2396:'well-order relation' make it sound like it's some special type of relation stand alone" doesn't have a clear meaning; it is indeed a special type of relation, but what are the words "stand alone" intended to convey? The only way I can make any sense out of your remarks is to assume that you have a misunderstanding of what the word "relation" means in mathematics. A relation is a set of ordered pairs from a set. For example, we can define a relation called "is less than" on the set of integers. That relation then includes, amongst others, the pairs (2, 87), (-5, -1), and so on. We can also define another relation, also called "is less than" on the set of positive real numbers. That will include the pair (2, 87) but not (-5, -1); it will also include (2.7, π). Note that these are two different relations. In a context where it is clear what particular relation we are referring to, we can just use a name for it such as "is less than" or "<", but in any context where order relations on different sets are used, we have to use a notation which distinguishes between them. I suspect from what you have said above that you had not realised that the standard interpretation of "relation" in mathematics is a specific relation on a specific set.
1444:
prove? Since we are assuming the set is well-ordered, it is already order-isomorphic to a unique ordinal, and thus has an infinitely decreasing sequence if and only if the ordinal does as well. However, it is easy to prove that any decreasing sequence of ordinals (and thus decreasing sequence of elements of an ordinal) is finite by transfinite induction without any form of the Axiom of Choice. If every ordinal smaller than alpha has no infinitely decreasing subset, then alpha has no infinitely decreasing subset, for its first element must be some alpha_0 < alpha. After this element, every other element is in alpha_0, so the remainder of the sequence must be finite. However, adding a term to a finite sequence does not yield an infinite sequence.
2440:"The relation you have referred to is a relation on the natural numbers ..." It's defined for all natural numbers, it's not necessarily a function over the natural numbers. "A relation is a relation on a particular set ..". No. When we say 'relation on a set' we typically mean a binary relation from/to the same set, but that's not true in general. Relations can also be n-ary. "not some abstract concept divorced from any set ...". A relation itself can be completely represented by a set of tuples. I think you have some fundamental misunderstanding about what a relation is :)
2079:, we should generally stick to the usual sources here, unless there are more mainstream sources that also worry about predicative definitions of well orderings. Analogously, the same holds for mathematical constructivism - few Knowledge math articles spend time on what would happen if their topics were studied in constructive mathematics, because that is not a very common perspective in the references. It would give a distorted perspective if all our math articles were written from the POV of constructivism or predicativism. — Carl
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ways. WP is a nice place to look up various terms and concepts and get a quick back-grounder; but once one starts reading about a topic in depth (i.e. from a book, journal articles) it becomes clear that WP is one giant ball of undue-weight. To turn that around and use it as an argument to NOT extend an article with useful information seems to be a preposterous absurdity. I suppose that sounds harsh, but really, just take a step back and take a look at what's going on in WP math and physics. Its a fair train-wreck.
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for all a,b in the set. Rosen doesn't even talk about "strict" posets. But the strictness or otherwise of the ordering doesn't effect the well-ordering condition anyway, since there is a least element whether or not the relation is strict or non-strict. So there is no need to talk about it? In fact I think doing so just add confusion. --
1849:. But it was worth a try :-) Anyway, the word "wohl" in German does have two different meanings, the relevant sense of which presumably does not correspond exactly to the English word well. So perhaps Torvatore's perspective is most natural, that one really ought to consider "well-order" as an indivisible unit.
