Talk:Inaccessible cardinal

Is aleph-null inaccessible?

Anonymous Coward: Hey, it sure seems to me that aleph-null fits the description of an inaccessible cardinal. Am I missing something?

No. $\aleph _{0}$ would satisfy the requirements for an inaccessible cardinal; that's why the definition specifically talks only about kardinals κ > $\aleph _{0}$. -- Schnee 05:59, 30 Sep 2004 (UTC)

This is a usage question. A few authors count $\aleph _{0}$ as an inaccessible, and even as a measurable. I think Drake does; don't have a copy here to check. --Trovatore 01:55, 14 July 2005 (UTC)

Suppes explicitly counts $\aleph _{0}$ as an inaccessible (Suppes, Patrick (1972) [1960]. "§ 8.3 Axioms Which Imply the Axiom of Choice". Axiomatic Set Theory (unabridged and corrected republication ed.). Dover Publications. p. 251. ISBN 0-486-61630-4.). — Tobias Bergemann 18:30, 26 June 2006 (UTC)

Rename or merge

By default, inaccessible means strongly inaccessible. The current page should be moved to "weakly inaccessible", and "strongly inaccessible" should be put here. Or better, the pages should be merged. --Trovatore 01:40, 14 July 2005 (UTC)

---

Merge complete. Result could still use some polishing--the two paragraphs on consistency strength could be usefully turned into one. --Trovatore 02:58, 14 July 2005 (UTC)

Consistency with ZFC

In fact, it cannot even be proven that the existence of strongly inaccessible cardinals is consistent with ZFC (as the existence of a model of ZFC + "there exists a strongly inaccessible cardinal" can be used to prove the consistency of ZFC)

I find this confusing. The existence of a model of ZFC + anything can be used to prove the consistency of ZFC, since such a model is also a model of ZFC. Josh Cherry 14:13, 31 July 2005 (UTC)

Good point. See fix. --Trovatore 15:03, 31 July 2005 (UTC)

OK, thanks. I'm not saying this is wrong, but I still don't get the argument. Proving that the existence of inaccessible cardinals is consistent with ZFC would, it seems from this, only show that the consistency of ZFC is consistent with ZFC, i.e., that if ZFC is consistent then ZFC + consis(ZFC) is consistent. Can't we prove that? Josh Cherry 15:14, 31 July 2005 (UTC)

No. If you could, then ZFC would prove that ZFC is consistent, contradicting Second Incompleteness. --Trovatore 15:31, 31 July 2005 (UTC)

Um, so I was slightly off here. If you could prove that Con(ZFC)-->Con(ZFC+Con(ZFC)), then ZFC+Con(ZFC) would prove its own consistency (and therefore be inconsistent). That's only minutely better or more plausible than ZFC itself being inconsistent. --Trovatore 16:01, 31 July 2005 (UTC)

As is the case for the existence of any inaccessible cardinal, the inaccessible cardinal axiom is independent of the axioms of ZFC.

I think there is something wrong about this. For the inaccessible cardinal axiom to be independent of ZFC it has to be consistent with it, but its relative consistency cannot be proved (in ZFC). It can only be shown that the negation of that axiom is consistent with ZFC. So what does exactly mean independet here? (Godelian (talk) 22:34, 17 April 2010 (UTC))

It means that no contradiction can in fact be proved from ZFC+"there exists an inaccessible cardinal". It does not mean (and it is not in fact true) that ZFC proves that no such contradiction can be proved.

It is true that there is often a sort of (usually unspoken) convention that, when a mathematician asserts a statement as fact, he/she implies that the statement is provable from ZFC. Here we have an example where that convention becomes cumbersome and is discarded. --Trovatore (talk) 04:55, 18 April 2010 (UTC)

Re Godelian: some authors use "independent" to just mean "unprovable", and that may be the source of this sentence. — Carl (CBM · talk) 13:52, 18 April 2010 (UTC)

Background: Let I be the statement "there is an inaccessible cardinal". It's a standard point that any proof that "Con(ZFC) implies Con(ZFC + I)" cannot be formalized in ZFC, because ZFC+I proves Con(ZFC), and so this would mean ZFC+I proves Con(ZFC+I), which is impossible.

