285: 278: 258: 229:on Knowledge's 226: 121: 116: 115: 114: 91: 61: 12: 11: 5: 6075: 6073: 6065: 6064: 6059: 6049: 6048: 6045: 6044: 6043: 6042: 6041: 6040: 6015: 6011: 5986: 5951: 5950: 5934: 5933: 5926: 5925: 5894: 5893: 5883: 5873: 5863: 5853: 5842: 5839: 5837: 5836: 5820:requested move 5814: 5813: 5812: 5781: 5747: 5737:Alexei Kopylov 5725: 5724: 5723: 5676: 5644: 5643: 5642: 5641: 5614: 5597: 5580: 5560: 5559: 5526: 5509: 5508: 5507: 5474: 5393: 5391: 5357: 5356: 5342:requested move 5336: 5334: 5331: 5330: 5329: 5315:Alexei Kopylov 5310: 5309: 5284:Alexei Kopylov 5270: 5267: 5252: 5251: 5221: 5218: 5217: 5216: 5215: 5214: 5213: 5212: 5211: 5210: 5209: 5208: 5207: 5206: 5205: 5204: 5203: 5202: 5201: 5200: 5199: 5198: 5197: 5196: 5195: 5194: 5193: 5192: 5191: 5190: 5178: 5152: 5132: 5105: 5085: 4999: 4987: 4958: 4914: 4881: 4880: 4879: 4813: 4798: 4797: 4796: 4756: 4736: 4676: 4653:proper classes 4640: 4637: 4634: 4619: 4607: 4568: 4567: 4566: 4565: 4564: 4552: 4541: 4529: 4518: 4506: 4469: 4447: 4427: 4397: 4385: 4374:"conglomerate" 4361: 4332: 4331: 4330: 4329: 4328: 4327: 4308: 4307: 4306: 4305: 4292: 4291: 4290: 4287:proper classes 4275: 4274: 4273: 4272: 4271: 4259: 4248: 4236: 4225: 4213: 4188: 4166: 4146: 4116: 4104: 4093:"conglomerate" 4080: 4057: 4056: 4055: 4054: 4048: 4047: 4017: 4016: 3975: 3971:proper classes 3959: 3958: 3957: 3956: 3944: 3933: 3921: 3910: 3898: 3853: 3819: 3803: 3800: 3799: 3798: 3797: 3796: 3795: 3794: 3793: 3792: 3791: 3790: 3789: 3788: 3787: 3786: 3785: 3784: 3783: 3782: 3715: 3693: 3690: 3687: 3684: 3681: 3676: 3671: 3668: 3663: 3639: 3615: 3610: 3605: 3581: 3557: 3524: 3523: 3522: 3521: 3520: 3519: 3518: 3517: 3516: 3515: 3514: 3513: 3512: 3511: 3510: 3509: 3508: 3507: 3506: 3459: 3421: 3418: 3415: 3395: 3392: 3389: 3368: 3365: 3362: 3315: 3301: 3208: 3160: 3159: 3158: 3157: 3156: 3155: 3140: 3117: 3097: 3077: 3065: 3052: 3051: 3050: 3047:proper classes 3035: 3034: 3033: 3021: 3010: 2998: 2987: 2975: 2953: 2933: 2896: 2846:virtualization 2819: 2818: 2817: 2816: 2802: 2801: 2800: 2793: 2789: 2766: 2760: 2754: 2753: 2752: 2728: 2727: 2704: 2703: 2702: 2701: 2684: 2683: 2669: 2668: 2667: 2664:proper classes 2652: 2651: 2650: 2643: 2636: 2619: 2605: 2590: 2589: 2574: 2573: 2572: 2571: 2553: 2552: 2551: 2550: 2493: 2492: 2474: 2473: 2457: 2456: 2429: 2428: 2427: 2426: 2416: 2415: 2414: 2413: 2391: 2390: 2389: 2388: 2371: 2370: 2369: 2368: 2346: 2343: 2341: 2338: 2337: 2336: 2335: 2334: 2333: 2332: 2331: 2330: 2329: 2328: 2327: 2326: 2310:Alexei Kopylov 2268: 2223: 2222: 2221: 2220: 2192: 2191: 2144: 2141: 2139: 2135: 2126: 2120: 2115:not a problem. 