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I don't entirely agree with the following passage:
Generally speaking, mathematicians who favor a "rich" and "large" universe of sets are against CH, while those favoring a "neat" and "controllable" universe favor CH.
See the Maddy reference I added, page 500, the sections entitled Not-CH is restrictive (in favor) and Modern forcing (in favor).
I'll try to come up with some wording that's not too awkward, illustrating the historical view reported in the existing Wiki article, while also pointing out that between models having all the same reals, it's the ones with more sets of reals that are more likely to satisfy CH. -- trovatore
My attempt is now in place Trovatore 05:04, 27 Jun 2005 (UTC)
User Martindowd has been arguing that the truth of CH, indeed V=L, should be taken more seriously. He suggests adding the following at the end of the second paragraph of the "Arguments for and against CH" section of the Continuum hypothesis page. If you have any opinion on this, please let me know.
Recently, arguments have appeared that this view should be re-evaluated, and that there are arguments in favor of both (Dowd 2011).
Dowd, Martin (2011) "Some New Axioms for Set Theory", http://www.ijpam.eu/contents/2011-66-2/1/1.pdf Martindowd ( talk) 15:04, 21 September 2011 (UTC)
I would add to Trovatore's point that it is questionable (as stated on the page) that Cantor himself believed CH. Did he not alternate back and forth during his manic (and depressive) phases, sometimes believing CH, sometimes believing ~CH? See the BBC TV documentary "Dangerous Knowledge" willbown. 16 June 2008. —Preceding comment was added at 22:37, 16 June 2008 (UTC)
I've removed this section:
It's a good thing to add some intuition, but I don't know that this passage helped much. CH isn't about whether such a set S can be found in any ordinary sense, but just about whether one exists. -- Trovatore 21:12, 22 October 2005 (UTC)
Let me add this thought:
Since CH cannot be disproven, a set that denies it cannot be found. This doesn't mean it does not exist, but we're close :-) Honnza ( talk) 06:01, 17 May 2008 (UTC)
The undecidability of CH begs the question: If there is a cardinality between and , then what sets might there be that have this cardinality?
In other words, is there any known set that is larger than , but such that you need to set the truth value of CH to determine whether or not it's equivalent to continuum?
Moreover, is there any evidence pointing to the existence or non-existence of such a set, or whether it would be possible to find it if it does exist? -- Smjg ( talk) 16:35, 28 February 2008 (UTC)
But in the context of ZF, j:V into V refutes choice (according to the article on Reinhardt cardinals anyway), and GCH implies it (according to this article). So then j:V into V would refute GCH. Right? -- Unzerlegbarkeit ( talk) 15:58, 27 May 2008 (UTC)
The form makes the continuum hypothesis meaningful even if the continuum isn't well-ordered. The article states:
Do we have any examples which would make sense without AC, and would they depend on the form of CH? Also, on the other side of the coin, my understanding is that there could be no purely arithmetic consequences, roughly because ZF + V=L proves CH anyway (and even GCH), and relativising everything to L makes no difference to (first-order) arithmetic. If correct, I believe this is worth stating. Also, the article states:
The "assuming the axiom of choice" bit should really attach to the "in turn equivalent to", because even from the axioms ZF without AC, aleph_1 exists and is an immediate successor of aleph_0, right? -- Unzerlegbarkeit ( talk) 02:39, 1 June 2008 (UTC)
I notice there aren't any references to Cantor's original writings. I'll try to dig some up, but has anyone seen any English translated letters, etc? Libertyblues ( talk) 00:03, 3 August 2008 (UTC)
This statement is revealing a transfinite-platonic bias. without a formalization like ZFC, it is not clear that it is possible to give a meaning to the statement that the continuum even has an ordinal size. Before Cantor insisted it was true, an infinite collection like the set of real numbers was not considered to have a definite size, let alone a definite ordinal size, and indeed, after Cohen, it is again a widespread point of view that the continuum does not have a definite size as an ordinal.
The idea that the set of all real numbers has definite properties in a platonic realm can be classified as a type of "transfinite platonism", which is just what platonism usually means nowadays. But transfinite Platonism can be logically separated from "computational platonism" (I made that up, but it needs a name)--- the position that all computer programs either halt or do not halt. The second position is what people mean by platonism in practice--- that the results of computations with symbols have a meaning, and questions about their outcome have a truth value. it was the position of Paul Cohen, shared by most mathematicians, that all questions about the integers/computer-programs are decided by a strong enough axiom system, which is a way of saying that they have a truth value in an axiom-system independent way.
But you can believe this, and still be a formalist regarding transfinite set theory. The position might be called "formalist", but "formalist" conflates two notions: "formalist regarding uncountable ordinals and sets the size of the continuum" and "formalist regarding countable infinity". A "formalist regarding countable infinity" would, for example, consider a nonstandard models of Peano arithmetic to be just as "true" a model of the integers as the standard model. Such a formalist would believe that some statements about diophantine equations like Fermat's Last Theorem, are undecidable in an absolute sense. This is the type of straw-man formalist that people argue against.
A "formalist regarding uncountable ordinals" on the other hand, would say that the truth value of the 3N+1 conjecture is well defined, but statements like the continuum hypothesis have no truth value except relative to an axiom system. This is the type of formalism which the forcing models foist upon you. If you accept this philosophy, which many people do, then you would not regard the continuum hypothesis as independent of ZFC.
This is the philosophical position which Cohen alludes to in his book, hopefully stated clearly enough there so that this exposition will be recognized as a clarification. I have been trying to figure out how to say it clearly for a while. Likebox ( talk) 17:22, 24 August 2008 (UTC)
(deindent) I am pretty ignorant about determinacy--- I only have only the most superficial heresay knowledge, and no clear understanding. But I have a somewhat negative opinion anyway, probably unjustified.
When I think about the continuum hypothesis, it is linked in my mind to the notion of measure and probability--- the reason that many people have intuitions about the cardinality of the real numbers is that it is consistent to talk about randomly choosing real numbers in a way that is different from randomly choosing integers or elements of a well ordered set. You can't pick an integer uniformly at random--- the concept doesn't make sense. You need a probability distribution, and certain integers are more likely than others. The same goes for any countable ordinal--- you need to weight different positions with different probability.
But for the interval [0,1], the notion of picking a number at random seems to be perfectly well defined, because you can roll dice to find each digit in sucession. This converges to a real number, and any property of that number that has probability zero is false. If a random number can be thought of as an element of the mathematical universe, with the property of belonging or not belonging to any previously specified set, it means that this previously specified set has a well defined Lebesgue measure. So not only is the continuum hypothesis false with random reals, it is meaningless, because the real numbers can't be well ordered. Any ordinal, no matter how large, can be imbedded in the reals by inductively mapping each successive element into [0,1] at random, and the probability of picking the same real twice is always zero (the last statement is only self consistently true, it's true in a countable model).
Making this precise is the job of forcing, which, given the political situation in mathematics, phrases everything in terms of the picking process, although at the end it talks about "random reals". But a random real number is an obviously consistent idea, even though it is incompatible with choice.
So when I look at post-forcing axioms like determinacy, I am always looking for the probability interpretation, and in this case I couldn't see it. The determinacy axiom, as I heard it, was exciting because it had a completely different character--- it asserted something about infinite sets that somebody thought was intuitively consistent (I don't know why, but they turned out to be right, so they must have had a good idea). But the justification for this for those with a formal view of higher infinity is that a large cardinal axiom proved that it is equiconsistent with set theory. So determinacy, as far as theorems about integers are concerned, is a moderate large cardinal axiom, and is not as interesting as a completely new axiom (although Cohen's "article of faith" says there are not going to be any consistent new axioms that are not equiconsistent by virtue of a large enough cardinal). So a superficial glance at Woodin's article was disappointing for me, because it didn't give a perspective which was probabalistic. But that's a very self-centered point of view, so I'll give it another shot. Likebox ( talk) 17:51, 26 August 2008 (UTC)
Edited out a minor error: ZFC stands for "Zermelo-Frankel with the axiom of choice", whereas ZF is simply "Zermelo-frankel". Would be nice to see some superscript references in this article, I'm afraid I don't know how to do that yet. 83.71.3.220 ( talk) 00:41, 1 October 2008 (UTC)
Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers. His proofs, however, give no indication of the extent to which the cardinality of the natural numbers is less than that of the real numbers.