Most authors I have seen take this as evidence that it is not possible to rigorously prove Con(ZFC) implies Con(ZFC+I) in the metatheory; for example Hrbacek and Jech say "... it is not possible to prove that inaccessible cardinals are consistent with ZFC by constructing an appropriate model...", citing the incompleteness theorem. Fraenkel, Bar-Hillel, and Lévy make an even stronger claim: "it is impossible to give a convincing proof" that Con(ZFC) implies Con(ZFC+I); they again cite the incompleteness theorem.

Hrbacek and Jech do go on to say, "The assumption of existence of inaccessible cardinals requires a `leap of faith' (...) but some intuitive justification for the plausibility of making it can be given." They then describe the standard informal proof of consistency of inaccessibles, namely that if we take any model of set theory then the class of all ordinal numbers of that model would be an inaccessible cardinal if only it was a set; so assuming that we can embed any model of set theory as an initial segment of another model gives Con(ZFC+I). The ability to do such extensions can be justified on the "multiple universe" viewpoint, which says that On can never be completed, which is not the same as the "V = all sets" viewpoint. The difficulty in making the proof rigorous is that the extension model will be so much larger than the base model, so it seems impossible to describe the extension from a point of view inside the base model. — Carl (CBM · talk) 12:55, 18 April 2010 (UTC)

That is very interesting, thanks for the details. I find now this use of 'independent' more clear whithin this context (it still sounds a little odd to me, but that's perhaps because I'm not used to it).(Godelian (talk) 15:20, 18 April 2010 (UTC))

While you wrote this, I edited the article to try to clarify things. I hope it's clearer now. — Carl (CBM · talk) 15:43, 18 April 2010 (UTC)

Yes, that sounds much better. Godelian (talk) 19:55, 18 April 2010 (UTC)

Godelian, I'm curious: You say you "find this usage of independent ... odd", but from the other things you write, I don't think it's really the usage that you object to, but more the epistemology. Your question was worded as one about what the phrase means, but all your comments have been more about how we can warrant the assertion that it's true. Do you in fact think there's any ambiguity about what it means, or is your question really about how one can know that it's true? --Trovatore (talk) 20:10, 18 April 2010 (UTC)

I did find the usage of 'independent' weird. If it means that neither the proposition in question nor its negation can be formally proved in ZFC, then the phrase was certainly not true, but as you have pointed out it was used here to mean something different. Yet to avoid ambiguity I would prefer to use 'unprovable', and point out explicitly any evidence of relative consistency. On the other hand, I was not aware that such kind of evidence could be offered by means of heuristic considerations as Hrbacek and Jech expose, and that caught my attention as well. Godelian (talk) 21:18, 18 April 2010 (UTC)

Wait a minute — we have to get this straight. It absolutely does mean that neither the proposition nor its negation can be formally proved in ZFC, and that fact absolutely is true. If that's how you read it, then you read it correctly; there is no "weird" usage that you need to understand. --Trovatore (talk) 07:11, 19 April 2010 (UTC)

Ok now I'm lost. We have on one hand that it cannot be shown by methods formalizable in ZFC that the existence of inaccessible cardinals is consistent with ZFC. How can it then be true that the existence of inaccessibles can be formally proved in ZFC? It doesn't make any sense to me. Godelian (talk) 23:22, 19 April 2010 (UTC)

Well, that shouldn't make sense, because it isn't true. The existence of inaccessibles cannot be proved in ZFC. Still not seeing what the difficulty is. --Trovatore (talk) 00:39, 20 April 2010 (UTC)

Sorry, my mistake; where it says 'the existence of inaccessibles can be formally proved' should have said 'negation of the existence of inaccessibles cannot be formally proved'. That is, how can it be true that the negation of the existence of inaccessibles cannot be formally proved in ZFC? Godelian (talk) 01:49, 20 April 2010 (UTC)

Well, it can be true just because there isn't any ZFC proof that there are no inaccessibles. I still don't see why this is problematic. Is there something else you're really asking? --Trovatore (talk) 01:53, 20 April 2010 (UTC)

I guess what I'm asking is how one can be sure that there isn't any ZFC proof that there are no inaccessibles, and if it is just possible that such a proof has not yet been found. Godelian (talk) 02:10, 20 April 2010 (UTC)

OK, that's a legitimate question. In some sense it's "possible" that there's a proof in Peano Arithmetic that 0=1, and that that proof has just not yet been found. However very few workers in the field consider this a serious possibility.