2111: 2110: 2109: 2108: 2107: 2106: 2105: 2104: 2103: 2102: 2101: 2100: 2099: 2098: 2083: 2082: 2054: 2053: 2052: 2051: 2050: 2049: 2048: 2047: 2046: 2045: 2044: 2043: 2009: 1972: 1971: 1970: 1969: 1968: 1967: 1966: 1965: 1964: 1963: 1949: 1929: 1906: 1895: 1872: 1852: 1832: 1829: 1826: 1806: 1803: 1800: 1775: 1760: 1738: 1737: 1694: 1688: 1670: 1669: 1659:Alexei Kopylov 1650:second chapter 1645: 1644: 1643: 1642: 1641: 1640: 1626: 1606: 1605: 1604: 1603: 1574: 1573: 1559: 1521: 1513: 1508: 1503: 1498: 1493: 1488: 1483: 1478: 1453: 1450: 1449: 1448: 1447: 1446: 1445: 1444: 1411: 1391: 1371: 1353: 1350: 1347: 1329: 1328: 1317: 1316: 1306:Alexei Kopylov 1283: 1282: 1268: 1250: 1249: 1248: 1242: 1241: 1240: 1234: 1233: 1232: 1159:Alexei Kopylov 1151: 1148: 1147: 1146: 1145: 1144: 1143: 1142: 1141: 1140: 1107: 1079: 1078: 1068:Alexei Kopylov 1026: 1023: 1020: 1017: 1014: 1011: 1008: 1005: 986: 985: 975:Alexei Kopylov 911:Alexei Kopylov 900: 897: 896: 895: 894: 893: 892: 891: 890: 889: 888: 887: 886: 885: 857: 774: 773: 772: 685: 682: 679: 660: 659: 645:interpretation 632: 629: 626: 606: 603: 600: 558: 555: 552: 532: 529: 526: 514: 513: 512: 488: 468: 465: 462: 442: 439: 436: 405: 402: 401: 400: 359: 356: 354: 350: 349: 346: 345: 342: 341: 330: 324: 323: 321: 304:the discussion 291: 290: 274: 262: 261: 253: 241: 240: 234: 223: 209: 208: 201:the discussion 193: 181: 180: 118: 117: 113: 112: 107: 102: 93: 92: 90: 89: 82: 77: 68: 62: 60: 59: 48: 39: 38: 35: 34: 28: 13: 10: 9: 6: 4: 3: 2: 6074: 6063: 6060: 6058: 6055: 6054: 6052: 6039: 6035: 6031: 6013: 6009: 6000: 5999: 5998: 5995: 5992: 5987: 5985: 5982: 5979: 5975: 5971: 5967: 5966: 5965: 5961: 5957: 5953: 5952: 5949: 5945: 5941: 5936: 5935: 5928: 5927: 5920: 5919: 5918: 5917: 5914: 5911: 5905: 5903: 5899: 5891: 5887: 5884: 5881: 5877: 5874: 5871: 5867: 5864: 5861: 5857: 5854: 5852: 5848: 5845: 5844: 5840: 5835: 5833: 5829: 5825: 5821: 5816: 5815: 5811: 5807: 5802: 5797: 5792: 5788: 5787: 5782: 5780: 5777: 5776: 5772: 5770: 5769: 5762: 5758: 5754: 5753: 5748: 5746: 5742: 5738: 5734: 5729: 5726: 5722: 5717: 5713: 5707: 5706: 5705: 5701: 5697: 5696:GeoffreyT2000 5693: 5689: 5685: 5681: 5677: 5675: 5672: 5669: 5665: 5661: 5657: 5653: 5649: 5646: 5645: 5640: 5637: 5634: 5630: 5626: 5622: 5618: 5615: 5613: 5609: 5605: 5601: 5598: 5596: 5591: 5587: 5581: 5579: 5575: 5571: 5567: 5564: 5563: 5562: 5561: 5558: 5554: 5550: 5546: 5542: 5538: 5534: 5530: 5527: 5525: 5521: 5517: 5513: 5510: 5506: 