Shouldn't natural numbers be rational numbers? Since the natural numbers is a subset of the integers, it's quite obvious that the cardinality of the natural numbers isn't greater than the integers. —Preceding unsigned comment added by 131.215.42.208 ( talk) 23:21, 20 January 2009 (UTC)
I think the article should include a section saying which alephs could be (and ) in the case CH is undecided. The article about the cardinality of the continuum has some indications about this (for example, it could be that , but not ), but I think this article here is the proper place to a full analysis. And I have never seen an upper limit to ; does it make sense if could be some monstrosity like ? Albmont ( talk) 19:11, 3 April 2009 (UTC)
You might want to look at Easton's theorem which deals with a more general question. JRSpriggs ( talk) 06:37, 4 April 2009 (UTC)
My question is quite simple: given ZFC, can we name a few values of α such that ? There's no need to expand the metalanguage to discuss models, etc - let's get to the basic, simple theory and see what we can get. For example, the first value that we can show is , the second is , etc. Of course, given that it is impossible to prove within ZFC that is and others, the next question would be quite obvious, but let's not do that leap. Not yet. Albmont ( talk) 03:13, 5 April 2009 (UTC)
Albmont: the informal "short answer" is that any uncountable κ with uncountable cofinality is a possible candidate for the cardinality of the continuum. But this statement is not entirely precise because "possible" here means "can be achieved by a forcing extension" (because the informal statement is really just a much easier special case of Easton's theorem), and there are some technical issues related to forcing that have to be mastered in order to fully understand what's going on. These issues are what Trovatore is alluding to. — Carl ( CBM · talk) 04:03, 6 April 2009 (UTC)
I don't believe the "object language"/"metalanguage" distinction is really the problem here. In these sorts of situations we work model theoretically: start with a model W and work with a language that has a constant for every element of W. So every ordinal in W is definable in this language. One thing that does matter is to distinguish between things that look like names but are actually meant to be formulas, and things that look like names that are actually meant to be names. This is typically achieved by writing dots over the things that are meant to be names; we ought to write .
So a slightly better way of stating the theorem at hand goes like this.
I don't see that it matters whether any formula φ defining α is absolute or not, because we will not actually use φ to identify a cardinal in the forcing extension, we use the forcing name. The absoluteness issue is only relevant to seeing why Trovatore's example does not give a contradiction to Easton's theorem. Easton's theorem should have a dot above the G(α), that's all. But I can see why it is usually not included. — Carl ( CBM · talk) 12:13, 6 April 2009 (UTC)
Yuck. This is the kind of issue I'd ordinarily prefer to just ignore. But the issue has been raised at axiom of determinacy and seems likely to recur.
The question is, which definition of CH should we give as primary, the one I call weak CH, which says every uncountable set of reals is equinumerous with the reals, or strong CH, which says the reals are equinumerous with the countable ordinals (that is, )?
It's true that most sources seem to state it in a weak form (I'm counting there is no cardinality strictly between and as an instance of the weak form; it's equivalent modulo the violent pathology of an infinite Dedekind-finite set of reals). Jech is a notable exception; he explicitly equates the continuum hypothesis with .
However, in the default set-theoretic context, the axiom of choice holds, so there is no need to make the distinction. Therefore I think it is unjustified to assume that this choice of statement reflects a preference as to the essential meaning of CH in a non-AC context. I would actually argue that strong CH is more natural to think of as capturing that meaning from the standpoint of contemporary set theory; I've given a couple of reasons at talk:axiom of determinacy.
The problem at the AD article is that an editor wanted to add the claim that AD implies CH, which I think is severely contrary to the usage of the terminology in the set-theoretic community. I've worked with set theorists who study AD, and I never recall any of them putting it that way. The problem for this article is that it puts the weak CH statement first, and then asserts that, given AC, this is equivalent to strong CH; this I think is misleading. Better would be to state them both, note that they're equivalent, but then note that in the absence of AC, they're not. -- Trovatore ( talk) 19:59, 24 April 2009 (UTC)
The section "As the first Hilbert problem" and "Impossibility of proof and disproof (in ZFC)" don't agree on Kurt Godel's work using the AC:
vs
The former only mentions that Godel did his work in ZF, but the latter section says that his original work was under the ZFC. Given the results, I think it was under ZFC. I think the former section needs to be made more explicit. -- B-Con ( talk) 08:11, 15 May 2009 (UTC)
I was just reading a book about Cantor. It mentioned that one of the main reasons Cantor believed in the continuum hypothesis was because he managed to show every nonempty perfect set is the same size as the reals, and then showed a closed set C is either countable or has cardinality of the reals by partitioning C into a perfect set and a countable set. And then he said "In future paragraphs it will be proven that this remarkable theorem has a further validity even for linear point sets which are not closed,..." Should this be in the article? Also I personally believe his reasoning is wrong: since the reals have a countable base, the number of open sets is the same as the reals, and thus so is the number of closed sets (because they are complements of open sets). But there are "so many more" subsets, because 2^(aleph0) is negligible compared to 2^(2^(aleph0)). Right??? Standard Oil ( talk) 14:28, 14 August 2009 (UTC)
Could someone clueful please check whether this edit (by me) is any good, and revert it if necessary. I started having doubts after making it. Thanks. 66.127.55.192 ( talk) 17:57, 17 February 2010 (UTC)
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.57.6594
an article explaining(?) Woodin's recent work. I don't understand it at all. Maybe someone here can make sense of it and possibly cite it or use something from it. 66.127.55.192 ( talk) 06:24, 18 February 2010 (UTC)
Why is this hypothesis called the "continuum" hypothesis? The proposition "There is no set whose cardinality is strictly between that of the integers and that of the real numbers." is asserting the lack of any set of size (or cardinality) between integers and real numbers, which sounds far more like the a "non-continuum hypothesis." or since it is saying that there is not a single set that stands in that range (far less a continuum of size/cardinalities of sets) the theory might even be called the "not-even-close-to-continuum hypothesis" or "discontinuum hypothesis" or "discrete hypothesis." Since continuum is a word in normal English, I guess it would help to know why this theory has the name it has. I can't understand the name, let alone the math :-) -- Timtak ( talk) 10:09, 14 October 2010 (UTC)
It's classical terminology. The real numbers were traditionally called "the continuum" and the continuum hypothesis is a hypothesis about the cardinality of the real numbers. — Carl ( CBM · talk) 10:51, 14 October 2010 (UTC)
From my classes we were taught that [0,2\pi] was called "the continuum", (which of course has the same cardinality as the Reals), but nevertheless Cantor was interested in Fourier series convergence, and all those fellas ever talk about is [0,2\pi] :P . Handsofftibet ( talk) 13:15, 13 January 2011 (UTC)
Australia and University of Newcastle Upon Tyne - England Handsofftibet ( talk) 11:35, 14 February 2011 (UTC)
70.51.177.249 ( talk · contribs) and 70.54.228.146 ( talk · contribs) added a reference to "Deciding the Continuum Hypothesis with the Inverse Power Set" by Patrick St-Amant. This paper begins by adding a new axiom to ZFC, namely
This axiom is inconsistent with the existence of the empty set. The existence of the empty set is an unavoidable consequence of the existence of any set and the axiom of separation. The empty set is defined by
The powerset of a set B is defined by
If the new axiom were true, then we could instantiate it for the empty set to give
for some set Y. Then putting Y in for B in the definition of powerset gives
In particular, if C were the emptyset, we would get
By the definition of the emptyset
for any D, and thus
holds vacously. Consequently, we would arrive at
which contradicts the definition of the empty set. Since the new "axiom" led to a contradiction (using only definitions and the existence of the empty set), it must be false. Thus Patrick St-Amant's whole project is an exercise in futility, in fact, an obvious hoax. JRSpriggs ( talk) 12:29, 6 January 2011 (UTC)
To Tkuvko: If you understand his theory, then tell me — Where am I wrong in my deduction of a contradiction from his theory? Which is the first step that fails and why? JRSpriggs ( talk) 11:08, 8 January 2011 (UTC)
Is it consistent with ZFC to have = some large cardinal? YohanN7 ( talk) 07:48, 9 May 2011 (UTC)
One intuitive argument against CH is said to be Freiling's Axiom of Symmetry. I wonder where the intuition comes from. I agree that the Axiom of Symmetry (AX) sounds quite convincing on first reading, but that's a little thin for mathematical reasoning. I thought about a way to base the intuition on provable facts using smaller sets than the reals. What about this?
