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Results tagged with set-theory
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user 41274
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.
6
votes
Transfer of results from one model of set theory to another
Being an isomorphism between two (first order) structures is a $\Delta_0$-property, and hence, it is absolute for transitive models.
Thus, being isomorphic is a $\Sigma_1$-property, and it is upwards …
- 1,688
10
votes
Accepted
Paradoxical Decompositions
A positive answer is proved in S. Wagon's book "The Banach-Tarski Paradox", Theorem 13.2. Specifically, the statement proved there is:
Con(ZF) $\leftrightarrow$ Con(ZF + DC + GM),
where GM is the ex …
- 1,688
7
votes
1
answer
576
views
Does OCA imply $2^{\aleph_0}=\aleph_2$?
Is it known whether Todorcevic's Open Coloring Axiom implies $2^{\aleph_0}=\aleph_2$?
The only consistency proofs for OCA that I know are the following:
1) PFA implies OCA (and also $2^{\aleph_0}=\a …
- 1,688
4
votes
1
answer
520
views
Iterated ultrapowers of L
If there exists a measurable cardinal, we can generate a sequence of iterated ultrapowers $\{Ult_U^\alpha(V)\}_{\alpha\in ON}$. If $0^\sharp$ exists, i.e. if there exists an elementary embedding $j:L\ …
- 1,688
7
votes
1
answer
255
views
Generic filters of inverse limits
Maybe this doubt is silly, but I do not understand the final step of the proof of Lemma 5.2 in Hamkins' paper Fragile measurability, Journal of Symbolic Logic 59 (1994) 262-282.
There, $\mathbb P_\la …
- 1,688
11
votes
1
answer
832
views
Erdős cardinals and ineffable cardinals
In Cantor's Attic it is stated that an $\omega$-Erdős cardinal is a stationary limit of ineffable cardinals, and Jech book is given as a reference, but I cannot find this result in that book. I have s …
- 1,688
13
votes
1
answer
859
views
Ultrafilter theorem and translation invariant measures
The usual Vitali construction of a non-Lebesgue measurable set generalizes to a proof that there are no (non-trivial) translation invariant measures on $\mathcal P\mathbb R$.
On the other hand, there …
- 1,688
10
votes
1
answer
343
views
On the definition of the $\alpha$-iterable cardinals
I am reading the paper Ramsey-like cardinals II by Victoria Gitman and Philip Welch (Journal of Symbolic Logic, vol. 76, no. 2. pp. 541-560, 2011) and maybe I am missing something.
According to the d …
- 1,688
3
votes
1
answer
303
views
Invariants of category in Polish spaces
Consider the invariants appearing in the Cichoń's diagram: $add(\mathcal I)$, $cov(\mathcal I)$, $non(\mathcal I)$, $cof(\mathcal I)$, where $\mathcal I$ is either the ideal of null sets for the Lebes …
- 1,688
8
votes
1
answer
168
views
Does being special on a club imply being special?
Let $T$ be an Aronszajn-tree, $C\subset \omega_1$ a club set and $f:\bigcup\limits_{\alpha\in C}T_\alpha\longrightarrow \mathbb Q$ a strictly increasing function (where $T_\alpha$ is the $\alpha$-leve …
- 1,688
6
votes
1
answer
237
views
On the definition of $\alpha$-proper poset
I am reading Uri Abraham's chapter on Proper Forcing in the Handbook of Set Theory and I have a quite trivial question on the definition of $\alpha$-proper forcing. Since there are many equivalent def …
- 1,688
3
votes
2
answers
419
views
Is not SH + not CH consistent?
I guess that $\lnot$SH + $\lnot$CH is consistent, but I have not found this question discussed anywhere. Is there any relatively simple model of $\lnot$SH + $\lnot$CH?
- 1,688
10
votes
1
answer
450
views
Erdős cardinals and $0^\sharp$
It is well-known that if the Erdős cardinal $\kappa(\omega_1)$ exists, then $0^\sharp$ exists, but what if $\kappa(\lambda)$ exists for a limit ordinal $\omega_1^L\leq \lambda<\omega_1$? Does this sti …
- 1,688
7
votes
2
answers
829
views
Is the forcing relation defined for mathematical formulas?
Meta-matematical formulas of the language of set-theory (which are not sets, but just sequences of signs) should not be confused with mathematical ones (i.e. formulas coded as sets, e.g. finite sequen …
- 1,688
4
votes
0
answers
161
views
What is the meaning of restricting a Boolean value to a subalgebra?
$\require{AMScd}$
I am studying the proof that the ordering principle does not imply the axiom of choice in Jech's book "The Axiom of Choice" (Section 5.5). Let $P$ be the set of finite partial functi …
- 1,688