5503: 5500: 5496: 5495: 5494: 5489: 5485: 5478: 5475: 5473: 5469: 5465: 5461: 5458: 5457: 5456: 5454: 5451: 5450: 5446: 5444: 5443: 5437: 5436: 5429: 5426: 5422: 5418: 5414: 5410: 5406: 5401: 5396: 5390: 5389: 5384: 5380: 5372: 5368: 5362: 5355: 5353: 5348: 5343: 5338: 5337: 5332: 5328: 5324: 5320: 5316: 5312: 5311: 5308: 5304: 5300: 5296: 5295: 5294: 5293: 5289: 5285: 5280: 5276: 5268: 5266: 5265: 5261: 5257: 5250: 5246: 5242: 5238: 5237: 5236: 5235: 5231: 5227: 5219: 5189: 5186: 5183: 5180:Received. — 5179: 5177: 5173: 5169: 5164: 5150: 5130: 5117: 5103: 5083: 5072: 5071: 5070: 5066: 5062: 5058: 5057: 5056: 5053: 5050: 5046: 5042: 5041: 5040: 5036: 5032: 5028: 5027: 5026: 5022: 5018: 5014: 5013: 5012: 5008: 5004: 5000: 4998: 4995: 4992: 4988: 4986: 4982: 4978: 4974: 4973: 4972: 4968: 4964: 4959: 4957: 4953: 4949: 4944: 4943: 4942: 4938: 4934: 4929: 4928: 4927: 4923: 4919: 4915: 4913: 4909: 4905: 4900: 4899: 4898: 4894: 4890: 4886: 4882: 4878: 4874: 4870: 4866: 4865: 4864: 4860: 4856: 4852: 4848: 4844: 4843: 4842: 4838: 4834: 4830: 4826: 4822: 4821:model (logic) 4818: 4814: 4812: 4808: 4804: 4799: 4794: 4790: 4787:" but not in 4786: 4782: 4778: 4774: 4770: 4754: 4734: 4726: 4722: 4718: 4714: 4710: 4706: 4702: 4698: 4694: 4690: 4674: 4666: 4662: 4658: 4654: 4638: 4635: 4632: 4624: 4620: 4605: 4597: 4593: 4589: 4585: 4581: 4577: 4573: 4569: 4550: 4542: 4527: 4519: 4504: 4496: 4495: 4494: 4493: 4491: 4487: 4483: 4467: 4460: 4456: 4452: 4451:"collections" 4448: 4445: 4441: 4425: 4418: 4414: 4410: 4406: 4402: 4398: 4383: 4375: 4359: 4352: 4348: 4344: 4343: 4340: 4339: 4338: 4337: 4336: 4335: 4334: 4333: 4326: 4322: 4318: 4314: 4313: 4312: 4311: 4310: 4309: 4304: 4300: 4296: 4293: 4288: 4284: 4280: 4276: 4257: 4249: 4234: 4226: 4211: 4203: 4202: 4201: 4200: 4186: 4179: 4175: 4171: 4170:"collections" 4167: 4164: 4160: 4144: 4137: 4133: 4129: 4125: 4121: 4117: 4102: 4094: 4078: 4071: 4067: 4063: 4062: 4061: 4060: 4059: 4058: 4052: 4051: 4050: 4049: 4046: 4042: 4038: 4033: 4032: 4031: 4030: 4026: 4022: 4014: 4010: 4006: 4002: 3998: 3994: 3993: 3992: 3989: 3988: 3984: 3980: 3974: 3972: 3968: 3964: 3942: 3934: 3919: 3911: 3896: 3888: 3887: 3886: 3885: 3884: 3882: 3878: 3877:"collections" 3873: 3871: 3867: 3851: 3844: 3840: 3836: 3832: 3828: 3824: 3818: 3817: 3813: 3809: 3801: 3781: 3777: 3773: 3769: 3768: 3767: 3763: 3759: 3755: 3754: 3753: 3749: 3745: 3741: 3740: 3739: 3735: 3731: 3685: 3666: 3608: 3545: 3544: 3543: 3542: 3541: 3537: 3533: 3529: 3525: 3505: 3501: 3497: 3493: 3488: 3483: 3478: 3474: 3473: 3472: 