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I don't know what you mean by "consider the same well-order relation on the reals". The relation you have referred to is a relation on the natural numbers; it is not a relation on the real numbers, so "the same well-order relation on the reals" has no meaning. A relation is a relation on a particular
1876:
Usually, when someone defines "good" in a proof, it's a way of saying, "this is just a notion I want to define for convenience in this particular proof. I understand that, of course, you can't really make 'good' a permanent technical term, but I don't expect this notion to ever be useful outside the
1644:, the ways that hyphens are used to join words are listed. The compound modifier, as in "well-ordered set" is given. Also given are combinations of nouns. When speaking of a "well ordering" neither of these is applicable (unless "well" is noun, such as a place where we get water). That leaves the
1899:
Although we usually use the hyphen in the article, we are not entirely consistent. Probably that should be fixed. Also, perhaps we should have a short section listing the alternatives, or at least a sentence somewhere near the "wellordering" sentence, to indicate that sometimes the separated words
1815:
Unlikely. Mathematicians rarely write about their terminology in that way, so finding references that talk about it is very unlikely. This is true for many aspects of mathematical terminology and notation, not just here. Of course, we also don't say anything about this in the article, which is where
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This is a consequence of ZF, but is independent of
Zermelo Set Theory (and if I recall correctly is the biggest reason for preferring ZF over Z set theory.) Is it fair to mention this? Given that Z is a common axiomitization I feel it's a valid thing to mention, unlike for example something that's a
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Actually, it is sufficient to discard the reals which are not constructible. That is, because the constructible universe is well-ordered by a specific formula, its intersection with the reals is also well-ordered by that formula. In other words, you can use the construction of the constructible real
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Easy. The set of real numbers . It has a least number, 0. What is the least element of the subset consisting of the interval (0,1] ? It is the smallest number greater than zero. But what is the smallest real number greater than zero? (Actually, if the Axiom of Choice is true then there is a way
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which gives different formulations of well-ordering and asserts that they are (provably) equivalent. Nik Weaver's definition seems to be the one referred to in that section as "Transfinite induction works for the entire ordered set". Can you give a reason to doubt that that definition is equivalent
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section. Is it your thinking that a "well-ordering" is like a "lily-of-the-valley", which distinguishes a specific kind of lily from an arbitrary lily growing in the valley? Is it common enough for someone to use "a well ordering" to be an ordering that was done well, but which is not necessarily
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Second, we should not insist on the order being strict, certainly not in the first paragraph of the article. There is no need to mention "strict" at all, as long as you use symbols like < and ≤ which make it clear whether you are talking about a reflexive or areflexive relation. Of course, in
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I'm not an expert and certainly don't understand all that is at stake (I have no idea what that
Groundhog's function is :P). But it does seem akward to me that the first sentence defines the well-order relation as strict, but then does nothing with this strictness: the concept of a least element is
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As far as I was aware, a poset is by definition is a "non-strict" relation. In my uni
Discrete Math text (K.Rosen, Discrete Mathematics) it's simply any reflexive, antisymmetric, transitive relation on a set (thus "non-strict"). A total ordering is a poset with one extra requirement: (a,b) or (b,a)
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above: I too am not really sure what you're getting at. Firstly, it is not true that "there is no such thing as a well-order relation"; the expression "well-order relation" exists, is commonly used by mathematicians, and has the meaning attributed to it in this
Knowledge article. Secondly, "Saying
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The fact that the adverb "well" remains "well" as an adjective instead of becoming "good" is common when irregularly transforming words are used in novel ways. Compare "Today, the bird flies; yesterday, the bird flew" with "This inning, the batter flies out; last inning, the batter flied out". I
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do not have a hyphen between them — if for no other reason than to prevent someone who knows the
English rules for hyphens from fixing the current article. (FWIW, I don't agree that "mathematicians rarely write about their terminology in that way", but it appears to be moot, so I won't argue it.)
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I was simply reading around, and was interested in alternative formulations and discussions. Weaver's quote seemed notable. Regarding "undue weight", that's just ridiculous: every single math article on WP gives undue weight to one or another topic, and is grossly deficient in a large variety of
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It's just technical jargon, and the question is not whether it conforms to broader
English usage (which clearly it doesn't, since an adverb can't modify a noun). "A well order" is pretty much ungrammatical, again, as you say, unless you mean an order inside a well, or perhaps an order that isn't
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JR has reverted Tobias's changes on the grounds that "well-founded only applies to strictly-less; total order only applies to less-or-equal". Those may be the definitions in the linked articles but I do not think either restriction is really standard. In general these things are understood from
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We could: (A) State that some authors take wellorders to be strict and others to be reflexive, and give the two corresponding definitions, (B) say that there are different definitions depending on context, or (C) give a definition that covers both cases (e.g. the irreflexive part has no infinite
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In the "Equivalent
Formulations" section, the article states that the Axiom of Dependent Choice is required to prove that a well-ordered set cannot have an infinite decreasing sequence. However, isn't the fact that the set is well-ordered precisely what the Axiom of Dependent Choice is there to
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This is a slightly annoying point — I think I brought it up some years ago but never followed through on it. In actual usage, I agree, both strict and nonstrict orders may be described as wellorders, and no one worries about it much. In an article, we probably have to worry about it at least a
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Both Jech's book and Kunen's books use "well-order", with a hyphen, as in "R is a well-ordering of A" and "A is well-ordered". This is very common in the field of set theory, which this article is about. Given that the article was already established using the hyphenated form, I am in favor of
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Can the wellorder of the reals as proved to exist under ZF+AC be explicitly stated, or described in any detail? I'm trying to wrestle with the implications of the wellordering theorem. I can conceptualize wellorders for all finite and countable sets. It would be nice to have an idea of what a
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In
English, a hyphen is not generally used to connect an adverb, like "well" to a verb like "ordered". However, when an adverb not ending in "ly" modifies a verb acting as participle and together they modify a subsequent noun then the hyphen is used, as in "well-ordered set". This article,
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The definition of a well ordered set says that every subset of a set has a least element. However, since the relational operator must be anti-symmetric, I'm having a hard time imagining a totally ordered set which itself has a least element but has a subset without a least element.