Probably a few more consider it a serious possibility that ZFC might prove the nonexistence of inaccessibles. But I think it's fair to say that it's not a mainstream worry, at least among set theorists.

Why, you might ask, is it not a mainstream worry? That's a much more complicated question. Part of the reason is that immensely more powerful large cardinal axioms are in regular use, and appear to present a coherent picture of the mathematical world. --Trovatore (talk) 02:19, 20 April 2010 (UTC)

Ok, that is really interesting and addresses my concern. Now I don't have much experience with large cardinal axioms as I have with Peano arithmetic and then I couldn't give '¬I' the same status as '0=1'. Although Con(PA) is something I'm ready to believe, I really can't feel the same about Con(ZFC+I) or even Con(ZFC) for epistemic reasons; while I think PA really models the real world, I can't say the same of ZFC+I or ZFC, and even the axiom of infinity (though formally provable independent), looks suspicious to me. Obviously these concerns are not mathematical, but I'd still prefer not to use 'independent' literally as it's been used in the assertion. Of course, that's only my opinion, most probably not shared by experts on the field. Godelian (talk) 03:00, 20 April 2010 (UTC)

The confusing thing for newcomerts is that the reasons to believe that ZFC does not disprove the existence of inaccessibles are not as strong as the reasons to believe ZFC does not prove their existence. It's certainly the case that few-to-none among the set theory community feel that ZFC might disprove the existence of inaccessibles. But this is not because of a finitary relative consistency proof, it's just because of developed intuition. — Carl (CBM · talk) 11:38, 19 April 2010 (UTC)

Yes, but again, that's not a question of what the assertion means. It's a question of how one can warrant the assertion, and how strong that warrant is. --Trovatore (talk) 19:15, 19 April 2010 (UTC)

But isn't it true that no matter how you warrant it, such a "proof" of relative consistency is not formalizable in ZFC? Godelian (talk) 23:22, 19 April 2010 (UTC)

Yes, that is true. That doesn't change anything I've said. I sense that there's still some disconnect here and I'm not sure what it is. You do understand that a statement (such as the relative consistency of the existence of inaccessibles) can be true, without being provable in ZFC, right? --Trovatore (talk) 00:30, 20 April 2010 (UTC)

I do agree that there are true statements not formally provable in ZFC, what I fail to see is that Con(ZFC+I) is such a statement. It is certainly not provable in ZFC (even assuming Con(ZFC)), in particular it is impossible to construct a model (from ZFC alone) in which there are inaccessibles. Then how can you warrant in a convincing way that Con(ZFC+I) is true? Shouldn't this appeal to something more than ZFC and then require, citing H&J, some 'leap of faith'? Godelian (talk) 01:24, 20 April 2010 (UTC)

Ah, well that's a whole nother discussion. The way you phrased your question initially was in terms of what the assertion meant, not whether it was true, or whether there was sufficient evidence to believe that it's true. Do you now understand what the assertion means, as opposed to whether you buy that sufficient evidence has been given for it? --Trovatore (talk) 01:40, 20 April 2010 (UTC)

I see; so the term 'independence' just means exactly that. The question is now whether there is enough evidence supporting that independence that allows to assert such a statement. Godelian (talk) 02:10, 20 April 2010 (UTC)

Just to make sure I don't mislead anyone, the argument I sourced to Hrbacek and Jech is very old, dating back to the early 20th century. I just used them as a reference for the sake of convenience, because they explicitly discuss the issue of relative consistency of inaccessibles. — Carl (CBM · talk) 22:33, 18 April 2010 (UTC)

In arithmetic, I believe that one can show in a systematic way that any finite subset of the axioms is consistent, but lengthier proofs are needed for larger subsets. So at a meta-mathematical level one can see that PA implies Con(PA) without it being derivable within the theory itself. Similarly, I suspect that one can show from ZFC that if S is any finite subset of ZFC, then ZFC |- Con(S+I) in a systematic way. Thus at a meta-mathematical level we can establish that ZFC implies Con(ZFC+I) even though no single proof within ZFC can show that. JRSpriggs (talk) 10:17, 20 April 2010 (UTC)

