Set theory

@MartinSleziak what are good book/papers on the history of set theory?

@MartinSleziak I went through bits of that talk, it seems interesting although I detected a few mistakes or things that the speaker say he didn't know but were kind of obvious to me. I also noticed that there was no bibliography slides. It just had references, [Kar20] is presumably my Zornian Functional Analysis write-up. But I was curious about others.

I have changed the feeds with bounties and HQs from messages to ticker feeds.

Can $|A|=|A\times\{0,1\}|$ for a well-orderable $A$ be proved in $\sf ZF$?

mathoverflow.net/a/29960/49381 this answer in particular seems to be an example of the kind I'm looking for

So my professor mentioned as a curiosity that there are some statements $\phi$ whose best known proof goes something like "Start with a model $M$ and build a forcing extension $M[G]$ designed to make $\phi$ true. But now note that $\phi$ is simple enough ($\Pi^1_2$ or lower), so by Shoenfield's absouteness $\phi$ was already true in $M$" but there are no known direct proofs staying inside $M$

Does anybody know examples of such $\phi$'s?

I'm looking for someone familiar with random forcing and in particular with Friedman's paper "a consistent Fubini-Tonelli theorem for nonmeasurable functions" for a couple of basic questions about the tools used in the paper

Quick sanity check: if $(M,\in_M)$ is an ill-founded model of $\mathsf{ZFC}$ then it has an infinite descending $\in_M$-chain in $\mathsf{Ord}^M$, since I can just take $(\mathrm{rank}(x_n))_{n\in_M\omega^M}$ where $(x_n)_{n\in_M\omega^M}$ is an infinite descending $\in_M$-chain, right?

A question which is supposed to be trivial but it's giving me an headache. I have a function $f:[\lambda]^\omega\to\lambda$ and I need to establish $f\in V_{lambda+2}$, but I can only establish $f\in V_{\lambda+3}$...

Sanity check: is the theory "ZFC-extensionality+not extensionality+every set has a unique powerset" consistent relatively to ZFC? If so is there a simple way to produce a model of this theory from one of ZFC?

I'm having some troubles following the proof of existence of $\omega$-Jónsson functions on every cardinal $\lambda$ as presented in Kanamori's book, is anyone here familiar with it (or with other proofs of the same fact?)

I just posted this as a question on main if you're interested to see the answer (hopefully I'll get one!)

Rowbottom's theorem says that infinite homogeneous subsets exist for every $F\colon[\kappa]^{<\omega}\to2$ for measurable $\kappa$ and it is a standard result that in $\mathsf{ZFC}$ for every infinite cardinal $\kappa$ there is $F\colon[\kappa]^\omega\to2$ that admits no infinite homogeneous set in $\kappa$

Is it consistent with $\mathsf{ZF}$ that there is a cardinal $\kappa$ such that for every $F\colon[\kappa]^\omega\to2$ there is an infinite $X\subseteq\kappa$ such that $F\upharpoonright[X]^\omega$ is constant? (an homogeneous set for $F$)

Possibly. His English is, anyway. : )

Theo is also damned near perfect.

Discovering Modern Set Theory: The Basics by W. Just and M. Weese.

I know Martin axioms and the GCH imply the continuum is regular

I have question which axioms in set theory implies the continuum is regular.

Just a reminder - to read MathJax/LaTeX in chat you can use the bookmarklet mentioned in this post on meta or go directly to robjohn's website: math.ucla.edu/~robjohn/math/mathjax.html

What's provable in $\sf ZFC$+"there exist a measurable cardinal" about the quantity of measurable cardinals? For example can there be only one? Or on the opposite side must there always be an unbounded class of measurable cardinals? Or neither?

I ended up asking a short question limited in scope to the relationship between $F\otimes G$ and the product of the associated measures @Martin

So I don't see the analogy with the product of ultrafilters :/

We say that a set is $O$-finite (order-finite, I don't know if there is a name for this property already) if it admits a well order $\lhd$ such that the reverse order $\lhd^{-1}$ is also a well order

Sure, $|\alpha\times\alpha|=|\alpha|$ is a theorem of $\sf ZF$ for $\alpha\ge\omega$ an ordinal

Why is 0=0 included in every diagram of the Consequences?

Is there a model of PA in which Goodstein's theorem doesn't hold?

@MartinSleziak That definitely settles the question with $\Bbb Q$

It's not the same, but in somewhat similar spirit: An order type $\tau$ equal to its power $\tau^n, n>2$ (MathOverflow).

Inspired by a question on main, is there an ordered set $X$, with $|X|>1$, such that $(X\times X,\leq_{\text{lex}})$ and $(X\times X\times X,\leq_{\text{lex}})$ are isomorphic as ordered sets?

I asked it just a few seconds ago, here's a link

Consider the set of aleph fixed point $\{\gamma:\gamma=\aleph_\gamma\}$ and let $\daleth_\alpha$ be the $\alpha$-th element in that set, let $f:\sf Ord\to\sf Ord$ be the function $\alpha\mapsto\daleth_\alpha$. I'm sure that $f$ can't be shown to be normal in $\sf ZFC$, because its fixed points would be weakly inaccessible cardinals, but I'm not sure whether it can be proved that it isn't normal. Is $\daleth_\omega=\sup\limits_{\alpha<\omega}\{\daleth_\alpha\}$?

room mode changed to Gallery: anyone may enter, but only approved users can talk

Anyone here familiar with the Church-Kleene ordinal?

ohhh, wait, in $G\upharpoonright\alpha$ only the values of $G(\beta)$ for $\beta<\alpha$ are needed, so we're getting $G(\alpha)$ in terms of the previous ones

Hi, I'm looking for someone familiar with the transfinite recursion theorem because I don't really understand what it's saying or what's its significance

any idea? For reference this is a step in the proof of the Sierpinski-Erdös duality theorem, if you know other sources discussing it that'd be also appreciated

So, I need to try and show that ZFC+IC proves that the least worldly cardinal is less than the least inaccessible.

The book (Google books preview)

There we get $\kappa$ many subsets of $\omega$ by the generic filter

In introductory courses of set theory, many times the definition of a set doesn't mention the distinctness of the elements. Any ideas why that happens?

The fact that on countable set there is an almost disjoint family of cardinality $\mathbb c$ seems like something that should have many applications.

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I didn't know this existed, otherwise I would have been in here all summer. I've been studying Introduction to Set Theory out of Jech and Hrbacek, and talking to anybody who's seen any real Set Theory (even an introductory amount) would have been nice this whole time.

Enderton seems very good, I'll look more into it and see if it fits my needs! (I want to study some set theory on my own, because I find it extremely interesting)

What is a good introductory book on logic and set theory? I have a copy of Shoenfield's mathematical logic but I find it a bit too harsh for a complete beginner