Talk:Inaccessible cardinal

message: 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)
replies: text: 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)
replies: text: Suppes explicitly counts \aleph_0 as an inaccessible (). — Tobias Bergemann 18:30, 26 June 2006 (UTC)
thread_title:  Is aleph-null inaccessible? 

message: 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)
thread_title: Rename or merge

message: 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)
replies: text: 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)
replies: text: 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)

text: 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)
replies: text: Yes. One could make a weak analogy between α-inaccessibles and multiples of \omega^{\omega^{\alpha}}. JRSpriggs 09:26, 28 May 2006 (UTC)
thread_title:  recursive definition of α-inaccessibility incomplete? 

message: 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)
replies: text: "∞" 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 Vκ. JRSpriggs 02:08, 5 June 2006 (UTC)
replies: text: I see.  Would you object to adding that to the article? -lethe talk [ +] 02:26, 5 June 2006 (UTC)

text: Done. And I already read that talk when you guys posted it. JRSpriggs 02:36, 5 June 2006 (UTC)
thread_title:  what does ∞ mean? 

message: It's used in the second sentence and not defined or linked to.  --Michael C. Price talk 13:39, 26 June 2006 (UTC)
replies: text: 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)
replies: text: 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)
replies: text: Thanks. The corresponding HTML entity ℵ is apparently not widely supported. —Tobias Bergemann 15:15, 26 June 2006 (UTC)
thread_title:  what does  ω  mean? 

message: 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)
replies: text: 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)
thread_title:  can a cardinal k be k-hyper-inaccessible? 

message: *"Vκ 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 Lκ 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)
replies: text: 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 Vα; 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)
replies: text: 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)
replies: text: Thanks for replies. --Acepectif (talk) 01:03, 16 January 2008 (UTC)
thread_title:  What is a standard model? 

message: Is there any way to make it simple for a high schooler or a teenager? The Potchero (talk) 13:00, 13 November 2022 (UTC)
replies: text: 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)
replies: text: Even worse. Finite things that are too large are already enough counter-intuitive, for example ZFC does not determine BB(745)! Also, although countable infinity is the smallest infinity, it can serve as the cardinality of a model of ZFC (if ZFC is consistent)! Inside the model there are as many sets as we want, but viewed outside the model everything is just countable! 129.104.241.193 (talk) 15:21, 10 May 2024 (UTC)
replies: text: In a way I sometimes think countable infinity is more intuitive than finite things that are too large. At least in the sense that more familiar objects have the former size than the latter. :) Double sharp (talk) 18:56, 10 May 2024 (UTC)
replies: text: Of course ;) 129.104.241.193 (talk) 22:28, 12 May 2024 (UTC)
thread_title:  Can we simply this? 

message: Where ZFC2 is the second-order ZFC, and κ is an inaccessible cardinal. The property is stated in the answer to this MSE question. Where could we find some reference as to cite them in this article? 129.104.241.193 (talk) 14:18, 5 May 2024 (UTC)
thread_title:  It is stated that the models of ZFC<sub>2</sub> are precisely ''V''<sub>κ</sub>