Modify AX the following way:
1.) We throw darts at a countable set instead of the unit interval (I). Say the natural numbers (N) for definiteness.
2.) The function f:I->{x:x is a countable subset of I} occuring in AX is exchanged for f:N->{x:x is a finite subset of N}
Leave everything else untouched in AX. Call this thing the Axiom of Countable Symmetry (ACX).
I think that ACX "sounds" just about as convincing as AX on "first reading". Lets call a counterexample of AX (or ACX) a Freiling function. ACX is false. Proof: f(n) = {m: m <= n} is a Freiling function.
I don't draw any conclusions from this. But provided I'm correct in the above [and I might very well not be, the night is not young;)], I can't see the validity of the claim that AX is an argument against CH. If anything, it might be an argument in favour of CH, but I wouldn't go as far as that. In the article Freiling's axiom of symmetry there are listed two objections against AX. My example above might just be an example of objection number two, but I can't see exactly what that number two says. At any rate one does not need CH to disprove ACX, and I can't see the need for AC either. For a countable set there exists a bijection between it and N. That bijection does provide a wellordering (with the same length as N). Or doesn't it? Correct me if I am wrong. YohanN7 ( talk) 00:00, 10 May 2011 (UTC)
That's the point call Indduction i and strong induction I
Then what I have proved it that ZFCI<=>ZFCi<=>ZFCiCH<=>ZFCCH hence following it back and forth ZFCI<=>ZFCCH Hence If we assume ZFC, and not induction, we have not CH.
Hence ZFCi<=>ZFCI<=>ZFCRH which implies trivially that ZFCi<=>ZFCRH you can just never right down all the maps from R to N and N to R at the same time, but you can approximate them by 1-1 function f_N:I->R and the function f(x)= x and f(x)=-x:R->R and hence I->I invertibaly.
you should probably understand now, thanks for looking :) — Preceding unsigned comment added by WhatisFGH ( talk • contribs) 03:46, 20 October 2011 (UTC)
I added a note [1] about an article by Solomon Feferman, though I think I may have messed up the explanation somewhat—I figured the main thing was to get the citation into the article. Any review/fixes would be appreciated. The EFI page (url in the Feferman citation) looks interesting in general and there's an overview by Koellner [2] of the current state of CH, that also seems worth summarizing in the article. I might try, but I'm not very knowledgeable, so it would be great if someone else did it. Feferman has some other slides about his "definiteness" theory [3] that are a little more detailed than his EFI slides, if that's of any interest. 64.160.39.72 ( talk) 01:57, 5 February 2012 (UTC)
Hugh Woodin appears to have changed his mind regarding the truth value of CH. This should be reflected in the article. (See e.g. the Hugh Woodin wiki page.) YohanN7 ( talk) 22:35, 9 April 2012 (UTC)
Never mind the article. It will not be changed before a publication comes anyway. The interesting thing for me is if he has changed his mind. Does anyone of you gurus have an idea? Ultimate L seems to be a model of ZFC + a bunch of large cardinal axioms (including Woodin cardinals). There are lecture notes on it from a set theory workshop in 2010 available on the net. YohanN7 ( talk) 20:20, 10 April 2012 (UTC)
Does anyone know, if;
What this evaluates to?
Edit: It won't let me format that how i wanted to - basically I mean what is 2 to the power of 2 to the power of 2 to the power of 2 to the power of 2 and so on. — Preceding unsigned comment added by 109.149.174.130 ( talk • contribs) 16:24, 23 October 2012
My apologies, cheers. — Preceding unsigned comment added by 86.151.16.138 ( talk) 20:37, 28 October 2012 (UTC)
Since a recent edit has implicitly raised the question of how the formulas in Continuum hypothesis#Implications of GCH for cardinal exponentiation are justified. I will provide proofs here.
First, take notice of the following facts:
Now, suppose that α ≤ β+1, then
which means
On the other hand, suppose that β+1 < α, then
which means
If we further suppose that where cf is the cofinality operation, then any function from to must be bounded above by some And γ has a cardinality where δ < α. The cardinality of the set of functions so bounded by γ is
Adding these together for the possible values of γ gives
which means
On the other hand, if we further suppose then by one of the corollaries of König's theorem we have
and thus
which means
These were what was to be proved. JRSpriggs ( talk) 10:05, 13 November 2012 (UTC)
Continuum hypothesis#The generalized continuum hypothesis contains the following statement, which is tagged as needing a citation:
"A recent result of Carmi Merimovich shows that, for each n≥1, it is consistent with ZFC that for each κ, 2κ is the nth successor of κ. On the other hand, Laszlo Patai proved, that if γ is an ordinal and for each infinite cardinal κ, 2κ is the γth successor of κ, then γ is finite."
Are those two results published? Using Google Scholar, I have been unable to locate a source by Laszlo Patai containing this statement. The source by Merimovich which the first part is apparently based on might be
Merimovich, Carmi (2007). "A power function with a fixed finite gap everywhere" (PDF). J. Symbolic Logic. 72 (2): 361–417. doi: 10.2178/jsl/1185803615. MR 2320282. Zbl 1153.03036.
Is there a source for the part attributed to Laszlo Patai? -- Toshio Yamaguchi 13:07, 11 July 2013 (UTC)
Actually, in ZF, I think the Aleph hypothesis does imply the axiom of choice. It doesn't in ZFU (with urelements) or ZF- (- regularity). Let me see. I'll have to check my references, but
implies PW:
implies AC. — Arthur Rubin (talk) 15:50, 3 September 2013 (UTC)
Obviously, ZF + AC -> ( GCH <-> AH ). And, as I showed at Talk:Axiom of choice/Archive 4#Another try, ZF + GCH -> AC. If one can also show that ZF + AH -> AC, then it follows that ZF -> ( GCH <-> AH ).
It suffices to show for all ordinals α that Vα can be injected into the ordinals. However, there is a difficulty with the argument by Arthur Rubin above because it does not deal with the case that α is a limit ordinal. If one simply tries to aggregate the sequence of injections together, one runs into the fact that one is implicitly using AC to select one injection (or well ordering) at each ordinal below α.
This problem can be fixed by restricting ourselves to just picking one item provided that it is sufficiently large. Choose a single bijection f from P(ωα+1) to ωα+2. We will use this to construct a bijection from Vω+α to an ordinal between ωα+1 and ωα+2.
Suppose S is any subset of ωα+1. Let g(S) = ωα+2·rank(S) + f(S). Let h be the result of collapsing g to squeeze out the holes in its image. Let l be defined inductively on Vω+α by l(T) = h( { l(U) | U ∈ T } ). Then l is the desired injection from Vω+α to the ordinals. We use AH again (no use of AC here though) to verify that the elements of rank ω+β are mapped into a range beginning between ωβ and ωβ+1 and ending between ωβ+1 and ωβ+2; which fact is needed to make sure that our original choice of f had a sufficiently large domain. JRSpriggs ( talk) 04:53, 6 September 2013 (UTC)
User:JRSpriggs made a short remark in a recent edit correcting (thank you!) my previous edit. I would like to expand this remark here for two reasons: I want to help others avoid the mistake that I had made; also, perhaps we can add a short form of this remark into the article itself.
I will write Z0 for ZF without the axiom of foundation and without the axiom of replacement.
There are two versions of GCH:
So the two versions are indeed equivalent over the base theory ZF. Still I think that the (apparently) "stronger" version GCHpower should be mentioned as the GCH. When we mention Sierpinski's proof, we should clarify that he proved AC from GCHpower.
(In ZF, AC also follows from GCHaleph, but the interesting part is the implication from "P(well-order)=well-orderable" to AC, which has nothing to do with cardinal arithmetic. Was Sierpinski also the author of this theorem?)
