Talk:Transfinite induction

2325:

having functions G as used in the above explanation to "doing something in steps" is what is needed to really "understand" transfinite induction, yet it is - for the most likely reader of this page, i.e. someone not yet familiar with how these things work - not trivial. Adding something like "so we have G defined as ... and then we apply the principles outlayed above" would help. To be clear: I'm not disputing the correctness of the statements there. I just don't think the example is particularly serving its purpose of showing where AoC is used and where the transfinite induction happens (and that AoC is not used there any more).

2273:

obvious. More dangerously, these matters need very precise attention especially when in a learning stage. But looking anywhere in the article, nowhere is "chose something" uncountably often mentioned - instead, either a restriction of a class function (no strict classes mentioned here), or a version with precise steps to follow regarding the treatment of successor and limit ordinals. Neither of those appear anywhere either. I would think that any example on this page should take great care to be exact and rigorous in its use of mathematical terms, and this section is anything but.

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information to the overall point of the article? The article is titled "Transfinite Induction" and this section isn't even about that. So now we've detoured into a related subject, and we're expecting the reader to know jargon that he apparently didn't have to know in order to understand the main topic. All of this, and it's not even the topic that brought him here... In addition to making it more accessible, perhaps we also need to better justify this section.

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accuracy. On the one hand, I think the topic is technical enough, that it is reasonable to expect the reader to have some interest in set theory. And they may appreciate the precision of our language later, once our meaning is clear. On the other hand, even the highly motivated and interested reader who isn't already accustomed to the jargon is going to find this presentation jarring.

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dismissing mathematicians with whom you are unfamiliar, such as myself, as "trivial".). I am not making an ad hoc argument as, in my experience, virtually all proofs by transfinite induction/recursion are simpler and more concise given this treatment, but clearly your experience is different (given a limited understanding of empty intersections).

2943:: "Section and subsection headings should preferably be unique within a page; otherwise section links may lead to the wrong place, and automatic edit summaries can be ambiguous." Anyway, I think I actually prefer the flow of the page before this change, with two sections "Transfinite induction" and "Transfinite recursion", see 789:; I do not see any need to create a separate article, it's adequately covered there. Some folk call well-founded relations "Noetherian", but a quick search confirms that "Noetherian induction" is by an order of magnitude less common than "well-founded induction", and I frankly despise the name. Anyway, 860:

is a set-like relation; that is, for any x, the collection of all y such that y R x must be a set.) However, frequently proofs or constructions using induction and recursion also use the axiom of choice to wellorder a set in order to be able to use the simple prescriptions given in this article. ..."

2730:

Going back to the harpoon. You probably haven't written science papers, so let me explain. If you write a paper, you do it for others, not for your self. So you should care about making it understandable for all readers. In a situation when you are using some notation that is not a total standard, it

2309:

Maybe what you're really looking for is some sort of explanation as to how to turn the argument into a formal deduction from ZFC or something? In that case I completely disagree. The important thing to get across here is what's really going on, "naively" or "Platonistically" if you will, and what's

2272:

To clarify a bit: The reason this example is in the article about transfinite induction is, presumably, to demonstrate it in use. But precisely this use is totally obscured in "chose at each step". For the reader who is not clear on what transfinite induction is, its use in that example is not at all

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I am not sure I understand the objection. As far as I can tell, once the original wellorder is chosen, after that everything is completely well-specified and constructive; at each step you choose the least possible element of the equivalence class, "least" being understood in the sense of the chosen

2241:

The example states that choice is only used to well-order the reals. But the actual "construction" doesn't seem a real "construction". Specifically, the claim to be able to just "countinue; at each step" to chose uncountably often does not seem well-defined at all. Of course it is possible to clearly

1833:

It is true that, formally, you can always treat 0 with the limit ordinals, and for that matter you can always treat all the ordinals together. That's kind of trivial, because all it really means is that, when you have a finite number of fixed cases, you can always combine them into a single function

1170:

This is true, formally, we can indeed state it like that. But it's not the way proofs by transfinite induction actually go, basically ever. In virtually all nontrivial cases we end up distinguishing at least the three cases mentioned (base, successor ordinal, limit ordinal). Many times those break

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2.) If one lives completely i ZF (and ZF only) suddenly want the luxory of having transfinite recursion/induction available for ALL sets, the one would have to add to ZF AC making it ZFC so that within ZFC (only) AC actually is required if transfinite recursion/induction available for all sets? I. e.

311:"Suppose whenever for all α < β, P (α) is true, then P (β) is also true." Should this instead read "Suppose that for all ordinals α and β satisfying α < β, then P (α) is true implies P (β) is also true." ? It's standard to note where variables are taken from before their first use, not after. 2114:

In practice, most arguments start with some arbitrary, complicated object as the base case. It's a given thing, coming from the hypotheses assumed for the theorem. It's usually only in the base step that it's useful to mention it, not in nonzero limit steps, and this makes the base step different.

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2nd point - Since that's a statement that can't really be cited in either direction, how about we say instead "0 is sometimes considered a limit ordinal and is then treated in proofs similarly as limit ordinals when possible". And, I would be VERY interested to see a proof where it cannot be treated

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defined for all ordinals α. Suppose that whenever P(β) is true for all β < α, then P(α) is also true. Then transfinite induction tells us that P is true for all ordinals. That is, if P(α) is true whenever P(β) is true for all β < α, then P(α) is true for all α. Or, more practically: in order

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There is a popular misconception that transfinite induction, or transfinite recursion, or both, require the axiom of choice (AC). This is incorrect. Transfinite induction can be applied to any wellordered set. However, frequently proofs or constructions using transfinite induction also use the axiom

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The page says "Notice that the second and third cases are identical except for the type of ordinal considered (...)", talking about the Successor Case and Limit Case of Transfinite induction. This doesn't seem right; the cases are different because in the Limit case, the limit ordinal is not in the

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YOU are imposing a normative aesthetic when you say "no one" does this, and when you claim that your way is the only way worth describing in the article. If one treats zero as a limit ordinal (as many do), then there is no reason to treat it differently, as I have demonstrated above. I do not claim

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is to represent all viewpoints fairly, not to force one particular one on readers. I don't ask that you accept my POV, only that it is represented in the article and not simply removed because someone dislikes it. (And by the way, the concept of empty intersection is a clear, natural, and necessary

