# Tagged Questions

forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations,...

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### all subsets borel

Assume Martin's axiom plus $\neg CH$. It is well known, via almost disjoint forcing, that every set of reals of size less than continuum is an example of a metric space whose subsets are all ...
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### Does $\mathfrak{P}(A)\cong\mathfrak{P}(B)$ imply $A\cong B$? [duplicate]

Does $\mathfrak{P}(A)\cong\mathfrak{P}(B)\implies A\cong B$ hold for arbitrary sets $A, B$? Notation: We write $\mathfrak{P}(M)$ to denote the power set of a set $M$. $M_1\cong M_2$ is just an ...
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### Is there a (first-order) sentence which admits $(\aleph_2,\aleph_0)$ iff a Kurepa tree exists?

In Chang and Keisler's Model Theory I came across the following theorem (Theorem 7.2.13): Theorem There exists a (first-order) sentence $\sigma$ such that for all infinite cardinals $\alpha$, $\sigma$...
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### Transitive models and CH

The following was asked on stackexchange but I think it also belongs here: http://math.stackexchange.com/questions/1513446/transitive-models-and-ch Suppose $M, N$ are two countable transitive models ...
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### Consistency Strength of “HC is elementary in V[G]”

Let $P$ be the Levy-collapse of the ordinals, so $P$ is a class forcing notion that makes every ordinal countable. Note that since $P$ is weakly homogeneous, for any formula $\phi(\overline{a})$ ...
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### Is Extensionality needed for the incompleteness of very weak set theories?

$ST$ is the weak set theory built upon identity theory and containing the axiom for empty set, the axiom for adjunction and the axiom for extensionality. It is known that $ST$ interprets ...
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### PCF conjecture and fixed points of the $\aleph$-function

Recently Moti Gitik refuted Shelah's PCF conjecture, by producing a countable set $a$ of regular cardinals with $|pcf(a)| \geq \aleph_1.$ See his papers Short extenders forcings I and Short ...
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### Is every ordinal potentially definable?

It is easy to see that, if $V\models\alpha>\omega_1^{CK}$, then $\alpha$ is not recursive in any forcing extension of $V$. The argument goes as follows: The relation "$\Phi_e=r$" is $\Pi^0_2$. ...
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### Large cardinals arising from alternate set theories

My question is whether there are model-theoretic large cardinal axioms associated with $NFU$ - or more generally, with set theories other than $ZFC$. Large cardinal properties generally come in one ...
### Idea behind the proof of consistency of club filter of $\omega_1$ is ultrafilter + ZF + DC
I've been trying to understand Radin Forcing and some of its applications, one of which is the use of it to prove the consistency of ''Club filter of $\omega_1$ is an ultrafilter + ZF + DC''. However, ...