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Results tagged with set-theory
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user 11145
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, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
10
votes
2
answers
386
views
Iteration of $\aleph_2$-properness
Let us say a forcing $P$ is proper for a class of models $\mathcal C$, if for large enough regular $\theta$ and $M \prec H_\theta$ in $\mathcal C$ with $P \in M$, every $p \in P \cap M$ can be extende …
- 1,291
12
votes
Accepted
What is the evidence for and against the HOD conjecture?
I believe it is fair to say that the HOD Conjecture has been refuted.
The HOD Conjecture is the statement that the theory ZFC + "There is an extendible cardinal" proves that there is a proper class of …
- 18.7k
14
votes
Why believe in the existence of large cardinals rather than just their consistency?
I think it can be reasonably argued that the large cardinal notions follow a common conceptual pattern, even a semi-formal template. Once we understood the characterization of measurable cardinals in …
- 18.7k
28
votes
3
answers
3k
views
Construction of nonmeasurable sets
I have a history question for which I've had trouble finding a good answer.
The common story about nonmeasurable sets is that Vitali showed that one existed using the Axiom of Choice, and Lebesgue et …
- 11.4k
9
votes
0
answers
162
views
Algebraic structures on spaces of ultrafilters
The space of ultrafilters on $\omega$ has a natural semigroup structure, and ultrafilters that are idempotent in that algebra have seen applications in combinatorics on the natural numbers, for exampl …
- 18.7k
9
votes
1
answer
247
views
Copy of $P(\omega)/\mathrm{fin}$ on $\omega_1$
$\newcommand{\fin}{\mathrm{fin}}$Under what hypotheses does there exist a uniform ideal $I$ on $\omega_1$ such that $P(\omega_1)/I \cong P(\omega)/\fin$? What is the consistency strength?
It follows …
- 11.4k
3
votes
Accepted
Is discriminative choice provable in ZFC?
Yes. Enumerate $F$ as $\langle F_\alpha : \alpha<\kappa\rangle$, where $\kappa$ is a cardinal. Inductively pick $x_\alpha \in F_\alpha$ such that $x_\alpha$ is not $\phi$-equivalent to any $x_\beta$, …
- 18.7k
6
votes
Accepted
Forcing $\neg\square_{\omega_1}$ from a Mahlo cardinal, reference
I'll give the proof, which was told to me by Martin Zeman.
Let $G \subseteq \mathrm{Col}(\omega_1,{<}\kappa)$ be generic, where $\kappa$ is Mahlo. Suppose towards contradiction that $\square_{\omega_ …
- 18.7k
5
votes
Accepted
Reference request: $\kappa^+$-saturated ideal from iterated Cohen forcing on measurable $\ka...
The key here is that you don't have to force beyond $\mathrm{Add}(\kappa,\kappa^+)$ in order to lift the embedding.
Suppose $G \subseteq \mathbb P$ is generic and $H \subseteq \mathrm{Add}(\kappa,\kap …
- 18.7k
6
votes
A reference for forcing projections
I don’t know if there’s a “canonical” writeup, but I taught a master’s course a few years ago and wrote up many details of these things here. But maybe this isn’t useful if you’re looking for somethi …
- 18.7k
3
votes
Accepted
Can a generic ultrafilter over $\mathrm{NS}^+_{\omega_1}$ witness $\omega_1$ is Ramsey-like?
The statement implies that for any $f : [\omega_1]^2 \to 2$, there is an uncountable $S$ such that $f$ is constant on $[S]^2$. This is impossible since $\omega_1$ is not weakly compact. See Jech, Le …
- 18.7k
13
votes
Truth in a different universe of sets?
In contrast to Joel's answer, I would like to point out that the notion of truth-in-a-structure, or whether $A \models \phi$ for a given $A,\phi$, is not so wild and capricious. Although the question …
- 18.7k
17
votes
6
answers
1k
views
Strategic vs. tactical closure
The Banach-Mazur game on a poset $\mathbb P$ is the $\omega$-length game where the players alternate choosing a descending sequence $a_0 \geq b_0 \geq a_1 \geq b_1 \geq \dots$. Player II wins when th …
- 2,163
3
votes
2
answers
339
views
Ultrafilter projections and critical points of factor maps
Suppose $j : V \to M$ is $\lambda$-supercompactness embedding derived from an $\kappa$-complete normal ultrafilter $U$ on $P_\kappa(\lambda)$, $\lambda$ regular. Suppose $\eta$ is an ordinal such tha …
- 8,099
4
votes
Existence of trees with height $\omega$, size $\aleph_1$ and $\aleph_2$ maximal branches
Suppose $T$ is a tree of height $\omega$ with $<2^\omega$-many branches. Then it must be the case that for all $t \in T$, there is $s \geq t$ such that all $x \geq s$ are comparable. Otherwise we co …
- 18.7k