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There is a popular misconception that induction, or recursion, or both, require the axiom of choice (AC). This is incorrect. One can use induction to prove things and one can define objects by recursion on any well-founded relation R. (R need not even be a set; it can be a proper class, provided it

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This last bit is a bit closer, but not fully. I don't mean a formal deduction from ZFC - that'd certainly be far over the top. But a somewhat more formal deduction from the principles actually laid out in the paragraph above is what I'd expect to see, or at least hinting at it. The transition from

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because it makes clear that the quantifier over alpha should be applied to the whole implication, not just its hypothesis. That the quantifier over beta is limited to the hypothesis I would assume due to unary operators (negation and quantifiers) taking priority over binary operators (implication,

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which yield an ordinal sequence into the powerset of the reals. Proof by induction seems to be a different matter since there we must exhaust the set X in a one-to-one manner. That cannot be done with the prescription in the article unless one in fact wellorders X - or X is wellordered as comes. I

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1.) Does one need ordinals at all to define transfinite recursion/induction in ZF? Well-ordering should be enough. Iv'e seen it done without ordinals. (Yes, in that horrible Halmos book since you ask;) Is that enough to make the article belong to the Set Theory category as well. Perhaps not, looks

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I am not trying to add to the article a claim that no one does it in practice, although it is the truth in nontrivial cases. You have demonstrated no such thing. What you have done is make an ad-hoc argument that, in retrospect, you can fit the form of the proof into your Procrustean mold. But

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This may seem silly to others, but upon reading, I noticed in the proof that the indexing set is integers. Perhaps this condition on α,β should be stated earlier. Based on the name of the page, I was complacent assuming that those were elements of an arbitrary indexing set, but if so, P(β+1) is

2076:

I object strongly to the assertion that 0 is "treated similarly when possible". No one strains to treat 0 the same as nonzero limit ordinals in actual practice, and there is no good reason to. I think you're trying to impose a normative aesthetic that you would like to see followed, but where

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I'm not familiar with the comparison lemma, if you can provide a proof for it I will gladly examine it. "I am not trying to add to the article a claim that no one does it in practice, although it is the truth in nontrivial cases." ok, so you admit your argument is based on a false claim (while

2832:

I am trying to think of the audience and I'm trying to be reasonable about what technical terms we should allow or not... I'm not sure the general public even thinks of functions as sets anyway... So from that perspective it seems as tho we're muddying up the article for the sake of pedantic

2828:

As a fresh eye on this, it is definitely awkward, to read this section. The term class function appears out of the blue without having it defined anywhere on the page. There is no link, no reference, nothing. What to do about this? Another question is, how important is this bit of technical

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mentions that this definition is sometimes used, and this is the more elegant definition. Anyways, I really don't care if you disagree; my point was only to mention that 0 can usually be treated exactly the same way as limit ordinals in proofs using transfinite induction (which incidentally

1884:

It is not silly, it makes proofs more concise and more elegant, and is almost always a reasonable thing to do. So much so that I can't even recall whether any proofs have necessitated treating it separately. Some readers may be interested in the benefits of this approach, even if you are

587:, any article in the former is implicitly also in the latter. An article should not be explicitly included simultaneously in a category and one of its ancestors. I have no idea what do you mean by "looks like double work unless ordinals are allowed in Set Theory Category". 2431:

If α is a limit ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X. This concept, a transfinite sequence or ordinal-indexed sequence, is a generalization of the concept of a sequence. An ordinary sequence corresponds to the case α =

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P(N) there must exist an M<N for which P does not hold. The same argument can be applied to M and so on; we have an infinite descending chain (it is infinite because it never reaches the minimal elements- P is true for them ). And that contradicts our well-founding.

2647:), so its meaning should be precised. Moreover, the type of G is wrong. Without fixing it V stands for any class. Thus the type V → V says that G may be even an empty function, what is surely wrong. Shouldn't V be, for example, the class of all sets? — K. 2014-02-25 832:"... More generally, one can define objects by transfinite recursion on any well-founded relation R. (R need not even be a set; it can be a proper class, provided it is a set-like relation; that is, for any x, the collection of all y such that y R x must be a set.) 733:

the name noetherian induction is mentioned for the general case of a well-founded relation. This seems to be the case when I search the net as well. Best thing might be to create a new page called Noetherian Induction and move the induction/recursion material from

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So it now reads "If P(a) follows from the truth of P(b) for all a < b, then it is simply a special case to say that P(0) is true, since it is vacuously true that P(b) holds for all b < 0."... Don't we mean to say "b < a" in the first part of the sentence?

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To chimpionspeak: Yes, the redundancy is clearly indicated by the use of "That is,". It is intentional. The concept is hard to render in English and we felt that expressing it in two different ways would give the reader a better chance of understanding it.

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transfinite induction, will generally require AC. In particular, one frequently wants to exhaust all the elements of some set, one at a time, in an inductive process. I think that's what Yohan was getting at with the "available for all sets" phrase.

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2) I have no idea what do you mean by "transfinite recursion/induction available for all sets" either. Transfinite induction has nothing to do with the axiom of choice, whether you do it for ordinals or for well-founded relations. For example,

2713:? If you do not write this, the standard convention says that V is quantified universally. In other words the standard semantics of "Given a class function G: V → V" is "Let V be any class and G by any class function of the type V → V". 728:

I did a quick "Google" on transfinite induction. It seems that either ordinals or well-ordering IS assumed. Of course it's technically correct that a well-founded relation is sufficient to do induction as Emil and you pointed out. In

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really going on is an uncountable sequence of discrete steps. There is nothing ill-specified about it; it's completely canonical. The details of how to create a formal ZFC argument are less canonical, and also less interesting. --

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But this is usually a fairly silly thing to do, which is why in practice no one ever does it, for nontrivial arguments. It's misleading to plop it in neutrally as though it were a reasonable thing to do, which it almost never is.

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Whether 0 is or is not a limit ordinal (like whether 0 is a natural number or whether finite sets are countable) is one of these points of terminology that people tend to get very exercised about, but on which there is no general

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Trovatore: This is a true statement, and it's something people might want to know. You also agreed with my claim that 0 can almost always be treated the same way as limit ordinals in proofs. So why did you remove from the page?

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If you think you can explain things better, then please be my guest and edit the article. But please make small localized changes and wait for a few days to see whether others make corrections or reversion before continuing.