Other descriptions

Is the first weakly inaccessible cardinal the same as $\aleph _{\aleph _{\ddots }}$? I've seen references to this, but it seems singular. Jim Apple 11:18, 18 August 2005 (UTC)

Well, depends on what you mean by that notation. Every weakly inaccessible cardinal is a fixed point of the ℵ function. I.e. if κ is weakly inaccessible then κ = ℵ_κ. But it's not the first fixed point, which is the limit of the ω-sequence

$\aleph _{0},\aleph _{\aleph _{0}},\aleph _{\aleph _{\aleph _{0}}},\ldots$

You're quite right that this limit has cofinality ω --Trovatore 21:28, 18 August 2005 (UTC)

In fact, the first weakly inaccesible cardinal τ isn't the κ'th fixed point of $\aleph$for any κ < τ, right? Jim Apple 06:05, 19 August 2005 (UTC)

Right again. --Trovatore 17:43, 20 August 2005 (UTC)

Why isn't the first fixed point of $\aleph$ inaccesible? Which requirement is missing?--lion 21.2.2006

It's not a regular cardinal; its cofinality is ω, as you can see by the existence of the ω-sequence above that approaches it. --Trovatore 20:55, 21 February 2006 (UTC)

recursive definition of α-inaccessibility incomplete?

The article defines α-inaccessibility to mean that the set of smaller β-inaccessibles is cofinal, for all β<α. This recursive definition seems incomplete to me. Don't we also need to define a base for the recursion? I would guess we should define 0-inaccessible to mean regular limit cardinal? -lethe ^{talk +} 00:55, 21 May 2006 (UTC)

In my definition, "For any ordinal α, a cardinal κ is α-inaccessible iff κ is inaccessible and for every ordinal β < α, the set of β-inaccessibles less than κ is unbounded in κ (and thus of cardinality κ, since κ is regular).", the base is "κ is inaccessible", i.e. an uncountable regular strong limit cardinal. When α=0, the other part of the right-hand side, which refers to β, contributes nothing. So 0-inaccessible is the same as inaccessible. JRSpriggs 08:20, 21 May 2006 (UTC)

I think I somehow failed to see that phrase "κ is inaccessible". It makes sense now, thank you for explaining. -lethe ^{talk +} 13:47, 21 May 2006 (UTC)

You are welcome. At the risk of belaboring the obvious, let me expand my explanation. α+1-inaccessible means an α-inaccessible limit of α-inaccessibles. And when λ is a limit, λ-inaccessible means α-inaccessible for all α < λ. JRSpriggs 06:13, 22 May 2006 (UTC)

On the contrary, your expansions are quite welcome, because it's an unfamiliar concept that I'm still digesting. Stated loosely, a number is 1-inaccessible if there are an inaccessible number of inaccessible numbers below it. And so on up the recursive ladder. -lethe ^{talk +} 15:01, 22 May 2006 (UTC)

Yes. One could make a weak analogy between α-inaccessibles and multiples of $\omega ^{\omega ^{\alpha }}$. JRSpriggs 09:26, 28 May 2006 (UTC)

what does ∞ mean?

In the sentence "This is a relatively weak large cardinal axiom since it amounts to saying that ∞ is 1-inaccessible in the language of the next section.", what does the symbol ∞ mean? -lethe ^{talk +} 04:22, 4 June 2006 (UTC)

"∞" is a symbol for the class of all ordinals. That is, it refers to the least ordinal which is not an element of your model, i.e. it is κ if your model is V_κ. JRSpriggs 02:08, 5 June 2006 (UTC)

I see. Would you object to adding that to the article? -lethe ^{talk +} 02:26, 5 June 2006 (UTC)

By the way, you might be interested to read the discussion about that section which I had last week with User:Trovatore at his talk page. -lethe ^{talk +} 02:28, 5 June 2006 (UTC)

Done. And I already read that talk when you guys posted it. JRSpriggs 02:36, 5 June 2006 (UTC)

what does ω mean?