-- Aleph4 ( talk) 12:48, 12 September 2013 (UTC)
Yes, I should have referenced the preceding discussion, and adapted my notation. Sorry. The preceding discussion gives the proof of (1→)2→3. My point is that the different character of the two implications between GCH (GCHpower) and AH (GCHaleph) is not made clear in the article. (But if I cannot make this clear on the discussion page, I won't even try on the article page.) -- Aleph4 ( talk) 15:58, 13 September 2013 (UTC)
The lead has been going back and forth for a few days w/o me being involved except for a revert. Does the lead need to mention NGB set theory? Perhaps so (I have no idea), but in that case, the relationship between ZF and NGB set theories must be described - otherwise it's just confusing. Please discuss it here, it's what the talk page is for. I will not interfere. YohanN7 ( talk) 22:27, 5 January 2014 (UTC)
JR's point seems to be that Goedel's contribution is listed first and Cohen's second (same as the chronological order). But the unusual word order "disproved nor proved" really does not get this across; it just looks weird. I suggest that if we want to say that Goedel showed it couldn't be disproved in ZFC and Cohen showed it couldn't be proved in ZFC, then that is what needs to be said. But at that level of the lead, I don't know that we need to get that specific, and I think we should just go back to "proved nor disproved".
There has been some back-and-forth on whether ZF is "a standard foundation" or "the standard foundation". I feel that using the definite article is a POV. Category theory provides another "standard" foundation, for example. There are other varieties of set theory that are similarly considered "standard" by subsets (no pun intended) of the mathematical public, such as MK or NBG. Tkuvho ( talk) 08:23, 9 January 2014 (UTC)
I have expressed qualified support for Eleuther's concerns, but I do not think this latest effort works very well:
The most problematic aspect is the second sentence, which is both of unclear meaning and seriously POV if given any substantive meaning. I have removed that sentence.
But I'm not all that thrilled with the rest of it either, though I agree it is simpler. Mainly I think the phrasing "independent of the other axioms of set theory" is problematic. First, "other" axioms of set theory suggests that CH is an axiom of set theory, which is an unusual position. Then too, " 'the' other axioms of set theory" suggests that there's a canonical list of axioms of set theory, which I thought was Eleuther's complaint in the first place. -- Trovatore ( talk) 00:31, 10 January 2014 (UTC)
The recent removal of the mention of ZFC from the lead is misguided. The question of CH can only be made precise in the context of ZFC (or ZF) and this should be mentioned. I have the impression that only one editor opposes the inclusion of such an explicit mention. It would be nice if the editors expressed themselves on this limited point. Tkuvho ( talk) 09:25, 10 January 2014 (UTC)
I may be old-fashioned, but I don't see the specific relevance of Hampkins' paper to CH; although I haven't yet read a the paper, it would seem to apply to any independent (in some sense) statement, not just CH.
Even assuming the paper is considered significant in the field. — Arthur Rubin (talk) 13:51, 27 January 2014 (UTC)
The book:
Cohen, P. J. (1966). Set Theory and the Continuum Hypothesis. W. A. Benjamin.
is not accessible: you can request it, pay for it and the there is no book and no refunds.
Suspicious? — Preceding unsigned comment added by 87.218.117.85 ( talk) 09:00, 30 June 2014 (UTC)
The first line of the article reads: "In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis, advanced by Georg Cantor in 1878, about the possible sizes of infinite sets."
While it's traditionally named the "continuum hypothesis," does it really make sense to refer to it explicitly as a "hypothesis" (including a link to the article for "hypothesis," no less)? "Hypothesis" generally refers to a testable predictive claim about a system - given that the "truth" of CH is provably independent of the standard axiomatization of mathematics (and thus we can, in a way, think of CH as a specification of a system we're working with, not a claim about the properties of an established system), this seems like a poor description. It certainly doesn't make sense to link to the article for "hypothesis," any more than it would to do the same from the description of the axiom of choice or the parallel line postulate - both of which, incidentally, are (correctly) referred to as "axioms" as I believe should be the case here.
173.66.29.191 ( talk) 20:29, 9 September 2014 (UTC)
It's not a big deal, but I think that the recent edits have been for the worse. The size of a set is, in informal terms, its cardinality. The lead can, at this point, stay informal, the term cardinality coming one sentence later. The article now links Set (mathematics), not infinite set. Worse, it speaks of sets with infinite elements. YohanN7 ( talk) 13:16, 14 November 2014 (UTC)
Moved to Talk:Cantor's diagonal argument/Arguments.
Can a better example for bijection be provided? The example given, "Hence, the set {banana, apple, pear} has the same cardinality as {yellow, red, green}." lists three fruits that exist in more than one of the colors given. Mjpollard ( talk) 14:17, 6 November 2016 (UTC)
The "independence" section is just a list of references, which should be moved to the history section. The independedence section should have an outline of the actual proof instead, Wikipedia is supposed to contain facts rather than just references. Can anyone knowledgeable make a go at doing this ? — Preceding unsigned comment added by 194.80.232.19 ( talk) 08:49, 5 May 2017 (UTC)
Is saying that really equivalent to the continuum hypothesis? There seems to be a bijection between the set of real numbers and powerset of the natural numbers here, which would imply the continuum hypothesis is true if they were equivalent. -- AkariAkaori ( talk) 00:03, 10 July 2017 (UTC)
There is a recent study by these authors which has been widely reported in the press, and seems highly relevant to the CH. I'm not a mathematician, so the technicalities are beyond me, and I was hoping to find a clear explanation in this article. There's a summary here: https://www.scientificamerican.com/article/mathematicians-measure-infinities-and-find-theyre-equal/ — Preceding unsigned comment added by 86.154.202.49 ( talk) 19:42, 16 September 2017 (UTC)
Should this "hypothesis" be categorized under Category:Conjectures rather than Category:Hypotheses? Or should the definitions of those categories include historical naming that goes against current convention. Perhaps the article should be renamed (with a redirect). Dpleibovitz ( talk) 21:49, 28 December 2017 (UTC)
I have limited math understanding, but can I risk asking a possibly pedantic question: should the sentence "That is, every set, S, of real numbers can either be mapped one-to-one into the integers or the real numbers can be mapped one-to-one into S." be changed to say "infinite set S" or should something about "subsets" be inserted somewhere? 103.27.161.6 ( talk) 11:33, 21 February 2019 (UTC)johnjPerth
[4] Explains Cohen's proof of independence of CH. It looks pretty self contained yet is short enough for one lecture. I'm going to try to read it since other places have made the proof sound very mysterious and I have never understood it. Leaving link here for future reference and for possible use in improving the article. 2602:24A:DE47:BB20:50DE:F402:42A6:A17D ( talk) 05:17, 18 August 2020 (UTC)
I've just seen this Quanta article, which lead me to the papers "Model theory and the cardinal numbers 𝔭 and 𝔱" and "General topology meets model theory, on p and t", and then "Cofinality spectrum theorems in model theory, set theory and general topology". All of this is way over my head, mathematically. I'm uncertain about exactly what the proof of 𝔭 = 𝔱 actually means, and its relationship to the topic of this article. Does this now-proven hypothesis have a name, and what is the relationship of this proof to the continuum hypothesis? -- The Anome ( talk) 13:18, 31 December 2020 (UTC)
I just came across this article about a new mathematical proof implying that the hypothesis is true. But I don't understand the subject matter well enough to add it to the page in a sensible manner, perhaps someone else can take a gander?
Here's the article that might be worth including: https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/
Cheers Creapuscularr ( talk) 13:42, 6 August 2021 (UTC)
Since this conjecture is "a conjecture which can be disproved using a counterexample", if this conjecture is in fact false, we can use this counterexample to disprove it, thus the conjecture is decidable, and if the conjecture is undecidable, then it must be true, thus, if we can prove that this conjecture is undecidable, then we can prove that this conjecture is true, which is a contradiction, like Goldbach conjecture (a counterexample is an even number ≥4 which cannot be written as sum of two primes), Fermat Last Theorem (a counterexample is an integer n≥3 and positive integers x, y, z such that xn + yn = zn), Riemann hypothesis (a counterexample is a complex number z such that Re(z) ≠ 0, Im(z) ≠ 1/2, but ζ(z) = 0), and four color theorem (a counterexample is a graph which cannot be drawn using ≤4 colors), a counterexample of continuum hypothesis is a subset S of R such that Card(S) > Card(N) and Card(S) < Card(R). 2402:7500:916:25D6:DD5D:1928:6166:9583 ( talk) 09:25, 20 August 2021 (UTC)
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I don't entirely agree with the following passage:
Generally speaking, mathematicians who favor a "rich" and "large" universe of sets are against CH, while those favoring a "neat" and "controllable" universe favor CH.