2159:. The first part of that is true; zero is indeed sometimes considered a limit ordinal. The second part is just false. Yes, some arguments are presented this way, but it is not the usual treatment, even when "possible". -- 1009:

All of the three cases—the zero case, the successor case, and the limit case—are cases of proving that if something is true of every ordinal in some initial segment of the ordered class of ordinals, then it is true of the

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I defined V and the harpoon. Even though it's apparently becoming more common, I'd never seen the harpoon. And being explicitly told that G is a function on sets (by defining V) would've saved me some puzzlement.

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The question is, should it be? This is a point of usage rather than mathematics. My linguistic intuition here is that induction on a wellfounded relation other than a linear order is not standardly described as

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Now, can you unify the treatment of 0 with the treatment of the limit ordinals? Sure. Instead of "take intersections", you say "take the intersection of everything that has gone before, and intersect that with

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Now wait a minute. I may be halfway off the wall. Take the case of recursion. Why the heck shouldn't definition by transfinite recursion - as described in the article - yield a perfectly valid function f:O-:

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of transfinite induction generalizes to well-founded relations. The name does not, "transfinite induction" is indeed only used for ordinals. The general concept is known as "well-founded induction", see e.g.

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to the new page. At any rate, if one does not know (like I didn't) that induction/recursion works if one has a well-founded relation then that page isn't the most natural place to search for an answer.

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set theories (among others), a "class function" is just a function which is not necessarily a "set". Also, transfinite recursion doesn't quite work in ZF. You can correct the conclusion as follows:

228:, so that it says "If P(b) follows from the truth of P(a) for all a < b, then it is simply a special case to say that P(0) is true, since it is vacuously true that P(b) holds for all b < 0." 1446:

The difficulty of translating verbal statements of Transfinite Induction into symbolic logic indicates how readers can easily be confused by versions of Transfinite Induction that do not treat the

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It is simply not correct to say that it is "usually not necessary" to treat 0 differently, unless you mean in the trivial sense I alluded to earlier, which is already dealt with in the text. --

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is a redirect here. That's not necessarily bad. But if it's to stay that way, then the page really ought to say something about transfinite recursion (a method of definition, not of proof). --

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X where O is (a subset of) the ordinals - the point being that O has a well-founded relation (the wellorder in this case) and X is quite arbitrary. An example is the standard definition of the

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This is not the case. Suppose you have the sequence {1, 0, 0, 0, ...} and the proposition P(n) that the series sum after n is S(n) = 0. Assuming S(k)=0 gives S(k+1)=0, but S(0) = 1 so P(0) is

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I still don't follow you. Is your objection that what happens at each step is not actually "choosing", because it's pre-determined by the chosen wellorder? Would you be happier if the word

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The similarity lies in the fact that one is allowed, in both cases, to use the property for all previous ordinals in showing it for the current ordinal. So a common proof is often possible.

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I just noticed that some time ago a section header titled "Transfinite induction" was removed by a bot, effectively moving the contents of the first section into the lede section, see

2676:

That some people use other symbols for restriction of a function does not change the fact that the harpoon is used for that purpose in this article (and in many papers on set theory).

2044: 1029:, then the anonymous editor is right. (Of course, there may still be practical reasons why the zero case must be treated separately from other limit cases in particular proofs.) 3108: 2458:

As in most parts of mathematics, there are certain things you have to know before you start. If you do not know them, then you will not be able to understand the subject at all.

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P(n) as well an P(x) for all minimal elements (for ordinals it would be 1). Now suppose there existed an N for which P(N) does not hold. Since we have proved P(m<N)=: -->

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Consider whether this section should be moved to its own page, which can then be linked to in the "See also" section. It would be less jarring i think in that fashion.

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are the archetypical well-ordered sets. Every well-ordered set is order-isomorphic to an ordinal, and ordinals are well-ordered sets. So there should be no problem.

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to reserve the phrase transfinite induction for induction on well-ordered sets, in particular ordinals or some set indexed by ordinals. In fact, the present article

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If we take the view that it was an editing mistake, then what we have is someone who has noticed that we can formally state transfinite induction in the form, if

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The tone of that section of the article is not very good anyway, which may be why you see it as "misleading". I will replace it with part of your text. — Carl

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applies to all sets, and it does not need the axiom of choice, only the axiom of foundation. You cannot do induction (in the form discussed here) for anything

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wellorder. Are you claiming that the iteration as stated can proceed in two different ways, even given the same wellorder? I think that that is not true. --

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Umm, right, that's a point. Emil is certainly correct that you can do induction on any wellfounded relation, and you're correct that it isn't described as

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Okay. I thought that it didn't add anything because the first sentence of the second paragraph and the second sentence of the first paragraph are the same.

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This is not a paper by me. It is the result of contributions by dozens of people. If you see something wrong, YOU should fix it. Do not tell me to fix it.

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You argue that this process must terminate at some countable ordinal, and then recover the countable set as the union of the points deleted along the way.

1960:? The only reason is to make the base case look formally like the limit step. But the base case is different from the limit step, as anyone can see. -- 2683:

used for the class of all sets here. There are no types in ZFC set theory, so I do not understand what you mean when you say "the type of G is wrong".

3226: 189:) is true. Then it is not necessary to prove separately that P(the first ordinal) is true, because a special case of the induction step is that if P( 1888:"for that matter you can always treat all the ordinals together." Now you're just being pedantic, you understand perfectly well what I'm getting at. 3236: 462: 117: 3251: 370:

We should add a proof for the transfinite induction principle here. indirect proof: Suppose we have proved the crucial step P(m<n )=: -->

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for its domain. So every function is a class function, but some class functions are not functions because their domains are proper classes.

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It is not arbitrary; X is the ambient space and one is considering subobjects (subsets) of X. Intersection is a limit in the powerset of X.

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I see an anonymous editor saying the zero case is a special case of the limit case. If the limit case is regarded as the case where the

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That puts the spotlight a bit more on what the issue is. Also, the question of what the standard terminology is is temporarily avoided.

518: 2937:. I think the reason for changes like this is that the section title was equal to the page's title, leading to conflicting <h1: --> 1171:

up into further cases (a common one is to need to distinguish between limit ordinals of cofinality ω and all other limit ordinals). --

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pretty trivial. I'd like to see what you'd do to, say, the proof of the comparison lemma, where your base step starts with two mice.

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OK, so first of all, in the state in which the anon left the article, it claimed that the induction could be done at limit ordinals

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Given that the other article at least mentions the alternative definition, I will add the more neutral sentence into the article.