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tag from the spelling note; of course this is not the sort of thing for which citations are likely to be available. It's the kind of thing you'd find in a dictionary, but no one writes dictionaries of contemporary set-theoretic usage. If you want a citation in which the spelling is
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in substance the same, whether the order is strict or not (and is actually easier to define with a reflexive order). True, using a reflexive order might make the second sentence a bit more complicated to state; but the readability of the first sentence seems more important to me.
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Consider a so called "well-order relation" on the natural numbers. The natural numbers are a subset of the real numbers. Now consider the same "well-order relation" on the reals. It's not longer well-order relation. The well-ordered-ness comes from the set not from the relation.
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In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is then called a well-ordered set.
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the classical concept of well-ordering has a variety of formulations which are not predicatively equivalent (see §1.4 and §2.4). In fact, previous discussions of predicativism have tended to ignore this distinction, and this will emerge as a crucial source of confusion (see
1649:"a well ordering" in the sense of the article? If not, but you think that there are other uses for hyphens (such as "as technical non-analyzable terms in which the word has no independent meaning") can you give a citation, Knowledge article, or whatever to support that?
1396:
Sorry about that, I really wasn't aware of that convention. Anyway, I would like to restore my other changes to the lead section which I hope are correct and uncontroversial. I am trying to improve the flow of the lead section which seems a bit unconnected to my eyes.
2354:
I was getting at that there is no such thing as a well-order relation, only a well-ordering (a set plus a relation). I.e. there is no property of a relation taken alone that distinguishes it as a "well-order" relation as distinct from a total ordering. AFAICT anyway.
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To
Trovatore: You are right that I was not addressing the specific question that was asked. But if his question was motivated by wanting to avoid having to postulate the existence of a set without having a way of constructing it, then I was pointing out that there
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To be sure, we can't be that sloppy in an article for general consumption, but interpreting whatever choice was made by whoever wrote the linked articles as somehow canonical is not the way to go either. Maybe we could use an explanatory footnote or something?
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Woodin has also placed a limit on the best lower bound on the complexity you can get just from large cardinals of the sort that we know about (the ones that are preserved under small forcing). As I understand it, he showed that small forcing suffices to add a
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uses the nonstrict notion, but the examples given seem to be strict. Among adepts this is one of those things you don't worry too much about because it's usually clear from context, but in an article like this maybe we should be a little more careful.
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I.e. I want to change the above quoted text to remove the first instance of "well-order relation" and replace the second one with just "relation". Saying "well-order relation" make it sound like it's some special type of relation stand alone.
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I added an example of how to well order the integers (all the integers), but my reference book is at home, I didn't write it well and it needs a proof it well orders the integers. Somebody please improve it. (marked for cleanup)
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It never occurred to me that it might be "well" in the sense of a space with a bottom! I have always assumed that "well" means that the ordering is good (in some special sense). Does anyone have a reference to resolve this?
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to well order the reals, but the proof, in addition to requiring AC, is non constructive so no one knows how to well order the reals. Unlike well ordering integers where there is a way to well order the negative numbers).
616:, every projective set is Lebesgue measurable, so there cannot be a projective well-ordering of the reals, because this would lead to the existence of a projective Vitali set. On the other hand, if V = L then there is a
788:, section "Von Neumann definition of ordinals", this point is made; maybe the reference to the AC should be made more explicit, since "intuition" yells that the AC is invoked whenever we talk about infinite sets :-)
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ZF does not prove that R can be wellordered. In fact the usual proof (or at least the first proof that I learned) that ZF does not prove AC, goes through the fact that ZF does not prove there's a wellordering of the
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a reference would be required. (For the record, as someone who has worked with these orders professionally for years, I have never thought of the "well" as referring to a hole in the ground. ) — Carl
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933:, which is totally ordered but is not well-ordered. Am I misreading this part of the article, or is this part incorrect? If it's correct, I think there needs to be a better explanation.