See the Maddy reference I added, page 500, the sections entitled Not-CH is restrictive (in favor) and Modern forcing (in favor).
I'll try to come up with some wording that's not too awkward, illustrating the historical view reported in the existing Wiki article, while also pointing out that between models having all the same reals, it's the ones with more sets of reals that are more likely to satisfy CH. -- trovatore
My attempt is now in place Trovatore 05:04, 27 Jun 2005 (UTC)
User Martindowd has been arguing that the truth of CH, indeed V=L, should be taken more seriously. He suggests adding the following at the end of the second paragraph of the "Arguments for and against CH" section of the Continuum hypothesis page. If you have any opinion on this, please let me know.
Recently, arguments have appeared that this view should be re-evaluated, and that there are arguments in favor of both (Dowd 2011).
Dowd, Martin (2011) "Some New Axioms for Set Theory", http://www.ijpam.eu/contents/2011-66-2/1/1.pdf Martindowd ( talk) 15:04, 21 September 2011 (UTC)
I would add to Trovatore's point that it is questionable (as stated on the page) that Cantor himself believed CH. Did he not alternate back and forth during his manic (and depressive) phases, sometimes believing CH, sometimes believing ~CH? See the BBC TV documentary "Dangerous Knowledge" willbown. 16 June 2008. —Preceding comment was added at 22:37, 16 June 2008 (UTC)
I've removed this section:
It's a good thing to add some intuition, but I don't know that this passage helped much. CH isn't about whether such a set S can be found in any ordinary sense, but just about whether one exists. -- Trovatore 21:12, 22 October 2005 (UTC)
Let me add this thought:
Since CH cannot be disproven, a set that denies it cannot be found. This doesn't mean it does not exist, but we're close :-) Honnza ( talk) 06:01, 17 May 2008 (UTC)
The undecidability of CH begs the question: If there is a cardinality between and , then what sets might there be that have this cardinality?
In other words, is there any known set that is larger than , but such that you need to set the truth value of CH to determine whether or not it's equivalent to continuum?
Moreover, is there any evidence pointing to the existence or non-existence of such a set, or whether it would be possible to find it if it does exist? -- Smjg ( talk) 16:35, 28 February 2008 (UTC)
But in the context of ZF, j:V into V refutes choice (according to the article on Reinhardt cardinals anyway), and GCH implies it (according to this article). So then j:V into V would refute GCH. Right? -- Unzerlegbarkeit ( talk) 15:58, 27 May 2008 (UTC)
The form makes the continuum hypothesis meaningful even if the continuum isn't well-ordered. The article states:
Do we have any examples which would make sense without AC, and would they depend on the form of CH? Also, on the other side of the coin, my understanding is that there could be no purely arithmetic consequences, roughly because ZF + V=L proves CH anyway (and even GCH), and relativising everything to L makes no difference to (first-order) arithmetic. If correct, I believe this is worth stating. Also, the article states:
The "assuming the axiom of choice" bit should really attach to the "in turn equivalent to", because even from the axioms ZF without AC, aleph_1 exists and is an immediate successor of aleph_0, right? -- Unzerlegbarkeit ( talk) 02:39, 1 June 2008 (UTC)
I notice there aren't any references to Cantor's original writings. I'll try to dig some up, but has anyone seen any English translated letters, etc? Libertyblues ( talk) 00:03, 3 August 2008 (UTC)
This statement is revealing a transfinite-platonic bias. without a formalization like ZFC, it is not clear that it is possible to give a meaning to the statement that the continuum even has an ordinal size. Before Cantor insisted it was true, an infinite collection like the set of real numbers was not considered to have a definite size, let alone a definite ordinal size, and indeed, after Cohen, it is again a widespread point of view that the continuum does not have a definite size as an ordinal.
The idea that the set of all real numbers has definite properties in a platonic realm can be classified as a type of "transfinite platonism", which is just what platonism usually means nowadays. But transfinite Platonism can be logically separated from "computational platonism" (I made that up, but it needs a name)--- the position that all computer programs either halt or do not halt. The second position is what people mean by platonism in practice--- that the results of computations with symbols have a meaning, and questions about their outcome have a truth value. it was the position of Paul Cohen, shared by most mathematicians, that all questions about the integers/computer-programs are decided by a strong enough axiom system, which is a way of saying that they have a truth value in an axiom-system independent way.
But you can believe this, and still be a formalist regarding transfinite set theory. The position might be called "formalist", but "formalist" conflates two notions: "formalist regarding uncountable ordinals and sets the size of the continuum" and "formalist regarding countable infinity". A "formalist regarding countable infinity" would, for example, consider a nonstandard models of Peano arithmetic to be just as "true" a model of the integers as the standard model. Such a formalist would believe that some statements about diophantine equations like Fermat's Last Theorem, are undecidable in an absolute sense. This is the type of straw-man formalist that people argue against.
A "formalist regarding uncountable ordinals" on the other hand, would say that the truth value of the 3N+1 conjecture is well defined, but statements like the continuum hypothesis have no truth value except relative to an axiom system. This is the type of formalism which the forcing models foist upon you. If you accept this philosophy, which many people do, then you would not regard the continuum hypothesis as independent of ZFC.
This is the philosophical position which Cohen alludes to in his book, hopefully stated clearly enough there so that this exposition will be recognized as a clarification. I have been trying to figure out how to say it clearly for a while. Likebox ( talk) 17:22, 24 August 2008 (UTC)
(deindent) I am pretty ignorant about determinacy--- I only have only the most superficial heresay knowledge, and no clear understanding. But I have a somewhat negative opinion anyway, probably unjustified.
When I think about the continuum hypothesis, it is linked in my mind to the notion of measure and probability--- the reason that many people have intuitions about the cardinality of the real numbers is that it is consistent to talk about randomly choosing real numbers in a way that is different from randomly choosing integers or elements of a well ordered set. You can't pick an integer uniformly at random--- the concept doesn't make sense. You need a probability distribution, and certain integers are more likely than others. The same goes for any countable ordinal--- you need to weight different positions with different probability.
But for the interval [0,1], the notion of picking a number at random seems to be perfectly well defined, because you can roll dice to find each digit in sucession. This converges to a real number, and any property of that number that has probability zero is false. If a random number can be thought of as an element of the mathematical universe, with the property of belonging or not belonging to any previously specified set, it means that this previously specified set has a well defined Lebesgue measure. So not only is the continuum hypothesis false with random reals, it is meaningless, because the real numbers can't be well ordered. Any ordinal, no matter how large, can be imbedded in the reals by inductively mapping each successive element into [0,1] at random, and the probability of picking the same real twice is always zero (the last statement is only self consistently true, it's true in a countable model).
Making this precise is the job of forcing, which, given the political situation in mathematics, phrases everything in terms of the picking process, although at the end it talks about "random reals". But a random real number is an obviously consistent idea, even though it is incompatible with choice.
So when I look at post-forcing axioms like determinacy, I am always looking for the probability interpretation, and in this case I couldn't see it. The determinacy axiom, as I heard it, was exciting because it had a completely different character--- it asserted something about infinite sets that somebody thought was intuitively consistent (I don't know why, but they turned out to be right, so they must have had a good idea). But the justification for this for those with a formal view of higher infinity is that a large cardinal axiom proved that it is equiconsistent with set theory. So determinacy, as far as theorems about integers are concerned, is a moderate large cardinal axiom, and is not as interesting as a completely new axiom (although Cohen's "article of faith" says there are not going to be any consistent new axioms that are not equiconsistent by virtue of a large enough cardinal). So a superficial glance at Woodin's article was disappointing for me, because it didn't give a perspective which was probabalistic. But that's a very self-centered point of view, so I'll give it another shot. Likebox ( talk) 17:51, 26 August 2008 (UTC)
Edited out a minor error: ZFC stands for "Zermelo-Frankel with the axiom of choice", whereas ZF is simply "Zermelo-frankel". Would be nice to see some superscript references in this article, I'm afraid I don't know how to do that yet. 83.71.3.220 ( talk) 00:41, 1 October 2008 (UTC)
Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers. His proofs, however, give no indication of the extent to which the cardinality of the natural numbers is less than that of the real numbers.