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Let's take a concrete example. Probably the simplest nontrivial proof that goes by transfinite induction is the proof of the

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Zero is sometimes considered a limit ordinal and is then treated in proofs in the same case as limit ordinals when possible.

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is assumed to have a domain which is a set. A class function is something which is defined like a function but may have a

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The set of new ordinals that are inferred to be true after transfinite induction: is that set the compliment of alpha?

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state this process in a formally correct way, but then it won't demonstrate what is meant to be demonstrated here.

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Thanks for the replies. They were accurate albiet Emils was a bit ironic in places. Both the present article and

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Yes, I have one in mind. It'll take a bit of energy to write it up though. I'll see if I can get motivated. --

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Isn't there some redundancy in the first two paragraphs of Transfinite Induction? The same thing being said ie.

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that this is even common practice, but it's ridiculous to exclude it simply because it's a minority viewpoint.

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or some such? As for proper classes, there are none in sight, so I don't see the point of mentioning them. --

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The lead section talks about well-ordered sets, the rest of the article about ordinals. That is not nice.

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OkOk redirects are fine with me, though not perfect. I STILL argue that the present article is misleading.

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to prove a property P for all ordinals α, one can assume that it is already known for all smaller β < α.

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Maybe the article should explicitly comment on the form seemingly suggested by the bare statement that if

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to the sequence of all X's below lambda when lambda is limit to get X_lambda, I can always say you apply

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are enough, as described in that article. You seem to misunderstand how Knowledge categories work. Since

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A limit ordinal is an ordinal such that it is the supremum of all ordinals preceding it. The article on

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I'm not arguing with the concept of empty intersection. The artificialness comes in intersecting with

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less than the smallest ordinal, then P(the smallest ordinal) is true. And it is vacuously true that P(

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the same way as a limit ordinal (in a suitable elegant way, by using empty intersections or whatnot).

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Unlike all limit ordinals, zero belongs to an open set, namely {0}, which contains no other ordinal.

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demonstrates why this definition is more elegant). You agreed with this. So why did you remove it?

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The first and last order signs should actually be angle brackets, meaning the sequence of elements

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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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I'm not happy with the footnotes. The second one is not entirely bad, but the first one assumes

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In the inductive proof, you iteratively apply the operation of removing all isolated points. So

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form β+1. I think this comment is misleading (at best), and should be removed. Any objections? --

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Sorry.Michael has correctly reverted my edit, because I was reading this explanation as P(0) is

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First cut in place. Could use an example or two, and possibly some a more rigorous treatment

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is in good taste to give a note about what this notation means. (cf. Theorem 1 at this page

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I can guess what's "transfinite sequence", but the link points to the following definition:

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What "limited experience of empty intersections"? I never objected to empty intersections.

302:. The article does say "ordinal" all over the place; do you think it should be emphasized? 2910:, but I don't think it actual uses transfinite induction. It also uses, of course, Morse- 809: 613: 1914:, and you want to show that it's the disjoint union of a perfect set and a countable set. 1582: 1449: 2436:

But this definition can't be used here (I think), since the index set is the whole Ord.

1419:{\displaystyle \forall \alpha (\forall \beta <\alpha \,P(\beta )\implies P(\alpha ))} 3198: 2903: 2895: 2814: 2773: 2684: 2606: 2415:, I'd guess it's some limit operation. However, V is not restricted to be ordered. (In 2385: 2311: 2296: 2259: 2223: 2188: 2160: 2116: 2078: 2048: 2001: 1975: 1961: 1942:

with all isolated points deleted, and so on. At limit ordinals you take intersections.

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Transfinite induction is an extension of mathematical induction to well-ordered sets...

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ordinals is not a set. That doesn't mean there's no such thing as a set of ordinals. --

445: 419: 405: 338: 3142: 1284:{\displaystyle \forall \alpha (\forall \beta <\alpha P(\beta ))\implies P(\alpha )} 1118:{\displaystyle \forall \alpha (\forall \beta <\alpha P(\beta ))\implies P(\alpha )} 629:, with the axiom of choice. In practice, though, the arguments for which you want to 3215: 2419:, V is the class of all sets, but then this article should mention it as well). From 1809: 936: 315: 303: 289: 278: 259: 2644: 909: 887: 865: 740: 673: 561: 537: 2813:

Thank you for preventing some other people from experiencing the same puzzlement.

2388:, since the latter has to do with group theory (though of course one never knows). 3206: 2996: 2973: 2958: 2921: 2845: 2822: 2807: 2781: 2750: 2692: 2662: 2614: 2452: 2358: 2334: 2319: 2304: 2282: 2267: 2251: 2231: 2210: 2196: 2182: 2168: 2146: 2124: 2101: 2086: 2071: 2056: 2045:

Knowledge talk:WikiProject Mathematics/Archive/2011/Sep#What is a limit ordinal?

2027: 2009: 1988: 1969: 1897: 1879: 1822: 1799: 1780: 1759: 1740: 1712: 1698: 1440: 1344: 1210: 1195: 1180: 1038: 998: 975: 941: 921: 895: 873: 812: 748: 707: 681: 643: 616: 569: 541: 490: 474: 448: 438: 422: 408: 346: 318: 306: 292: 113: 2732: 2416: 882: 806: 610: 90: 2376:

Too {{technical}}: missing prerequisites in section on transfinite recursion

1048:. That was clearly nonsense. It's possible that it was an editing mistake. 2643:

is not the sole standard denotation for the restriction of functions (cf.

2413:

https://en.wikipedia.org/Transfinite_sequence#Successor_and_limit_ordinals

2940: 932: 173:

A needless redundancy may be here. Suppose we show that for any ordinal

2440: 2412: 533: 214: 601:

than well-founded relations, as transfinite induction over a relation

2380:

This section is not self-contained. I'm marking it as {{technical}}.

2339:

Actually, a fine example of what I'd expect to be explained is here:

1979:

one when taken from the point of view of limits in category theory.)

1572:{\displaystyle \forall \beta ((\beta <\alpha )\implies P(\beta ))} 556:

like double work unless ordinals are allowed in Set Theory Category?

2492:

means that the domain of the function on the left of the harpoon is

2441:

https://en.wikipedia.org/Transfinite_sequence#Transfinite_induction

1475:

case separately. I take that in above discussion the expression

552:

I think I understand the section. However, I have two questions.

2983:

It says that if P(alpha) is true, then P(all ordinals) is true.