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The term "well order", "well-order", or (my preferred version) "wellorder", are better thought of as technical non-analyzable terms in which the word "well" has no independent meaning.
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can be proven to have a well-ordering using only the ZF axioms (without AC)? I guess that means using some conventional construction of R, with
Dedekind cuts or whatever. Thanks.
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Consider the infinitive form. You would write "to well-order a set", not "to order a set well". If "well" were acting as an adverb, the first form would be a split infinitive.
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well ordering of the reals. So although AC implies that there is a well-ordering of the reals, there may or may not be a nice formula that defines this well-ordering. — Carl
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descending sequence). For (C) we need a source; is there any source that does that? (A) and (B) are more honest but harder to figure out how we deal with the sourcing. --
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don't find it alarming that "well" fails to be irregular when the adverb-verb phrase "well order" transforms to the adjective-noun phrase "well order" or "well ordering".
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If ≤ is a (non-strict) well-ordering, then < is a strict well-ordering. A relation is a strict well-ordering if and only if it is a well-founded strict total order.
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suggests that noun adjuncts may or may not be hyphenated. My impression is that in English, such things are generally governed by practice rather than fixed rules.
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the other versions. Without some version of the axiom of choice, there could be a non-well-ordered linear ordering which lacks any infinite decreasing sequences. If
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Yes, this is intentional. The example you are referring to uses an ordering different from the standard ordering. See also the blue example in the picture shown at
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This would be understandable if the set were defined in accordance with historical convention (i.e. starting with 1), but this explicitly states 0 is in the set.
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wellorder of the reals; therefore, no known large cardinals can refute the existence of such a wellorder. I don't know whether that's lightface or boldface. --
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I'm not opposed to mentioning after the fact that you need Replacement to prove this property. But we should still state the property without waffling. --
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How is it equivalent to say if a set is totally ordered then it is well-ordered? The way wolfram math world defines this makes this statement incorrect.
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Is there such a thing as a well-order relation? Will not any total-ordering relation suffice, with the well-ordering arising from the nature set only? --
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statement: The way of thinking about infinite sets that we've found most useful, makes it intuitively clear that AC is true. Is that what you meant? --
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The thing is, "well" is not actually an adverb here. If it were, then the noun form would be "good order" (ordinarily, an adverb cannot modify a noun).
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I'll briefly cite some: "The natural numbers are a well-order." "The set {1/n : n =1,2,3,...} has no least element and is therefore not a well-order."
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Third: Personally, I prefer strict orders when talking about well-orders, because then I can use the symbol ∈ to denote the order relation on ordinals.
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Leegrc, I didn't mean that "good ordering" wasn't the intent, etymologically. I think it was. I was saying that it can't be read that way, now.
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Those suggestions make sense, I think. I would note though that the usual English rules also forbid "a well order" (or even "a well ordering").
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an ordering done well. There's nothing "better" about a wellordering than any other ordering (though, certainly, sometimes it is more useful).
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is the least limit ordinal for which Weaver has not yet chosen a unique preferred (and predicatively acceptable) fundamental sequence at time
289:. The third possibility, namely a poset in which every nonempty subset has a minimal element, is not the same. To prevent confusion between
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Not really sure what you're getting at here. An ordering is a kind of binary relation, but not all total orderings are wellorderings. --
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First, {x} always has a minimal element, whether you take the order to be strict or not. You just have to define "minimal" properly.
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was a little confusing. Any help in making the demonstration clearer is welcome, as you can probably see I'm not a native speaker.
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This discussion seems predicated on the notion that "well" is an adverb. But, in the context, it seems equally likely that it is
1547:"Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. "
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Is there a reason the section in examples about the natural numbers (defined with 0 as an element) claims 1 has no predecessor?
2006:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
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Also note that there are no hyphens in "fly out", "flied out", "linear order", "total order", or "simple order". Just saying.
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If these authors deliberately-wrote textbooks that clearly-use "well-order" as a verb then that counts as well-written by me.
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That's one reason I like "wellorder", as it seems to make this especially clear. But in any case, all three forms are used. --
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Although I doubt that Weaver would accept this characterization, it seems to me that he is implicitly modeling mathematics as
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Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set.