Shouldn't natural numbers be rational numbers? Since the natural numbers is a subset of the integers, it's quite obvious that the cardinality of the natural numbers isn't greater than the integers. —Preceding unsigned comment added by 131.215.42.208 ( talk) 23:21, 20 January 2009 (UTC)
I think the article should include a section saying which alephs could be (and ) in the case CH is undecided. The article about the cardinality of the continuum has some indications about this (for example, it could be that , but not ), but I think this article here is the proper place to a full analysis. And I have never seen an upper limit to ; does it make sense if could be some monstrosity like ? Albmont ( talk) 19:11, 3 April 2009 (UTC)
You might want to look at Easton's theorem which deals with a more general question. JRSpriggs ( talk) 06:37, 4 April 2009 (UTC)
My question is quite simple: given ZFC, can we name a few values of α such that ? There's no need to expand the metalanguage to discuss models, etc - let's get to the basic, simple theory and see what we can get. For example, the first value that we can show is , the second is , etc. Of course, given that it is impossible to prove within ZFC that is and others, the next question would be quite obvious, but let's not do that leap. Not yet. Albmont ( talk) 03:13, 5 April 2009 (UTC)
Albmont: the informal "short answer" is that any uncountable κ with uncountable cofinality is a possible candidate for the cardinality of the continuum. But this statement is not entirely precise because "possible" here means "can be achieved by a forcing extension" (because the informal statement is really just a much easier special case of Easton's theorem), and there are some technical issues related to forcing that have to be mastered in order to fully understand what's going on. These issues are what Trovatore is alluding to. — Carl ( CBM · talk) 04:03, 6 April 2009 (UTC)
I don't believe the "object language"/"metalanguage" distinction is really the problem here. In these sorts of situations we work model theoretically: start with a model W and work with a language that has a constant for every element of W. So every ordinal in W is definable in this language. One thing that does matter is to distinguish between things that look like names but are actually meant to be formulas, and things that look like names that are actually meant to be names. This is typically achieved by writing dots over the things that are meant to be names; we ought to write .
So a slightly better way of stating the theorem at hand goes like this.
I don't see that it matters whether any formula φ defining α is absolute or not, because we will not actually use φ to identify a cardinal in the forcing extension, we use the forcing name. The absoluteness issue is only relevant to seeing why Trovatore's example does not give a contradiction to Easton's theorem. Easton's theorem should have a dot above the G(α), that's all. But I can see why it is usually not included. — Carl ( CBM · talk) 12:13, 6 April 2009 (UTC)
Yuck. This is the kind of issue I'd ordinarily prefer to just ignore. But the issue has been raised at axiom of determinacy and seems likely to recur.
The question is, which definition of CH should we give as primary, the one I call weak CH, which says every uncountable set of reals is equinumerous with the reals, or strong CH, which says the reals are equinumerous with the countable ordinals (that is, )?
It's true that most sources seem to state it in a weak form (I'm counting there is no cardinality strictly between and as an instance of the weak form; it's equivalent modulo the violent pathology of an infinite Dedekind-finite set of reals). Jech is a notable exception; he explicitly equates the continuum hypothesis with .
However, in the default set-theoretic context, the axiom of choice holds, so there is no need to make the distinction. Therefore I think it is unjustified to assume that this choice of statement reflects a preference as to the essential meaning of CH in a non-AC context. I would actually argue that strong CH is more natural to think of as capturing that meaning from the standpoint of contemporary set theory; I've given a couple of reasons at talk:axiom of determinacy.
The problem at the AD article is that an editor wanted to add the claim that AD implies CH, which I think is severely contrary to the usage of the terminology in the set-theoretic community. I've worked with set theorists who study AD, and I never recall any of them putting it that way. The problem for this article is that it puts the weak CH statement first, and then asserts that, given AC, this is equivalent to strong CH; this I think is misleading. Better would be to state them both, note that they're equivalent, but then note that in the absence of AC, they're not. -- Trovatore ( talk) 19:59, 24 April 2009 (UTC)
The section "As the first Hilbert problem" and "Impossibility of proof and disproof (in ZFC)" don't agree on Kurt Godel's work using the AC:
vs
The former only mentions that Godel did his work in ZF, but the latter section says that his original work was under the ZFC. Given the results, I think it was under ZFC. I think the former section needs to be made more explicit. -- B-Con ( talk) 08:11, 15 May 2009 (UTC)
I was just reading a book about Cantor. It mentioned that one of the main reasons Cantor believed in the continuum hypothesis was because he managed to show every nonempty perfect set is the same size as the reals, and then showed a closed set C is either countable or has cardinality of the reals by partitioning C into a perfect set and a countable set. And then he said "In future paragraphs it will be proven that this remarkable theorem has a further validity even for linear point sets which are not closed,..." Should this be in the article? Also I personally believe his reasoning is wrong: since the reals have a countable base, the number of open sets is the same as the reals, and thus so is the number of closed sets (because they are complements of open sets). But there are "so many more" subsets, because 2^(aleph0) is negligible compared to 2^(2^(aleph0)). Right??? Standard Oil ( talk) 14:28, 14 August 2009 (UTC)
Could someone clueful please check whether this edit (by me) is any good, and revert it if necessary. I started having doubts after making it. Thanks. 66.127.55.192 ( talk) 17:57, 17 February 2010 (UTC)
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.57.6594
an article explaining(?) Woodin's recent work. I don't understand it at all. Maybe someone here can make sense of it and possibly cite it or use something from it. 66.127.55.192 ( talk) 06:24, 18 February 2010 (UTC)
Why is this hypothesis called the "continuum" hypothesis? The proposition "There is no set whose cardinality is strictly between that of the integers and that of the real numbers." is asserting the lack of any set of size (or cardinality) between integers and real numbers, which sounds far more like the a "non-continuum hypothesis." or since it is saying that there is not a single set that stands in that range (far less a continuum of size/cardinalities of sets) the theory might even be called the "not-even-close-to-continuum hypothesis" or "discontinuum hypothesis" or "discrete hypothesis." Since continuum is a word in normal English, I guess it would help to know why this theory has the name it has. I can't understand the name, let alone the math :-) -- Timtak ( talk) 10:09, 14 October 2010 (UTC)
It's classical terminology. The real numbers were traditionally called "the continuum" and the continuum hypothesis is a hypothesis about the cardinality of the real numbers. — Carl ( CBM · talk) 10:51, 14 October 2010 (UTC)
From my classes we were taught that [0,2\pi] was called "the continuum", (which of course has the same cardinality as the Reals), but nevertheless Cantor was interested in Fourier series convergence, and all those fellas ever talk about is [0,2\pi] :P . Handsofftibet ( talk) 13:15, 13 January 2011 (UTC)
Australia and University of Newcastle Upon Tyne - England Handsofftibet ( talk) 11:35, 14 February 2011 (UTC)
70.51.177.249 ( talk · contribs) and 70.54.228.146 ( talk · contribs) added a reference to "Deciding the Continuum Hypothesis with the Inverse Power Set" by Patrick St-Amant. This paper begins by adding a new axiom to ZFC, namely
This axiom is inconsistent with the existence of the empty set. The existence of the empty set is an unavoidable consequence of the existence of any set and the axiom of separation. The empty set is defined by
The powerset of a set B is defined by
If the new axiom were true, then we could instantiate it for the empty set to give
for some set Y. Then putting Y in for B in the definition of powerset gives
In particular, if C were the emptyset, we would get
By the definition of the emptyset
for any D, and thus
holds vacously. Consequently, we would arrive at
which contradicts the definition of the empty set. Since the new "axiom" led to a contradiction (using only definitions and the existence of the empty set), it must be false. Thus Patrick St-Amant's whole project is an exercise in futility, in fact, an obvious hoax. JRSpriggs ( talk) 12:29, 6 January 2011 (UTC)
To Tkuvko: If you understand his theory, then tell me — Where am I wrong in my deduction of a contradiction from his theory? Which is the first step that fails and why? JRSpriggs ( talk) 11:08, 8 January 2011 (UTC)
Is it consistent with ZFC to have = some large cardinal? YohanN7 ( talk) 07:48, 9 May 2011 (UTC)
One intuitive argument against CH is said to be Freiling's Axiom of Symmetry. I wonder where the intuition comes from. I agree that the Axiom of Symmetry (AX) sounds quite convincing on first reading, but that's a little thin for mathematical reasoning. I thought about a way to base the intuition on provable facts using smaller sets than the reals. What about this?