It's used in the second sentence and not defined or linked to. --Michael C. Price ^talk 13:39, 26 June 2006 (UTC)

I have removed the inline definitions of limit cardinal, regular cardinal and uncountable, linking to the respective pages instead. I actually think this makes the article more readable. I also replaced the reference to ω (the first infinite ordinal number) with references to ℵ₀ (alef-null), the first infinite cardinal number. —Tobias Bergemann 15:04, 26 June 2006 (UTC)

Thanks. BTW the aleph symbol displays as a square on my browser, so I changed it to the form used at the beginning of this talk page: $\aleph$ --Michael C. Price ^talk 15:12, 26 June 2006 (UTC)

Thanks. The corresponding HTML entity ℵ is apparently not widely supported. —Tobias Bergemann 15:15, 26 June 2006 (UTC)

can a cardinal k be k-hyper-inaccessible?

can a cardinal k be k-hyper-inaccessible? —The preceding unsigned comment was added by 80.41.84.38 (talk) 17:44, 6 April 2007 (UTC).

Yes, or rather, it is not known to be inconsistent that such cardinals exist. See also Mahlo cardinal#Example: showing that Mahlo cardinals are hyper-inaccessible. But, of course, κ cannot be κ+1-hyper-inaccessible. JRSpriggs 08:20, 7 April 2007 (UTC)

What is a standard model?

• "V_κ is a standard model of ZFC which contains no strong inaccessibles."
• "Similarly, either V contains no weak inaccessible or, taking κ to be the smallest ordinal which is weakly inaccessible relative to any standard sub-model of V, then L_κ is a standard model of ZFC which contains no weak inaccessibles."

These two sentences are from the Models and consistency section of this article. What is meant by a 'standard model' here? I can't find the definition at model theory article, and it seems obvious that it is not the physical standard model. --Acepectif (talk) 06:50, 15 January 2008 (UTC)

It probably means a model whose membership relation is the true membership relation. I'm not sure there's a completely *ahem* standard meaning for the term, though. Some people might use it to mean a model that's a V_α; others might use it to mean any model whose membership relation is wellfounded. From a model whose membership relation is wellfounded you can recover an isomorphic one whose membership relation is the true one, so the first and third definitions aren't very different -- this is by the Mostowski collapsing lemma; do we have an article on that? --Trovatore (talk) 07:25, 15 January 2008 (UTC)

It's at Mostowski collapse lemma; I'll redirect the other name. I also was concerned about the terminology "a standard model" at some point. I accept "the standard model" as commonly understood terminology, but I would prefer to say "well-founded transitive" if that is what is meant by "a standard model". But I have never had strong enough feelings to follow up on it. — Carl (CBM · talk) 20:14, 15 January 2008 (UTC)

Thanks for replies. --Acepectif (talk) 01:03, 16 January 2008 (UTC)

Truth & Satisfaction

The final section reads:

"It is provable in ZF that ∞ satisfies a somewhat weaker reflection property, where the substructure (Vα, ∈, U ∩ Vα) is only required to be 'elementary' with respect to a finite set of formulas. Ultimately, the reason for this weakening is that whereas the model-theoretic satisfaction relation $\models$ can be defined, truth itself cannot, due to Tarski's theorem."

Do you mean that $\models$ can be defined in ZF? I thought that model-theoretic satisfaction was basically a `paraphrasing' of the truth predicate, in the sense that

a satisfies $\phi$ in A iff $A\models \phi (a)$
$\phi (a)$ is true in A iff $A\models \phi (a)$

Actually, this can't be right as I recall seeing a satisfaction predicate used in formulae within models - so, could anybody please explain how model-theoretic satisfaction differs from truth (maybe adding a line on this to the article)? Is truth satisfaction by the "intended" model? Thankyou Dr satsuma (talk) 15:18, 6 March 2009 (UTC)

What we might informally refer to as "satisfaction in V" is the same as truth. The reason that this doesn't let you define truth in the language of set theory is that, in the language of set theory, you can't talk about V. (Well, at least you can't talk about it as an individual — you can define it as a predicate, of course, but that doesn't help). --Trovatore (talk) 18:01, 6 March 2009 (UTC)

$A\models \phi (a)$ holds if and only if the formula resulting from restricting the quantifiers in $\phi (a)\,$ to $A\,$ is true. This is not a problem. It would only be a problem, if $A\,$ were a proper class instead of a set.