2894:

This article needs specific references. I don't have a copy of

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to the sequence of all X's below X_alpha to get X_alpha, where

15: 962:

But induction is valid only if the property P(α) is true for

2927:

Sections "Transfinite induction" and "Transfinite recursion"

2341:

http://en.wikipedia.org/Ordinal_number#Transfinite_induction

2599:

In this case the function may actually be a class function.

2077:

actual workers in the field simply don't agree with you. --

1834:

or predicate. Instead of saying I start with X_0, apply

1186:

I think maybe some serious concrete examples would help.

3139:

is also true. Then transfinite induction tells us that

2421:

http://www.proofwiki.org/Transfinite_Recursion/Theorem_1

1350:

I think that the better placement of the parentheses is

981:

If there are no smaller elements, then they all satisfy

243:

The redundancy you're seeing is that the induction step

2945: 2933: 1608:

case explcitly intends the "induction hypothesis" be ::

3168: 3145: 3116: 3090: 3061: 3041: 3008: 2898:'s book. I do have a copy of my late mother's book, 2645:

http://en.wikipedia.org/Restriction_%28mathematics%29

2629: 2506: 2478: 2397: 1614: 1585: 1522: 1481: 1452: 1358: 1303: 1225: 1137: 1059: 277:< 0. IOW I saw it saying P(0) was vacuously true. 2871:(Traditional) for any ordinal λ there is a function 418:(not instead of) the informal descriptions given. -- 112:, a collaborative effort to improve the coverage of 2963:The final inference: that P(all ordinals) is true. 3189: 3151: 3131: 3102: 3076: 3047: 3023: 2635: 2587: 2484: 2403: 1679: 1600: 1571: 1509:{\displaystyle \forall \beta <\alpha P(\beta )} 1508: 1467: 1418: 1324: 1283: 1158: 1117: 2496:to the class to the right of the harpoon. That is 575:1) One needs neither ordinals nor well-ordering; 3242:Knowledge level-5 vital articles in Mathematics 560:one must have those magic words "Well-order!"? 966:element: α=0. I quess it should be mentioned. 2173:I've changed the wording to be more neutral. 548:Regarding Relationship to the axiom of choice 8: 2577: 2556: 2544: 2539: 2534: 2522: 2519: 2384:What's a class function? I don't guess it's 1956:But this is artificial. Why intersect with 1431:conjunction, disjunction, and equivalence). 1335:and the form of proofs in actual practice. 989:(0) from nothing and the induction begins. 2733:http://planetmath.org/transfiniterecursion 2711:"V" is used for the class of all sets here 2417:http://planetmath.org/transfiniterecursion 58: 3190:{\displaystyle \forall \alpha P(\alpha )} 3167: 3144: 3115: 3089: 3060: 3040: 3007: 2628: 2505: 2477: 2396: 1838:to X_alpha to get X_{alpha+1}, and apply 1613: 1584: 1521: 1480: 1451: 1357: 1325:{\displaystyle \forall \alpha P(\alpha )} 1302: 1224: 1159:{\displaystyle \forall \alpha P(\alpha )} 1136: 1058: 828:How about this for a start: The passage 625:Transfinite induction has nothing to do, 2882:(Advanced) There is a "class function" 1862:of a sequence without a last element is 1663: 1658: 1641: 1636: 1552: 1547: 1399: 1394: 1267: 1262: 1101: 1096: 797:. For good measure, I created redirects 3232:Knowledge vital articles in Mathematics 2580: 2542: 2537: 1380: 60: 19: 3002:You are misinterpreting the line "Let 255:. You still have to prove it to be so. 3247:C-Class vital articles in Mathematics 2201:Thanks, I'm glad we worked this out. 1854:of a sequence with a last element is 7: 258:As such I've deleted the paragraph. 106:This article is within the scope of 1516:is regarded as being equivalent to 855:Relationship to the axiom of choice 841:of choice to wellorder a set. ..." 836:Relationship to the axiom of choice 49:It is of interest to the following 3257:High-priority mathematics articles 3169: 1615: 1523: 1482: 1368: 1359: 1304: 1235: 1226: 1138: 1069: 1060: 247:P(0) is true as well as all other 14: 3103:{\displaystyle \beta <\alpha } 2439:(BTW, this seems to overlap with 668:in the very first scentence that 126:Knowledge:WikiProject Mathematics 3227:Knowledge level-5 vital articles 2900:Set Theory for the Mathematician 2879:such that for all α < λ, .... 2636:{\displaystyle \upharpoonright } 2623:This seems not true. The symbol 2485:{\displaystyle \upharpoonright } 2404:{\displaystyle \upharpoonright } 205:less than the smallest ordinal! 129:Template:WikiProject Mathematics 93: 83: 62: 29: 20: 2914:set theory, rather than ZF.) — 2237:Vitali set example problematic? 