Modify AX the following way:
1.) We throw darts at a countable set instead of the unit interval (I). Say the natural numbers (N) for definiteness.
2.) The function f:I->{x:x is a countable subset of I} occuring in AX is exchanged for f:N->{x:x is a finite subset of N}
Leave everything else untouched in AX. Call this thing the Axiom of Countable Symmetry (ACX).
I think that ACX "sounds" just about as convincing as AX on "first reading". Lets call a counterexample of AX (or ACX) a Freiling function. ACX is false. Proof: f(n) = {m: m <= n} is a Freiling function.
I don't draw any conclusions from this. But provided I'm correct in the above [and I might very well not be, the night is not young;)], I can't see the validity of the claim that AX is an argument against CH. If anything, it might be an argument in favour of CH, but I wouldn't go as far as that. In the article Freiling's axiom of symmetry there are listed two objections against AX. My example above might just be an example of objection number two, but I can't see exactly what that number two says. At any rate one does not need CH to disprove ACX, and I can't see the need for AC either. For a countable set there exists a bijection between it and N. That bijection does provide a wellordering (with the same length as N). Or doesn't it? Correct me if I am wrong. YohanN7 ( talk) 00:00, 10 May 2011 (UTC)
That's the point call Indduction i and strong induction I
Then what I have proved it that ZFCI<=>ZFCi<=>ZFCiCH<=>ZFCCH hence following it back and forth ZFCI<=>ZFCCH Hence If we assume ZFC, and not induction, we have not CH.
Hence ZFCi<=>ZFCI<=>ZFCRH which implies trivially that ZFCi<=>ZFCRH you can just never right down all the maps from R to N and N to R at the same time, but you can approximate them by 1-1 function f_N:I->R and the function f(x)= x and f(x)=-x:R->R and hence I->I invertibaly.
you should probably understand now, thanks for looking :) — Preceding unsigned comment added by WhatisFGH ( talk • contribs) 03:46, 20 October 2011 (UTC)
I added a note [1] about an article by Solomon Feferman, though I think I may have messed up the explanation somewhat—I figured the main thing was to get the citation into the article. Any review/fixes would be appreciated. The EFI page (url in the Feferman citation) looks interesting in general and there's an overview by Koellner [2] of the current state of CH, that also seems worth summarizing in the article. I might try, but I'm not very knowledgeable, so it would be great if someone else did it. Feferman has some other slides about his "definiteness" theory [3] that are a little more detailed than his EFI slides, if that's of any interest. 64.160.39.72 ( talk) 01:57, 5 February 2012 (UTC)
Hugh Woodin appears to have changed his mind regarding the truth value of CH. This should be reflected in the article. (See e.g. the Hugh Woodin wiki page.) YohanN7 ( talk) 22:35, 9 April 2012 (UTC)
Never mind the article. It will not be changed before a publication comes anyway. The interesting thing for me is if he has changed his mind. Does anyone of you gurus have an idea? Ultimate L seems to be a model of ZFC + a bunch of large cardinal axioms (including Woodin cardinals). There are lecture notes on it from a set theory workshop in 2010 available on the net. YohanN7 ( talk) 20:20, 10 April 2012 (UTC)
Does anyone know, if;
What this evaluates to?
Edit: It won't let me format that how i wanted to - basically I mean what is 2 to the power of 2 to the power of 2 to the power of 2 to the power of 2 and so on. — Preceding unsigned comment added by 109.149.174.130 ( talk • contribs) 16:24, 23 October 2012
My apologies, cheers. — Preceding unsigned comment added by 86.151.16.138 ( talk) 20:37, 28 October 2012 (UTC)
Since a recent edit has implicitly raised the question of how the formulas in Continuum hypothesis#Implications of GCH for cardinal exponentiation are justified. I will provide proofs here.
First, take notice of the following facts:
Now, suppose that α ≤ β+1, then
which means
On the other hand, suppose that β+1 < α, then
which means
If we further suppose that where cf is the cofinality operation, then any function from to must be bounded above by some And γ has a cardinality where δ < α. The cardinality of the set of functions so bounded by γ is
Adding these together for the possible values of γ gives
which means
On the other hand, if we further suppose then by one of the corollaries of König's theorem we have
and thus
which means
These were what was to be proved. JRSpriggs ( talk) 10:05, 13 November 2012 (UTC)
Continuum hypothesis#The generalized continuum hypothesis contains the following statement, which is tagged as needing a citation:
"A recent result of Carmi Merimovich shows that, for each n≥1, it is consistent with ZFC that for each κ, 2κ is the nth successor of κ. On the other hand, Laszlo Patai proved, that if γ is an ordinal and for each infinite cardinal κ, 2κ is the γth successor of κ, then γ is finite."
Are those two results published? Using Google Scholar, I have been unable to locate a source by Laszlo Patai containing this statement. The source by Merimovich which the first part is apparently based on might be
Merimovich, Carmi (2007). "A power function with a fixed finite gap everywhere" (PDF). J. Symbolic Logic. 72 (2): 361–417. doi: 10.2178/jsl/1185803615. MR 2320282. Zbl 1153.03036.
Is there a source for the part attributed to Laszlo Patai? -- Toshio Yamaguchi 13:07, 11 July 2013 (UTC)
Actually, in ZF, I think the Aleph hypothesis does imply the axiom of choice. It doesn't in ZFU (with urelements) or ZF- (- regularity). Let me see. I'll have to check my references, but
implies PW:
implies AC. — Arthur Rubin (talk) 15:50, 3 September 2013 (UTC)
Obviously, ZF + AC -> ( GCH <-> AH ). And, as I showed at Talk:Axiom of choice/Archive 4#Another try, ZF + GCH -> AC. If one can also show that ZF + AH -> AC, then it follows that ZF -> ( GCH <-> AH ).
It suffices to show for all ordinals α that Vα can be injected into the ordinals. However, there is a difficulty with the argument by Arthur Rubin above because it does not deal with the case that α is a limit ordinal. If one simply tries to aggregate the sequence of injections together, one runs into the fact that one is implicitly using AC to select one injection (or well ordering) at each ordinal below α.
This problem can be fixed by restricting ourselves to just picking one item provided that it is sufficiently large. Choose a single bijection f from P(ωα+1) to ωα+2. We will use this to construct a bijection from Vω+α to an ordinal between ωα+1 and ωα+2.
Suppose S is any subset of ωα+1. Let g(S) = ωα+2·rank(S) + f(S). Let h be the result of collapsing g to squeeze out the holes in its image. Let l be defined inductively on Vω+α by l(T) = h( { l(U) | U ∈ T } ). Then l is the desired injection from Vω+α to the ordinals. We use AH again (no use of AC here though) to verify that the elements of rank ω+β are mapped into a range beginning between ωβ and ωβ+1 and ending between ωβ+1 and ωβ+2; which fact is needed to make sure that our original choice of f had a sufficiently large domain. JRSpriggs ( talk) 04:53, 6 September 2013 (UTC)
User:JRSpriggs made a short remark in a recent edit correcting (thank you!) my previous edit. I would like to expand this remark here for two reasons: I want to help others avoid the mistake that I had made; also, perhaps we can add a short form of this remark into the article itself.
I will write Z0 for ZF without the axiom of foundation and without the axiom of replacement.
There are two versions of GCH:
So the two versions are indeed equivalent over the base theory ZF. Still I think that the (apparently) "stronger" version GCHpower should be mentioned as the GCH. When we mention Sierpinski's proof, we should clarify that he proved AC from GCHpower.
(In ZF, AC also follows from GCHaleph, but the interesting part is the implication from "P(well-order)=well-orderable" to AC, which has nothing to do with cardinal arithmetic. Was Sierpinski also the author of this theorem?)