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to set theory, as to any mathematics; you can't make progress without it. The important (and difficult) thing is to get the
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There are various names attached to relations that satisfy some of these conditions and we put them down here for the record.
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The intro para was stated in such a way that the linearity of the ordering was a consequence of the proposed definition (a
1056:, as otherwise the empty relation is an exemplar of both the order types 0 and 1, making the proof that Hartog's function
999:? Specifying a total order, or a strict total order, is essentially the same thing, and this insistence is distracting.
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In general, the well-ordering of the reals may not be definable in any reasonable way. For example, in a model of ZFC +
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857:; there is the natural place to explore in more details what constructions do and what constructions don't require AC.
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I think this would be clearer with an example. Does a well-order require the definitions of mathematical sets? - anon
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is a finite sequence of elements above the end of the well-ordered initial segment of the ordering, then there is an
1018:", in the definition in the lede, also presumes strict, at least in its formal definition. If you look closely, if
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Can anyone give an example of why this definition is necessary? It might be something nice to include on the page.
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more difficult. Also, all the examples in this article are reflexive. Perhaps we should rewrite, after all. —
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real is ordinal-definable, and therefore the canonical wellorder of the OD reals is a wellorder of all the reals.
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The article is not very specific about whether it's discussing strict or nonstrict partial orders. The link to
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is a noun, not a verb (though it can also be used as a verb, our article titles are supposed to be nouns; see
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set, not some abstract concept divorced from any set; a relation on a different set is a different relation.
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context, and the appropriate modifications made silently and probably without even noticing. For example
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I think the previous paragraph in the Reals section about the countability of the well-ordered subsets of
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How come these errors weren't caught by anyone? This is mis-information and should be deleted or amended.
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It seems that this article ignores these distinctions as well, and perhaps that could be remedied?
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sick. "A well-order" at least gets you over the WTF-adverb-before-noun-what-is-this? reaction. --
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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wellorder of an uncountable set looks like or how it acts. Thanks! - UC Berkeley math student.
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the AC-characterization we have to say "no strictly decreasing sequence" (which we do already).
330:, I can find that for you once I get home (I'd put it on the talk page, not in the article). --
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A countably infinite subset of the reals may or may not be a well-order with the standard "≤".
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there's a formula at all. That's because there's a canonical, definable, wellorder of the
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which is also above the end. But we need DC to make an infinite sequence of such choices.
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numbers themselves to construct a well-ordering of the set of constructible real numbers.
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That, and the fact that well-order was very likely an English translation of the German
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goes — no one bothers AFAIK to state this as "the strict part of Wadge reducibility".
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Weaver's perspective is not a common one in mathematics. Because of the concept of
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On the other hand, requiring the relation to be reflexive allows identification of
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I would prefer no hyphen, but also no space. A wellordering is very specifically
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be wellordered, but some larger set (say, the powerset of the reals) cannot be. --
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But what I think is definitive, did I mention that adverbs can't modify nouns? --
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Last edited at 18:32, 11 May 2007 (UTC). Substituted at 02:41, 5 May 2016 (UTC)
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including its title, confuses this distinction and is using too many hyphens.
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I think we also ought to make it clear that "well-order" in the sense of the
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None the less, I added a few words which I hope will make it clearer to you.
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Axiom of Dependent Choice and "Every Strictly Descending Sequence is Finite"
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order which is well founded in the sense that every nonempty subset has a
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1233:. Sorry about the confusion. I can't help with sourcing, though. —
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You don't need choice to prove the above statement. You do need the
281:
element). The alternative way of defining a well ordering is as a
2504:
2485:
2449:
2435:
2420:
2405:
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1985:
1962:
1943:
1909:
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491:
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334:
305:
263:
1706:
Oh, I agree. I wasn't actually suggesting moving the article to
665:
There is however a limit to how bad the simplest formula can be,
501:
1229:, used "Hartogs' function" as a representation of what we call
15:
1754:: as in, every well has a bottom. More precisely, this is a
2283:{\displaystyle \Gamma _{0}\leq W_{t}<\omega _{1}^{CK}\,.}
1469:
You are looking at the wrong direction of implication. In
1998:
952:
say that totally ordered and well ordered are the same.
