Talk:Continuum hypothesis

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CH debunkment

So if 2^w is w1. Then w^w should be bigger than w1 which it is obviously not. 2^5 = 32 but 5^5 is a lot bigger. Which means that CH is false. If CH is true. Then w^w, e0 and EBO are all bigger than w1 which it is not. 2^3 = 8 3^3 = 27 2^4 = 16 4^4 = 256 2^5 = 32 5^5 = 3125. This is obviously why CH is not true. Jaajsisjkakmjikj (talk) 09:08, 16 July 2024 (UTC)

You are confusing cardinal exponentiation and ordinal exponentiation. The ordinal power ω^ω is of course countable. Not to mention that for ordinal exponentiation we have 2^ω = sup{2ⁿ : n ∈ ω} = ω. 129.104.241.231 (talk) 13:47, 21 September 2024 (UTC)

Here the professional set theorists are lying. If you do the logic w^w is not countable and is not as they say the set of all sets of w size alphabet of 0 to w (or w-1) many characters, since then the definition of ordinal exponentiation would have 2 definitions, one for the finite cases and another for the infinite, since saying say 10^7 means all sets of 10 character alphabet of up to 7 characters doesn't give the size 10^7 many one level lower sets/atoms.

In reality different definitions of exponentiation have different results, countable or not if the first operand is w, or second, increasing first operand lengthens set from the beginning, or end. Increasing second operand lengthens result set from end, or beginning so changing result is w or is larger Victor Kosko (talk) 02:01, 22 September 2024 (UTC)

CH is of particular interest

In my eyes, CH is of particular interest because, for a math learner who has never been taught/doesn't care about set theory or mathematical logic, they would likely take the axioms of ZFC and their consequences for granted, and CH would likely be the first proposition to challenge their intution. I mean, Con(ZFC) is also a statement independent of ZFC, but who other than set theorists cares about it? BB(745) is also independent of ZFC, but that's a far too advanced topic and only a few people care about it. However, CH is much more natural, and it would definitely appear once one learned that there are more reals than natural numbers. That's also why I like better the formulation "all uncountable subsets of $\mathbb {R}$ are equinumerous to $\mathbb {R}$" than "$2^{\aleph _{0}}=\aleph _{1}$": few people do care about $\omega _{1}$ ($\aleph _{1}$) even if it has a place in general topology. 129.104.241.231 (talk) 13:37, 21 September 2024 (UTC)

Doesn't the existence of Vitali sets (a result of AC) already challenge intuition?

Also, is there actually anything at the undergraduate level that depends on taking a position on CH? I can't actually think of anything that would seem particularly natural to include in a pretty normal sequence of courses, at least if we're not talking about someone who decided to specialise in set theory early. As Joel D. Hamkins recently wrote, though, this would probably have been different had nonstandard analysis become the first standard framework for rigourising the calculus. (Though, disclaimer: that is one of the reasons why I'm a GCH fan.) Double sharp (talk) 15:28, 21 September 2024 (UTC)

I don't want to be a spoilsport but the talk page is supposed to be for discussing improvements to the article. 29.104.241.23, you might ask a question at WP:RD/Math, if you can think of something that can be phrased in the form of a question and can potentially be answered with references. I love seeing set-theory questions at the refdesk. --Trovatore (talk) 19:19, 21 September 2024 (UTC)

Doesn't Solovay's result require an upper bound on cardinality?

The section Independence from ZFC contains this sentence:

"A result of Solovay, proved shortly after Cohen's result on the independence of the continuum hypothesis, shows that in any model of ZFC, if $\kappa$ is a cardinal of uncountable cofinality, then there is a forcing extension in which $2^{\aleph _{0}}=\kappa$."

But the uncountable cofinality of such a 𝜅 must be ≤ $2^{\aleph _{0}}$, because otherwise 𝜅 > $2^{\aleph _{0}}$, and so 𝜅 ≠ $2^{\aleph _{0}}$.

If this is right, I hope that someone knowledgeable about this subject will fix this omission. — Preceding unsigned comment added by 98.36.148.11 (talk • contribs)

See Forcing (mathematics). "in any model of ZFC" refers to a situation where the model is a set, not the actual universe. So an "uncountable cardinal" Κ is only uncountable in the model, not in the actual universe. So you are mistaken. JRSpriggs (talk) 14:18, 23 November 2024 (UTC)