220:I've now reversed the order of 146:This article has been rated as 3237:C-Class level-5 vital articles 3184: 3178: 3126: 3120: 3071: 3065: 3018: 3012: 2630: 2510: 2479: 2398: 1850:of the empty sequence is X_0, 1674: 1668: 1660: 1655: 1652: 1646: 1638: 1624: 1621: 1595: 1589: 1566: 1563: 1557: 1549: 1544: 1532: 1529: 1503: 1497: 1462: 1456: 1413: 1410: 1404: 1396: 1391: 1385: 1365: 1319: 1313: 1278: 1272: 1264: 1259: 1256: 1250: 1232: 1153: 1147: 1112: 1106: 1098: 1093: 1090: 1084: 1066: 459:Ordinals form class, not set. 1: 2782:10:50, 27 February 2014 (UTC) 2751:09:45, 27 February 2014 (UTC) 2693:08:25, 26 February 2014 (UTC) 2663:13:14, 25 February 2014 (UTC) 2472:In this context, the harpoon 120:and see a list of open tasks. 3252:C-Class mathematics articles 2615:15:19, 16 October 2013 (UTC) 2453:13:20, 16 October 2013 (UTC) 2359:21:53, 14 January 2012 (UTC) 2335:16:10, 12 January 2012 (UTC) 2320:06:30, 12 January 2012 (UTC) 2305:05:11, 12 January 2012 (UTC) 2283:05:03, 12 January 2012 (UTC) 2268:23:28, 11 January 2012 (UTC) 2252:22:50, 11 January 2012 (UTC) 2232:14:18, 8 December 2011 (UTC) 2211:05:10, 8 December 2011 (UTC) 2197:05:06, 8 December 2011 (UTC) 2183:04:57, 8 December 2011 (UTC) 2169:04:50, 8 December 2011 (UTC) 2147:04:39, 8 December 2011 (UTC) 2125:04:21, 8 December 2011 (UTC) 2102:04:12, 8 December 2011 (UTC) 2087:03:06, 8 December 2011 (UTC) 2072:02:13, 8 December 2011 (UTC) 2057:02:01, 8 December 2011 (UTC) 2028:01:46, 8 December 2011 (UTC) 2010:01:31, 8 December 2011 (UTC) 1989:01:27, 8 December 2011 (UTC) 1970:01:17, 8 December 2011 (UTC) 1898:01:10, 8 December 2011 (UTC) 1880:00:58, 8 December 2011 (UTC) 1823:00:58, 8 December 2011 (UTC) 1800:00:48, 8 December 2011 (UTC) 1781:00:39, 8 December 2011 (UTC) 1760:10:23, 23 October 2011 (UTC) 1741:04:55, 23 October 2011 (UTC) 1703:(I edited my above comment) 1441:21:50, 16 January 2011 (UTC) 1345:19:24, 16 January 2011 (UTC) 1211:05:12, 16 January 2011 (UTC) 1196:05:08, 16 January 2011 (UTC) 1181:20:45, 15 January 2011 (UTC) 1039:19:05, 15 January 2011 (UTC) 781:. Note that we already have 429:Successor Case vs Limit Case 3207:19:18, 24 August 2020 (UTC) 3159:is true for all ordinals.". 2997:15:25, 24 August 2020 (UTC) 2974:15:25, 24 August 2020 (UTC) 2959:10:37, 8 October 2015 (UTC) 2922:07:50, 23 August 2015 (UTC) 2423:, I see it's a restriction. 1027:smallest (weak) upper bound 1023:smallest strict upper bound 1012:smallest strict upper bound 999:07:20, 10 August 2010 (UTC) 976:06:58, 10 August 2010 (UTC) 201:) is true for all ordinals 193:) is true for all ordinals 3273: 3132:{\displaystyle P(\alpha )} 3024:{\displaystyle P(\alpha )} 1713:17:50, 26 April 2013 (UTC) 1699:17:30, 26 April 2013 (UTC) 948:How about minimal element? 542:12:25, 18 April 2008 (UTC) 520:11:52, 18 April 2008 (UTC) 491:18:33, 15 April 2007 (UTC) 475:11:05, 14 April 2007 (UTC) 347:08:34, 10 March 2010 (UTC) 3077:{\displaystyle P(\beta )} 3035:defined for all ordinals 2823:07:53, 5 March 2014 (UTC) 2808:06:00, 5 March 2014 (UTC) 2187:OK, that's acceptable. -- 1858:of the last element, and 1721:well-ordered vs. ordinals 942:04:01, 22 July 2009 (UTC) 922:21:01, 21 July 2009 (UTC) 896:15:31, 19 July 2009 (UTC) 874:11:05, 19 July 2009 (UTC) 813:10:27, 20 July 2009 (UTC) 773:What I meant is that the 749:08:06, 19 July 2009 (UTC) 708:07:39, 19 July 2009 (UTC) 682:06:18, 19 July 2009 (UTC) 644:21:34, 17 July 2009 (UTC) 617:10:28, 17 July 2009 (UTC) 570:20:24, 16 July 2009 (UTC) 449:07:47, 5 April 2007 (UTC) 439:16:39, 4 April 2007 (UTC) 423:07:03, 16 July 2005 (UTC) 409:01:51, 16 July 2005 (UTC) 145: 78: 57: 3055:. Suppose that whenever 2155:The text you added was, 1904:Cantor–Bendixson theorem 985:vacuously. Thus you get 581:Category:Ordinal numbers 262:20:11, 3 Nov 2004 (UTC) 209:00:41 Mar 3, 2003 (UTC) 152:project's priority scale 3048:{\displaystyle \alpha } 2846:07:18, 6 May 2014 (UTC) 532:indexed by α < c. — 319:05:26, 7 May 2006 (UTC) 307:00:42, 3 May 2006 (UTC) 293:00:37, 3 May 2006 (UTC) 281:10:03, 4 Nov 2004 (UTC) 232:21:34 8 Jun 2003 (UTC) 217:20:45 8 Jun 2003 (UTC) 109:WikiProject Mathematics 3222:C-Class vital articles 3191: 3153: 3133: 3104: 3078: 3049: 3025: 2709:Where is written that 2637: 2589: 2486: 2405: 1681: 1602: 1573: 1510: 1469: 1420: 1326: 1285: 1160: 1119: 960: 799:well-founded recursion 783:well-founded induction 577:well-founded relations 3192: 3154: 3134: 3105: 3079: 3050: 3026: 2638: 2590: 2487: 2406: 1682: 1603: 1574: 1511: 1470: 1421: 1327: 1286: 1161: 1120: 951: 795:well-founded relation 787:well-founded relation 736:well-founded relation 731:well-founded relation 696:transfinite induction 688:transfinite induction 658:well-founded relation 465:comment was added by 402:transfinite recursion 396:Transfinite recursion 36:level-5 vital article 3166: 3143: 3114: 3088: 3059: 3039: 3006: 2627: 2504: 2476: 2395: 1766:0 is a limit ordinal 1612: 1601:{\displaystyle P(0)} 1583: 1520: 1479: 1468:{\displaystyle P(0)} 1450: 1356: 1301: 1223: 1135: 1057: 803:Noetherian recursion 791:Noetherian induction 583:is a subcategory of 359:00:23, 12 March 2010 332:02:26, 9 March 2010 132:mathematics articles 1025:is the same as the 609:is well-founded. — 585:Category:Set theory 455:Sets of ordinals??? 