518:
I don't really see how that relates to the question. --
802:
It does? What intuition is that? I would agree with
2230:
2162:
1595:"a set that is well ordered" vs. "a well-ordered set"
1122:
1066:
917:
825:
Of course, "intuition" and "set theory" are disjoint
691:
622:
478:
However it is also consistent with ZF that the reals
441:
397:
1036:
are always strictly ordered by the element relation.
112:, a collaborative effort to improve the coverage of
1926:.) So the first sentence should start with just "
1688:maintaining it, personal preferences aside. — Carl
1277:
Many fundamental mistakes in the "Reals" subsection
2282:
2182:
1171:
1108:
925:
709:
640:
451:
403:
1351:is nonstrict, and is wellfounded for as high as
1303:so these examples are using that relationship.
909:http://mathworld.wolfram.com/WellOrderedSet.html
217:template from here since it has been cleanedup.
2540:Knowledge level-5 vital articles in Mathematics
2002:, and are posted here for posterity. Following
1996:The comment(s) below were originally left at
1543:Choice of Set theory and independence result.
8:
948:I think that you are misreading it. It does
1299:Right above this list of examples it says
1172:{\displaystyle 2^{H(X)}\leq 2^{2^{X^{2}}}}
58:
2266:
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2248:
2235:
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1319:Countability of well-ordered subsets of R
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396:
2466:Natural Numbers - Predecessor of 1 is 0?
2530:Knowledge vital articles in Mathematics
2275:
1551:consequence of the axiom of infinity.
973:Ok, it makes more sense to me, thanks.
677:definable wellorder of the reals, then
60:
19:
1109:{\displaystyle H(X)\leq ^{*}2^{X^{2}}}
368:
358:
2545:B-Class vital articles in Mathematics
1642:Joining section of the Hyphen article
822:Humour... it's a difficult concept...
7:
2038:different formulations of well-order
106:This article is within the scope of
49:It is of interest to the following
2555:High-priority mathematics articles
2316:Definition? :: well-order relation
2232:
2102:Well-order#Equivalent formulations
2100:This article contains the section
2049:(2009) I see that he has written:
1471:well-order#Equivalent formulations
693:
624:
277:order in which every subset has a
14:
2004:several discussions in past years
126:Knowledge:WikiProject Mathematics
2525:Knowledge level-5 vital articles
1374:The existing lead already says "
1227:Set Theory for the Mathematician
416:if it is a wellfounded ordering.
344:Moschovakis, Yiannis N. (1980).
129:Template:WikiProject Mathematics
93:
83:
62:
29:
20:
2042:While reading this: Nik Weaver
710:{\displaystyle \Delta _{2}^{2}}
641:{\displaystyle \Delta _{2}^{1}}
431:well-ordering of reals under ZF
146:This article has been rated as
2535:B-Class level-5 vital articles
1137:
1131:
1076:
1070:
452:{\displaystyle {\mathbb {R} }}
229:The new version looks great!
1:
2505:18:43, 7 September 2024 (UTC)
2486:00:24, 5 September 2024 (UTC)
2380:07:54, 24 December 2023 (UTC)
2365:07:47, 24 December 2023 (UTC)
2350:18:05, 23 December 2023 (UTC)
2335:09:18, 23 December 2023 (UTC)
1720:23:15, 29 February 2016 (UTC)
1701:23:10, 29 February 2016 (UTC)
1682:21:18, 29 February 2016 (UTC)
1659:21:10, 29 February 2016 (UTC)
1635:20:03, 29 February 2016 (UTC)
1610:18:29, 29 February 2016 (UTC)
1429:09:28, 12 December 2012 (UTC)
1407:08:23, 12 December 2012 (UTC)
1388:04:56, 12 December 2012 (UTC)
1369:04:34, 12 December 2012 (UTC)
911:There it uses the example of
889:17:07, 12 December 2008 (UTC)
867:11:05, 12 December 2008 (UTC)
839:11:04, 12 December 2008 (UTC)
816:20:14, 11 December 2008 (UTC)
798:10:40, 11 December 2008 (UTC)
780:18:46, 10 December 2008 (UTC)
757:15:52, 10 December 2008 (UTC)
578:05:29, 15 December 2007 (UTC)
563:08:06, 14 December 2007 (UTC)
528:07:06, 14 December 2007 (UTC)
514:05:51, 14 December 2007 (UTC)
492:18:33, 13 December 2007 (UTC)
469:11:32, 13 December 2007 (UTC)
348:. North Holland. pp. p. 104.