3468: 3464: 3460: 3458: 3454: 3450: 3445: 3444: 3443: 3439: 3435: 3419: 3416: 3413: 3393: 3390: 3387: 3366: 3363: 3360: 3352: 3351: 3350: 3346: 3342: 3338: 3330: 3329: 3328: 3324: 3320: 3316: 3314: 3310: 3306: 3302: 3300: 3296: 3292: 3287: 3283: 3278: 3277: 3276: 3272: 3268: 3264: 3263: 3262: 3258: 3254: 3250: 3249: 3248: 3244: 3240: 3236: 3232: 3231: 3230: 3226: 3222: 3206: 3197: 3195: 3187: 3186: 3185: 3181: 3177: 3173: 3168: 3167: 3166: 3165: 3164: 3163: 3162: 3161: 3154: 3150: 3146: 3141: 3139: 3135: 3131: 3115: 3095: 3075: 3066: 3064: 3060: 3056: 3053: 3048: 3044: 3040: 3036: 3019: 3011: 2996: 2988: 2973: 2965: 2964: 2962: 2958: 2957:"collections" 2954: 2951: 2947: 2931: 2924: 2920: 2916: 2912: 2908: 2904: 2900: 2899: 2897: 2895: 2891: 2887: 2881: 2880:) are classes 2879: 2875: 2871: 2864: 2863: 2862: 2858: 2854: 2850: 2847: 2843: 2839: 2835: 2831: 2827: 2823: 2822: 2821: 2820: 2815: 2811: 2807: 2803: 2798: 2794: 2788: 2785:(rather than 2784: 2780: 2777:(quoted from 2776: 2772: 2765: 2761: 2758: 2757: 2755: 2751: 2747: 2743: 2739: 2735: 2734: 2732: 2731: 2730: 2729: 2726: 2722: 2718: 2714: 2710: 2706: 2705: 2700: 2696: 2692: 2688: 2687: 2686: 2685: 2682: 2678: 2674: 2670: 2665: 2661: 2657: 2653: 2648: 2644: 2641: 2637: 2634: 2630: 2629: 2627: 2623: 2622: 2620: 2618: 2614: 2610: 2606: 2604: 2600: 2596: 2592: 2591: 2588: 2584: 2580: 2576: 2575: 2569: 2565: 2561: 2557: 2556: 2555: 2554: 2548: 2544: 2540: 2536: 2533: 2529: 2528:"collections" 2526: 2522: 2519: 2515: 2511: 2508: 2504: 2501: 2497: 2496: 2495: 2494: 2490: 2489: 2488: 2487: 2483: 2479: 2471: 2467: 2463: 2459: 2458: 2454: 2450: 2446: 2442: 2438: 2434: 2433: 2432: 2424: 2420: 2419: 2418: 2417: 2412: 2408: 2404: 2399: 2395: 2394: 2393: 2392: 2387: 2383: 2379: 2375: 2374: 2373: 2372: 2366: 2365: 2364: 2363: 2362: 2361: 2357: 2353: 2344: 2339: 2325: 2321: 2317: 2311: 2305: 2304: 2303: 2299: 2295: 2289: 2283: 2282: 2281: 2276: 2272: 2267: 2265: 2258: 2252: 2247: 2246: 2245: 2241: 2237: 2233: 2229: 2228: 2227: 2226: 2225: 2224: 2219: 2215: 2211: 2207: 2201: 2196: 2195: 2194: 2193: 2190: 2186: 2182: 2178: 2173: 2172: 2171: 2170: 2166: 2162: 2158: 2153: 2151: 2146: 2134: 2130: 2124: 2117: 2116: 2097: 2093: 2089: 2085: 2084: 2081: 2077: 2073: 2068: 2067: 2066: 2065: 2064: 2063: 2062: 2061: 2060: 2059: 2058: 2057: 2056: 2055: 2042: 2038: 2034: 2030: 2026: 2022: 2018: 2014: 2010: 2008: 2004: 2000: 1996: 1992: 1988: 1984: 1983: 1982: 1981: 1980: 1979: 1978: 1977: 1976: 1975: 1974: 1973: 1962: 1958: 1954: 1950: 1948: 1944: 1940: 1935: 1930: 1928: 1924: 