177:, for all ordinals 3187: 3162:The conclusion is 3149: 3129: 3100: 3074: 3045: 3021: 2633: 2585: 2581: 2543: 2538: 2482: 2401: 1677: 1664: 1659: 1642: 1637: 1598: 1569: 1553: 1548: 1506: 1465: 1416: 1400: 1395: 1381: 1322: 1281: 1268: 1263: 1156: 1115: 1102: 1097: 793:also redirects to 300:Ordinal arithmetic 101:Mathematics portal 45:content assessment 3152:{\displaystyle P} 3084:is true for all 2741:comment added by 2735:) K. 2014-02-27 2653:comment added by 2349:comment added by 2291:were replaced by 1909: 1866:of that sequence. 1790:a limit ordinal. 1731:comment added by 940: 925: 908:comment added by 785:as a redirect to 627:formally speaking 478: 392: 379:comment added by 166: 165: 162: 161: 158: 157: 3264: 3196: 3194: 3193: 3188: 3158: 3156: 3155: 3150: 3138: 3136: 3135: 3130: 3109: 3107: 3106: 3101: 3083: 3081: 3080: 3075: 3054: 3052: 3051: 3046: 3030: 3028: 3027: 3022: 2991: 2979:question about P 2968: 2951:Tobias Bergemann 2948: 2936: 2908:General Topology 2753: 2665: 2642: 2640: 2639: 2634: 2594: 2592: 2591: 2586: 2491: 2489: 2488: 2483: 2410: 2408: 2407: 2402: 2361: 1907: 1743: 1686: 1684: 1683: 1678: 1607: 1605: 1604: 1599: 1578: 1576: 1575: 1570: 1515: 1513: 1512: 1507: 1474: 1472: 1471: 1466: 1425: 1423: 1422: 1417: 1331: 1329: 1328: 1323: 1290: 1288: 1287: 1282: 1165: 1163: 1162: 1157: 1124: 1122: 1121: 1116: 930: 924: 902: 690:in this article. 460: 374: 134: 133: 130: 127: 124: 103: 98: 97: 87: 80: 79: 74: 66: 59: 42: 33: 32: 25: 24: 16: 3272: 3271: 3267: 3266: 3265: 3263: 3262: 3261: 3212: 3211: 3164: 3163: 3141: 3140: 3112: 3111: 3086: 3085: 3057: 3056: 3037: 3036: 3004: 3003: 2989: 2981: 2966: 2944: 2938:and <h2: --> 2932: 2929: 2853: 2851:Problems (2015) 2736: 2648: 2625: 2624: 2502: 2501: 2474: 2473: 2393: 2392: 2378: 2344: 2239: 1940: 1933: 1922: 1768: 1726: 1723: 1610: 1609: 1581: 1580: 1518: 1517: 1477: 1476: 1448: 1447: 1354: 1353: 1299: 1298: 1221: 1220: 1133: 1132: 1055: 1054: 1007: 950: 903: 886:need to sleep. 550: 531: 507: 503: 461:—The preceding 457: 431: 398: 368: 298:Not quite; see 269:by assuming P( 171: 131: 128: 125: 122: 121: 99: 92: 72: 43:on Knowledge's 40: 30: 12: 11: 5: 3270: 3268: 3260: 3259: 3254: 3249: 3244: 3239: 3234: 3229: 3224: 3214: 3213: 3210: 3209: 3186: 3183: 3180: 3177: 3174: 3171: 3160: 3148: 3128: 3125: 3122: 3119: 3099: 3096: 3093: 3073: 3070: 3067: 3064: 3044: 3020: 3017: 3014: 3011: 2980: 2977: 2928: 2925: 2892: 2891: 2890:such that .... 2880: 2852: 2849: 2826: 2825: 2793: 2792: 2791: 2790: 2789: 2788: 2787: 2786: 2785: 2784: 2761: 2760: 2759: 2758: 2757: 2756: 2755: 2754: 2743:79.186.221.235 2721: 2720: 2719: 2718: 2717: 2716: 2715: 2714: 2700: 2699: 2698: 2697: 2696: 2695: 2677: 2669: 2668: 2667: 2666: 2632: 2618: 2617: 2602: 2601: 2600: 2596: 2595: 2584: 2579: 2576: 2573: 2570: 2567: 2564: 2561: 2558: 2555: 2552: 2549: 2546: 2541: 2536: 2533: 2530: 2527: 2524: 2521: 2518: 2515: 2512: 2509: 2498: 2497: 2481: 2470: 2434: 2433: 2428: 2427: 2424: 2400: 2389: 2386:class function 2377: 2374: 2373: 2372: 2371: 2370: 2369: 2368: 2367: 2366: 2365: 2364: 2363: 2362: 2351:217.84.234.165 2327:217.84.242.179 2238: 2235: 2220: 2219: 2218: 2217: 2216: 2215: 2214: 2213: 2203:Thehotelambush 2175:Thehotelambush 2153: 2139:Thehotelambush 2134: 2133: 2132: 2131: 2130: 2129: 2128: 2127: 2112: 2107:this proof is 2094:Thehotelambush 2064:Thehotelambush 2041: 2040: 2039: 2038: 2037: 2036: 2035: 2034: 2033: 2032: 2031: 2030: 2020:Thehotelambush 2015: 1998: 1981:Thehotelambush 1954: 1946: 1943: 1938: 1931: 1920: 1915: 1906:. You have a 1890:Thehotelambush 1886: 1867: 1831: 1827: 1826: 1825: 1815:Thehotelambush 1810:limit ordinals 1803: 1802: 1773:Thehotelambush 1767: 1764: 1763: 1762: 1722: 1719: 1717: 1691:216.223.227.93 1676: 1673: 1670: 1667: 1662: 1657: 1654: 1651: 1648: 1645: 1640: 1635: 1632: 1629: 1626: 1623: 1620: 1617: 1597: 1594: 1591: 1588: 1568: 1565: 1562: 1559: 1556: 1551: 1546: 1543: 1540: 1537: 1534: 1531: 1528: 1525: 1505: 1502: 1499: 1496: 1493: 1490: 1487: 1484: 1464: 1461: 