It is OK to talk about the truth of smaller weaker structures. It is only a problem to talk about the truth of yourself or something which includes you. JRSpriggs (talk) 04:51, 7 March 2009 (UTC)

Ok, thankyou. Can V model ZFC? I thought that just V$_{\kappa }\,$ (inaccesible kappa) models ZFC, and V is smaller than this, isn't it?

Incidentally do you think mathematicians consider V to be the "intended model" of ZFC? I once asked my tutor this and was told that this wasn't the case because you can prove an upward Lowenheim-skolem result for proper class theory so that V is elementarily equivalent to lots of bigger, equally "intended" models of ZFC. However, nothing makes V any more intended for ZFC than V$_{\kappa }\,$, for any inaccessible $\kappa$.

But this seems to me a bit unmotivated since V is, after all, the set-theoretic hierarchy and contains all and only sets. (I hope this isn't too much of an off-the-wall question.) Dr satsuma (talk) 19:25, 7 March 2009 (UTC)

I disagree with your tutor; I think you have the correct response. V is not a completed totality at all, which is why it doesn't make sense to talk about going "upward" from it. If it were a completed totality, it would have to be a set; there's nothing else to distinguish it from sets.

In a way this is reminiscent of Errett Bishop's notion that the natural numbers, though existing individually, cannot be gathered into a completed whole. I think Bishop just stops too soon. While it is possible to imagine such an ontology, there is no obvious reason that it must be so limited, so why limit it? But for the sets, there is a reason, because if you could gather all sets together into a completed whole, then it would have to be a set, and it would then follow you could continue beyond there, contrary to hypothesis. --Trovatore (talk) 19:47, 7 March 2009 (UTC)

I would add, in response to your questions about whether V is a model of ZFC, and whether it's the intended model, that both answers are "yes", subject to the minor annoyance that V doesn't exist (as an object; it exists as a predicate).

The notion that all V_κ, for κ inaccessible, are equally "intended" models of ZFC, is clearly wrong, and a simple example suffices to refute it. Suppose we write κ₀ for the smallest inaccessible, and κ₁ for the second. Are V_κ0 and V_κ1 of equal status? Clearly not — the first model thinks there are no inaccessibles, which by hypothesis is not true. The second model is superior to the first one, in that it at least knows about the first inaccessible. --Trovatore (talk) 08:06, 10 March 2009 (UTC)

OK, yes that makes sense. My point about Lowenheim-Skolem was that if V is, in some way, the `smallest' proper class then it should be possible to prove in vNGB proper class theory that there are many larger proper class models of ZFC too, and I guess those models would also think that inaccessibles exist. The reason why I'm asking this is not so much from any mathematical uncertainties (when I think of set theory, I don't worry which model I'm working in), but a philosophical one: we (well, Hilary Putnam's realist) want our language to fix an interpretation of mathematical terms. It looks like V is a good candidate, for several philosophical reasons, not least because smaller transitive models than V can assert that some set has cardinality K when the object of the domain witnessing this fact is actually of cardinality <K because the domain of the model is <K, and the domain is transitive. (This is just Skolem's paradox, which while not mathematically problematic, does have nontrivial philosophical ramifications.) However, *if* there are upward copies of V then it doesn't end up being unique, so big problem for the position claiming it uniquely interprets usage of mathematical terms.

This was intended as a possible response to the claim that V is the intended model, because having assumed the existence of V, nothing is barring the existence of other proper classes. Which leads me to ask - what do you mean by V existing as a predicate? Would this need second order logic? Dr satsuma (talk) 16:03, 12 March 2009 (UTC)

Can we simply this?

Is there any way to make it simple for a high schooler or a teenager? The Potchero (talk) 13:00, 13 November 2022 (UTC)

I doubt it. Infinities are characterized by properties which do not occur in the finite world of everyday life. The larger the infinity, the more violently it conflicts with our intuition. Do you have a specific question? JRSpriggs (talk) 15:36, 13 November 2022 (UTC)