1920: 1904: 1896: 1894: 1890: 1886: 1870: 1850: 1830: 1827: 1824: 1804: 1801: 1798: 1790: 1789: 1788: 1784: 1780: 1776: 1774: 1770: 1766: 1761: 1759: 1755: 1751: 1747: 1742: 1741: 1740: 1739: 1736: 1732: 1728: 1722: 1717: 1716: 1715: 1711: 1707: 1703: 1699: 1698:corresponding 1693: 1689: 1687: 1683: 1679: 1674: 1673: 1672: 1671: 1668: 1664: 1660: 1656: 1651: 1647: 1646: 1639: 1635: 1631: 1627: 1625: 1621: 1617: 1612: 1611: 1610: 1609: 1608: 1607: 1602: 1598: 1594: 1590: 1586: 1582: 1578: 1577: 1576: 1575: 1572: 1568: 1564: 1560: 1558: 1554: 1550: 1546: 1542: 1538: 1537: 1536: 1535: 1531: 1527: 1523: 1518: 1515: 1510: 1505: 1500: 1495: 1490: 1485: 1480: 1475: 1472: 1471: 1467: 1463: 1451: 1443: 1439: 1435: 1431: 1427: 1423: 1409: 1389: 1369: 1359: 1354: 1351: 1348: 1345: 1340: 1338: 1331: 1330: 1326: 1321: 1320: 1319: 1318: 1315: 1311: 1307: 1299: 1295: 1291: 1287: 1286: 1285: 1284: 1281: 1277: 1273: 1269: 1267: 1263: 1259: 1255: 1251: 1246: 1245: 1243: 1238: 1237: 1235: 1230: 1229: 1227: 1223: 1219: 1215: 1210: 1208: 1204: 1200: 1192: 1190: 1186: 1179: 1175: 1171: 1170: 1169: 1168: 1164: 1160: 1156: 1149: 1139: 1135: 1131: 1127: 1123: 1119: 1105: 1097: 1093: 1089: 1083: 1082: 1081: 1080: 1077: 1073: 1069: 1065: 1060: 1056: 1052: 1048: 1047: 1046: 1042: 1038: 1033: 1027: 1024: 1021: 1018: 1015: 1012: 1009: 1006: 1003: 999: 995: 990: 989: 988: 987: 984: 980: 976: 972: 962: 954: 945: 944: 943: 942: 938: 934: 930: 926: 922: 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to edit 49: 46: 41: 40: 37: 36: 32: 26: 22: 18: 17: 5991:Arthur Rubin 5978:Arthur Rubin 5969: 5910:Arthur Rubin 5906: 5901: 5897: 5895: 5823: 5817: 5783: 5774: 5767: 5766: 5749: 5727: 5684:Conglomerate 5668:Arthur Rubin 5663: 5659: 5647: 5633:Arthur Rubin 5616: 5599: 5565: 5528: 5511: 5499:Arthur Rubin 5476: 5459: 5448: 5441: 5440: 5434: 5433: 5425:Arthur Rubin 5416: 5412: 5408: 5407:used in set 5404: 5392: 5360: 5358: 5346: 5339: 5272: 5253: 5223: 5182:Arthur Rubin 5122: 5075: 5049:Arthur Rubin 4991:Arthur Rubin 4884: 4847:Model theory 4829:Model theory 4623:proper class 4450: 4373: 4169: 4092: 4018: 3990: 3976: 3960: 3876: 3874: 3820: 3805: 3486: 3481: 3285: 3282:this article 3191: 2956: 2877: 2869: 2867: 2841: 2837: 2833: 2825: 2786: 2763: 2646: 2639: 2632: 2625: 2567: 2563: 2559: 2534: 2527: 2524: 2520: 2517: 2509: 2506: 2499: 2475: 2469: 2465: 2461: 2430: 2398:One Universe 2397: 2348: 2263: 2156: 2154: 2149: 2147: 2132: 2128: 2122: 2118: 2114: 2112: 1933: 1720: 1691: 1588: 1584: 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