1458: 1455: 1444: 1443: 1428: 1427: 1426: 1415: 1412: 1409: 1406: 1403: 1398: 1393: 1390: 1387: 1384: 1379: 1376: 1373: 1370: 1367: 1364: 1361: 1333: 1332: 1321: 1318: 1315: 1312: 1309: 1306: 1292: 1291: 1280: 1277: 1274: 1271: 1266: 1261: 1258: 1255: 1252: 1249: 1246: 1243: 1240: 1237: 1234: 1231: 1228: 1214: 1213: 1184: 1183: 1168: 1167: 1166: 1155: 1152: 1149: 1146: 1143: 1140: 1127: 1126: 1125: 1114: 1111: 1108: 1105: 1100: 1095: 1092: 1089: 1086: 1083: 1080: 1077: 1074: 1071: 1068: 1065: 1062: 1049: 1019: 1018: 1006: 1003: 1002: 1001: 953:Let P(α) be a 949: 946: 945: 944: 899: 898: 863: 850: 847:is changed to 844: 831: 826: 825: 824: 823: 822: 821: 820: 819: 818: 817: 816: 815: 760: 759: 758: 757: 756: 755: 754: 753: 752: 751: 717: 716: 715: 714: 713: 712: 711: 710: 691: 649: 648: 647: 646: 620: 619: 589: 588: 549: 546: 545: 544: 529: 511:68.161.161.206 506: 504:| α<c : --> 501: 495: 494: 493: 456: 453: 452: 451: 430: 427: 426: 425: 416:in addition to 397: 394: 381:217.232.29.149 367: 364: 363: 362: 361: 360: 350: 349: 326: 325: 324: 323: 322: 321: 314:Well, fit it! 285: 284: 283: 282: 256: 241: 170: 167: 164: 163: 160: 159: 156: 155: 144: 138: 137: 135: 118:the discussion 105: 104: 88: 76: 75: 67: 55: 54: 48: 26: 13: 10: 9: 6: 4: 3: 2: 3269: 3258: 3255: 3253: 3250: 3248: 3245: 3243: 3240: 3238: 3235: 3233: 3230: 3228: 3225: 3223: 3220: 3219: 3217: 3208: 3204: 3200: 3181: 3175: 3172: 3161: 3146: 3123: 3117: 3097: 3094: 3091: 3068: 3062: 3042: 3034: 3015: 3009: 3001: 3000: 2999: 2998: 2995: 2994: 2984: 2978: 2976: 2975: 2972: 2971: 2961: 2960: 2956: 2952: 2947: 2942: 2935: 2926: 2924: 2923: 2920: 2917: 2913: 2909: 2905: 2901: 2897: 2889: 2885: 2881: 2878: 2874: 2870: 2869: 2868: 2866: 2862: 2858: 2857:ZF set theory 2850: 2848: 2847: 2843: 2839: 2838:24.110.50.184 2834: 2830: 2824: 2820: 2816: 2812: 2811: 2810: 2809: 2805: 2801: 2800:173.25.54.191 2797: 2783: 2779: 2775: 2771: 2770: 2769: 2768: 2767: 2766: 2765: 2764: 2763: 2762: 2752: 2748: 2744: 2740: 2734: 2729: 2728: 2727: 2726: 2725: 2724: 2723: 2722: 2712: 2708: 2707: 2706: 2705: 2704: 2703: 2702: 2701: 2694: 2690: 2686: 2682: 2678: 2675: 2674: 2673: 2672: 2671: 2670: 2664: 2660: 2656: 2655:79.184.244.27 2652: 2646: 2622: 2621: 2620: 2619: 2616: 2612: 2608: 2603: 2598: 2597: 2582: 2574: 2571: 2568: 2565: 2562: 2559: 2553: 2550: 2547: 2531: 2528: 2525: 2516: 2513: 2507: 2500: 2499: 2495: 2471: 2468: 2464: 2460: 2459: 2457: 2456: 2455: 2454: 2450: 2446: 2442: 2437: 2430: 2429: 2425: 2422: 2418: 2414: 2390: 2387: 2383: 2382: 2381: 2375: 2360: 2356: 2352: 2348: 2342: 2338: 2337: 2336: 2332: 2328: 2323: 2322: 2321: 2317: 2313: 2308: 2307: 2306: 2302: 2298: 2294: 2290: 2286: 2285: 2284: 2280: 2276: 2275:217.84.235.60 2271: 2270: 2269: 2265: 2261: 2256: 2255: 2254: 2253: 2249: 2245: 2244:217.84.235.60 2236: 2234: 2233: 2229: 2225: 2212: 2208: 2204: 2200: 2199: 2198: 2194: 2190: 2186: 2185: 2184: 2180: 2176: 2172: 2171: 2170: 2166: 2162: 2158: 2154: 2151: 2150: 2149: 2148: 2144: 2140: 2126: 2122: 2118: 2113: 2110: 2105: 2104: 2103: 2099: 2095: 2090: 2089: 2088: 2084: 2080: 2075: 2074: 2073: 2069: 2065: 2061: 2060: 2059: 2058: 2054: 2050: 2046: 2029: 2025: 2021: 2016: 2013: 2012: 2011: 2007: 2003: 1999: 1996: 1992: 1991: 1990: 1986: 1982: 1977: 1973: 1972: 1971: 1967: 1963: 1959: 1955: 1952: 1947: 1944: 1941: 1934: 1927: 1923: 1916: 1913: 1910:set of reals 1905: 1901: 1900: 1899: 1895: 1891: 1887: 1883: 1882: 1881: 1877: 1873: 1868: 1865: 1861: 1857: 1853: 1849: 1845: 1841: 1837: 1832: 1828: 1824: 1820: 1816: 1811: 1807: 1806: 1805: 1804: 1801: 1797: 1793: 1789: 1785: 1784: 1783: 1782: 1778: 1774: 1765: 1761: 1757: 1753: 1749: 1746: 1745: 1744: 1742: 1738: 1734: 1730: 1720: 1718: 1715: 1714: 1710: 1706: 1701: 1700: 1696: 1692: 1688: 1671: 1665: 1649: 1643: 1633: 1630: 1627: 1618: 1592: 1586: 1560: 1554: 1541: 1538: 1535: 1526: 1500: 1494: 1491: 1488: 1485: 1459: 1453: 1442: 1438: 1434: 1429: 1407: 1401: 1388: 1382: 1377: 1374: 1371: 1362: 1352: 1351: 1349: 1348: 1347: 1346: 1342: 1338: 1337:Michael Hardy 1316: 1310: 1307: 1297: 1296: 1295: 1275: 1269: 1253: 1247: 1244: 1241: 1238: 1229: 1219: 1218: 1217: 1212: 1208: 1204: 1200: 1199: 1198: 1197: 1193: 1189: 1188:Michael Hardy 1182: 1178: 1174: 1169: 1150: 1144: 1141: 1131: 1130: 1128: 1109: 1103: 1087: 1081: 1078: 1075: 1072: 1063: 1053: 1052: 1050: 1047: 1043: 1042: 1041: 1040: 1036: 1032: 1031:Michael Hardy 1028: 1024: 1017: 1016: 1015: 1013: 1004: 1000: 996: 992: 988: 984: 980: 979: 978: 977: 973: 969: 965: 959: 956: 947: 943: 938: 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