-- Aleph4 ( talk) 12:48, 12 September 2013 (UTC)
Yes, I should have referenced the preceding discussion, and adapted my notation. Sorry. The preceding discussion gives the proof of (1→)2→3. My point is that the different character of the two implications between GCH (GCHpower) and AH (GCHaleph) is not made clear in the article. (But if I cannot make this clear on the discussion page, I won't even try on the article page.) -- Aleph4 ( talk) 15:58, 13 September 2013 (UTC)
The lead has been going back and forth for a few days w/o me being involved except for a revert. Does the lead need to mention NGB set theory? Perhaps so (I have no idea), but in that case, the relationship between ZF and NGB set theories must be described - otherwise it's just confusing. Please discuss it here, it's what the talk page is for. I will not interfere. YohanN7 ( talk) 22:27, 5 January 2014 (UTC)
JR's point seems to be that Goedel's contribution is listed first and Cohen's second (same as the chronological order). But the unusual word order "disproved nor proved" really does not get this across; it just looks weird. I suggest that if we want to say that Goedel showed it couldn't be disproved in ZFC and Cohen showed it couldn't be proved in ZFC, then that is what needs to be said. But at that level of the lead, I don't know that we need to get that specific, and I think we should just go back to "proved nor disproved".
There has been some back-and-forth on whether ZF is "a standard foundation" or "the standard foundation". I feel that using the definite article is a POV. Category theory provides another "standard" foundation, for example. There are other varieties of set theory that are similarly considered "standard" by subsets (no pun intended) of the mathematical public, such as MK or NBG. Tkuvho ( talk) 08:23, 9 January 2014 (UTC)
I have expressed qualified support for Eleuther's concerns, but I do not think this latest effort works very well:
The most problematic aspect is the second sentence, which is both of unclear meaning and seriously POV if given any substantive meaning. I have removed that sentence.
But I'm not all that thrilled with the rest of it either, though I agree it is simpler. Mainly I think the phrasing "independent of the other axioms of set theory" is problematic. First, "other" axioms of set theory suggests that CH is an axiom of set theory, which is an unusual position. Then too, " 'the' other axioms of set theory" suggests that there's a canonical list of axioms of set theory, which I thought was Eleuther's complaint in the first place. -- Trovatore ( talk) 00:31, 10 January 2014 (UTC)
The recent removal of the mention of ZFC from the lead is misguided. The question of CH can only be made precise in the context of ZFC (or ZF) and this should be mentioned. I have the impression that only one editor opposes the inclusion of such an explicit mention. It would be nice if the editors expressed themselves on this limited point. Tkuvho ( talk) 09:25, 10 January 2014 (UTC)
I may be old-fashioned, but I don't see the specific relevance of Hampkins' paper to CH; although I haven't yet read a the paper, it would seem to apply to any independent (in some sense) statement, not just CH.
Even assuming the paper is considered significant in the field. — Arthur Rubin (talk) 13:51, 27 January 2014 (UTC)
The book:
Cohen, P. J. (1966). Set Theory and the Continuum Hypothesis. W. A. Benjamin.
is not accessible: you can request it, pay for it and the there is no book and no refunds.
Suspicious? — Preceding unsigned comment added by 87.218.117.85 ( talk) 09:00, 30 June 2014 (UTC)
The first line of the article reads: "In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis, advanced by Georg Cantor in 1878, about the possible sizes of infinite sets."
While it's traditionally named the "continuum hypothesis," does it really make sense to refer to it explicitly as a "hypothesis" (including a link to the article for "hypothesis," no less)? "Hypothesis" generally refers to a testable predictive claim about a system - given that the "truth" of CH is provably independent of the standard axiomatization of mathematics (and thus we can, in a way, think of CH as a specification of a system we're working with, not a claim about the properties of an established system), this seems like a poor description. It certainly doesn't make sense to link to the article for "hypothesis," any more than it would to do the same from the description of the axiom of choice or the parallel line postulate - both of which, incidentally, are (correctly) referred to as "axioms" as I believe should be the case here.
173.66.29.191 ( talk) 20:29, 9 September 2014 (UTC)
It's not a big deal, but I think that the recent edits have been for the worse. The size of a set is, in informal terms, its cardinality. The lead can, at this point, stay informal, the term cardinality coming one sentence later. The article now links Set (mathematics), not infinite set. Worse, it speaks of sets with infinite elements. YohanN7 ( talk) 13:16, 14 November 2014 (UTC)
Moved to Talk:Cantor's diagonal argument/Arguments.
Can a better example for bijection be provided? The example given, "Hence, the set {banana, apple, pear} has the same cardinality as {yellow, red, green}." lists three fruits that exist in more than one of the colors given. Mjpollard ( talk) 14:17, 6 November 2016 (UTC)
The "independence" section is just a list of references, which should be moved to the history section. The independedence section should have an outline of the actual proof instead, Wikipedia is supposed to contain facts rather than just references. Can anyone knowledgeable make a go at doing this ? — Preceding unsigned comment added by 194.80.232.19 ( talk) 08:49, 5 May 2017 (UTC)
Is saying that really equivalent to the continuum hypothesis? There seems to be a bijection between the set of real numbers and powerset of the natural numbers here, which would imply the continuum hypothesis is true if they were equivalent. -- AkariAkaori ( talk) 00:03, 10 July 2017 (UTC)
There is a recent study by these authors which has been widely reported in the press, and seems highly relevant to the CH. I'm not a mathematician, so the technicalities are beyond me, and I was hoping to find a clear explanation in this article. There's a summary here: https://www.scientificamerican.com/article/mathematicians-measure-infinities-and-find-theyre-equal/ — Preceding unsigned comment added by 86.154.202.49 ( talk) 19:42, 16 September 2017 (UTC)
Should this "hypothesis" be categorized under Category:Conjectures rather than Category:Hypotheses? Or should the definitions of those categories include historical naming that goes against current convention. Perhaps the article should be renamed (with a redirect). Dpleibovitz ( talk) 21:49, 28 December 2017 (UTC)
I have limited math understanding, but can I risk asking a possibly pedantic question: should the sentence "That is, every set, S, of real numbers can either be mapped one-to-one into the integers or the real numbers can be mapped one-to-one into S." be changed to say "infinite set S" or should something about "subsets" be inserted somewhere? 103.27.161.6 ( talk) 11:33, 21 February 2019 (UTC)johnjPerth
[4] Explains Cohen's proof of independence of CH. It looks pretty self contained yet is short enough for one lecture. I'm going to try to read it since other places have made the proof sound very mysterious and I have never understood it. Leaving link here for future reference and for possible use in improving the article. 2602:24A:DE47:BB20:50DE:F402:42A6:A17D ( talk) 05:17, 18 August 2020 (UTC)
I've just seen this Quanta article, which lead me to the papers "Model theory and the cardinal numbers 𝔭 and 𝔱" and "General topology meets model theory, on p and t", and then "Cofinality spectrum theorems in model theory, set theory and general topology". All of this is way over my head, mathematically. I'm uncertain about exactly what the proof of 𝔭 = 𝔱 actually means, and its relationship to the topic of this article. Does this now-proven hypothesis have a name, and what is the relationship of this proof to the continuum hypothesis? -- The Anome ( talk) 13:18, 31 December 2020 (UTC)
I just came across this article about a new mathematical proof implying that the hypothesis is true. But I don't understand the subject matter well enough to add it to the page in a sensible manner, perhaps someone else can take a gander?
Here's the article that might be worth including: https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/
Cheers Creapuscularr ( talk) 13:42, 6 August 2021 (UTC)
Since this conjecture is "a conjecture which can be disproved using a counterexample", if this conjecture is in fact false, we can use this counterexample to disprove it, thus the conjecture is decidable, and if the conjecture is undecidable, then it must be true, thus, if we can prove that this conjecture is undecidable, then we can prove that this conjecture is true, which is a contradiction, like Goldbach conjecture (a counterexample is an even number ≥4 which cannot be written as sum of two primes), Fermat Last Theorem (a counterexample is an integer n≥3 and positive integers x, y, z such that xn + yn = zn), Riemann hypothesis (a counterexample is a complex number z such that Re(z) ≠ 0, Im(z) ≠ 1/2, but ζ(z) = 0), and four color theorem (a counterexample is a graph which cannot be drawn using ≤4 colors), a counterexample of continuum hypothesis is a subset S of R such that Card(S) > Card(N) and Card(S) < Card(R). 2402:7500:916:25D6:DD5D:1928:6166:9583 ( talk) 09:25, 20 August 2021 (UTC)