264:08:25, 24 December 2023 (UTC)
120:and see a list of open tasks.
2550:B-Class mathematics articles
2105:to the others listed there?
1313:16:10, 4 February 2012 (UTC)
1294:14:07, 4 February 2012 (UTC)
926:{\displaystyle \mathbb {Z} }
2450:02:40, 6 January 2024 (UTC)
2436:22:19, 5 January 2024 (UTC)
2421:23:54, 4 January 2024 (UTC)
2406:15:35, 4 January 2024 (UTC)
2222:below level α. Notice that
2571:
1537:08:37, 26 March 2013 (UTC)
1463:22:10, 25 March 2013 (UTC)
853:link to the article about
784:Ok, I saw that in article
727:08:01, 19 April 2010 (UTC)
661:01:38, 19 April 2010 (UTC)
606:01:13, 19 April 2010 (UTC)
335:23:22, 30 April 2007 (UTC)
233:13:43, Jun 24, 2005 (UTC)
221:13:46, Jun 24, 2005 (UTC)
205:15:22, Mar 25, 2005 (UTC)
2304:21:17, 11 July 2016 (UTC)
2183:{\displaystyle L_{W_{t}}}
2011:
1986:11:49, 6 March 2016 (UTC)
1963:19:54, 1 March 2016 (UTC)
1944:17:58, 1 March 2016 (UTC)
1910:17:48, 1 March 2016 (UTC)
1890:17:41, 1 March 2016 (UTC)
1867:16:22, 1 March 2016 (UTC)
1829:15:28, 1 March 2016 (UTC)
1803:14:29, 1 March 2016 (UTC)
1780:14:10, 1 March 2016 (UTC)
1738:13:00, 1 March 2016 (UTC)
1589:21:33, 15 July 2014 (UTC)
1572:21:23, 15 July 2014 (UTC)
1447:Am I missing something?
983:23:15, 20 June 2010 (UTC)
965:08:14, 18 June 2010 (UTC)
943:04:50, 18 June 2010 (UTC)
197:05:18, 25 Mar 2005 (UTC)
181:21:46, Feb 7, 2005 (UTC)
145:
78:
57:
2138:19:16, 9 July 2016 (UTC)
2115:05:26, 9 July 2016 (UTC)
2092:22:55, 8 July 2016 (UTC)
2070:21:45, 8 July 2016 (UTC)
2027:18:32, 11 May 2007 (UTC)
1999:Talk:Well-order/Comments
1337:14:39, 21 May 2012 (UTC)
1271:13:50, 12 May 2011 (UTC)
1241:00:46, 12 May 2011 (UTC)
1221:19:45, 11 May 2011 (UTC)
1202:15:40, 11 May 2011 (UTC)
1187:13:20, 11 May 2011 (UTC)
1032:-minimal element. And,
1009:09:22, 11 May 2011 (UTC)
871:Intuition is absolutely
404:{\displaystyle \preceq }
306:13:10, 1 July 2006 (UTC)
250:8 July 2005 00:04 (UTC)
152:project's priority scale
903:Equivalent Formulations
426:05:17, 1 May 2007 (UTC)
109:WikiProject Mathematics
2520:B-Class vital articles
2284:
2184:
2044:Predicativity Beyond Γ
1173:
1110:
927:
739:How does the sentence
711:
642:
614:projective determinacy
551:constructible universe
453:
405:
380:The relevant quote is
346:Descriptive Set Theory
2476:Is this intentional?
2285:
2185:
2014:Well-founded relation
1174:
1111:
928:
712:
643:
454:
406:
311:unhyphenated spelling
36:level-5 vital article
2391:I echo the words of
2228:
2160:
1120:
1064:
1034:von Neumann ordinals
915:
768:von Neumann ordinals
764:axiom of replacement
689:
673:reals. If there is
620:
439:
435:Is it known whether
395:
237:Strict or nonstrict?
132:mathematics articles
2294:What do you think?
2274:
1028:} does not have an
706:
637:
421:Hope this helps, --
315:I have removed the
2280:
2276:
2257:
2220:constructible sets
2180:
2012:Some overlap with
1992:Assessment comment
1877:current argument."
