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The crux of the matter is that (people claim) we have evidence that PA is consistent, but we do not have similar evidence that CH is true. Note that "true in the standard model of set theory" is (basically) synonymous with "true." Why is this? Well, let's begin with: why do we believe PA is consistent? Usually, actually, a stronger assertion is made: PA is ...

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As Noah points out, in the context of $\sf PA$ we have a unique "very nice" model which has very nice properties. $\Bbb N$, we can show that any well-founded model of $\sf PA$ is isomorphic to it, and we know that this model exists, if we assume a sufficient meta-theory. So in the context of arithmetic, we can say "the standard model" and confuse between ...

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I don't think the question is unclear. You want to know if most mathematicians still work with set theory, roughly as formulated by Cantor, and you want to know if most mathematicians worry about issues raised by using the axiom of choice. For any reasonable definition of "most," the answer is that most mathematicians still work with set theory, roughly as ...

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In my experience, I never seen a single mathematician even mention the Axiom of Choice during lecture/talk. If they choose elements in every open cover, then they are perfectly okay with that, and never even make a big deal about it. But after two months of choosing elements in open covers, they decide to give a proof of Tychonoff theorem, and announce ...

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No. $2\omega$ is a limit ordinal and infinite, but not initial as it is still countable.

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As you mention, addition of ordinals is not commutative. And you gave the example $$\omega=1+\omega \neq \omega +1$$ Based on that you get $\omega^\omega \cdot \omega = \omega^{\omega+1}$ while $\omega \cdot \omega^\omega = \omega^{1+\omega}=\omega^\omega$. And those last two ordinals are different. More precisely \omega^\omega < ...

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I think that you’ve misinterpreted Definition $\mathbf{8.18}$. I’ll quote it and the sentence that precedes it. In the context of closed unbounded and stationary sets we use the phrase for almost all $\alpha\in S$ to mean that the set of all contrary $\alpha\in S$ is nonstationary. Definition $\mathbf{8.18.}$ Let $S$ and $T$ be stationary subsets of ...

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See this post where Monroe explained this. Let $cf(\lambda) = \theta < \lambda$. I think the problem is missing the assumption $2^{< \lambda} > \lambda$. It is easy to check that this forcing is $\theta$-closed (so it preserves all cardinals $\leq \theta$) and that it collapses $\lambda$ to $\theta$. As you noted, if $\lambda$ is a strong limit, ...

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Here are two contributions, one regarding complex analysis, the other number theory. But first some information to OPs question regarding Cantor and transfinite induction. Historical Notes: The following is mainly based upon The Search for Mathematical Roots 1870-1940 by I. Grattan-Guinness. The author mentions in chapter 3: Cantor: Mathematics as ...

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It is way too late to post an answer but let me take a crack at it anyways. So let $M$ be an inner model of $\text{ZF}$. We want to show that there exists a least inner model $M[X]$ of $\text{ZF}$, where $X\subseteq M$ and such that $M\subseteq M[X]$ and $X\in M[X]$. We use relative constructibility over sets. For every $\alpha\in \text{ORD}$, define ...

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