1349:Wadge reducibility
1169:
1106:
923:
707:
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638:
623:
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401:
101:Mathematics portal
45:content assessment
2090:
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1984:
1865:
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1778:
1699:
1575:
1558:comment added by
1453:comment added by
671:ordinal-definable
659:
596:comment added by
166:
165:
162:
161:
158:
157:
2562:
2497:Jochen Burghardt
2390:
2289:
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2240:
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2189:
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2177:
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2080:
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2009:
2008:
2001:
1983:
1981:
1969:
1864:
1862:
1850:
1817:
1777:
1775:
1763:
1689:
1574:
1560:James Waddington
1552:
1465:
1399:Tobias Bergemann
1178:
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932:
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922:
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846:
716:
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713:
708:
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649:
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639:
636:
631:
608:
458:
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455:
450:
448:
447:
410:
408:
407:
402:
376:
371:has extra text (
370:
366:
364:
356:
339:OK, here we go.
324:
318:
134:
133:
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127:
124:
103:
98:
97:
87:
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79:
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66:
59:
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33:
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25:
24:
16:
2570:
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2559:
2510:
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2384:
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2231:
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2217:
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2168:
2163:
2158:
2157:
2047:
2040:
1997:
1994:
1979:
1974:
1970:
1860:
1855:
1851:
1773:
1768:
1764:
1646:Other compounds
1597:
1553:
1545:
1526:
1517:
1507:
1497:
1490:
1483:
1455:140.180.243.190
1448:
1441:
1344:
1321:
1286:212.149.212.108
1279:
1155:
1150:
1145:
1123:
1118:
1117:
1094:
1089:
1079:
1062:
1061:
1040:
1019:
993:
913:
912:
905:
879:intuitions. --
855:ordinal numbers
850:
844:
843:BTW, I added a
745:axiom of choice
737:
735:Axiom of choice
687:
686:
618:
617:
591:
437:
436:
433:
393:
392:
367:
357:
343:
322:
316:
313:
287:minimal element
271:
239:
227:
211:
187:
171:
131:
128:
125:
122:
121:
99:
92:
72:
43:on Knowledge's
40:
30:
12:
11:
5:
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2279:
2272:
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2256:
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2238:
2234:
2215:
2202:
2192:
2191:
2190:
2175:
2171:
2166:
2145:
2144:
2143:
2142:
2141:
2140:
2120:
2119:
2118:
2117:
2095:
2094:
2058:
2057:
2045:
2039:
2036:
2030:
2029:
1993:
1990:
1989:
1988:
1977:
1972:
1947:
1946:
1916:
1897:
1896:
1895:
1894:
1893:
1892:
1878:
1874:
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1870:
1869:
1858:
1853:
1836:
1835:
1834:
1833:
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1831:
1808:
1807:
1806:
1805:
1787:
1786:
1785:
1784:
1783:
1782:
1771:
1766:
1743:
1742:
1741:
1740:
1723:
1722:
1685:
1684:
1669:
1638:
1637:
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1621:
1617:
1616:
1596:
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1591:
1544:
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1522:
1512:
1503:
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1440:
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1273:
1259:
1256:
1252:
1249:
1248:
1247:
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1245:
1244:
1243:
1231:Hartogs number
1225:My mother, in
1208:
1162:
1158:
1153:
1148:
1144:
1139:
1136:
1133:
1130:
1126:
1101:
1097:
1092:
1086:
1082:
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1075:
1072:
1069:
1037:
992:
989:
988:
987:
986:
985:
968:
967:
953:
921:
904:
901:
900:
899:
898:
897:
896:
895:
894:
893:
892:
891:
841:
786:ordinal number
743:relate to the
736:
733:
732:
731:
730:
729:
704:
699:
695:
682:
635:
630:
626:
598:98.210.233.159
587:
586:
585:
584:
583:
582:
581:
580:
565:
535:
534:
533:
532:
531:
530:
495:
494:
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446:
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419:
418:
400:
389:
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378:
377:
312:
309:
270:
267:
238:
235:
226:
223:
213:I removed the
210:
207:
186:
183:
170:
169:Example needed
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164:
163:
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159:
156:
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144:
138:
137:
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118:the discussion
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104:
88:
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55:
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48:
26:
13:
10:
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6:
4:
3:
2:
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2515:
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2502:
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2487:
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2422:
2418:
2414:
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2408:
2407:
2403:
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2214:
2210:
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2201:
2197:
2193:
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2108:
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2019:
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1991:
1987:
1982:
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1951:
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1934:relation". --
1933:
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1199:
1195:
1194:David Olivier
1190:
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1128:
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1932:well-order
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