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Results tagged with set-theory
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user 23835
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
PFA and saturated ideals
No. It is a theorem of Shelah that PFA is consistent with the failure of weak Chang's conjecture: there exists $\langle f_i\in \omega_1^{\omega_1}: i < \omega_2+1\rangle$ that is increasing mod $NS_{\ …
- 3,038
6
votes
2
answers
321
views
A question regarding strong cardinals and measure sequence
Let $E$ be a $(\kappa, \lambda)$-extender such that $j: V\to M\simeq Ult(V,E)$ is the corresponding elementary embedding with critical point $\kappa$, $M\supset V_{\kappa+2}$, $M^\kappa\subset M$. Let …
- 3,038
1
vote
A question regarding strong cardinals and measure sequence
I believe the argument could be saved by considering the following ``sub-extender'': For each $\beta<(2^\kappa)^+$, let $a=\bigcup \max\{\beta, \kappa^+\} \cup \bigcup\{a_\gamma: \gamma<\beta\}$ wher …
- 3,038
1
vote
A proof of $ZF \vdash AC^L$
The usual strategy is to define inductively a well-ordering on $L(\alpha)$ for each ordinal $\alpha$. Since under the assumption $V=L$, any set $x \in L (or$ $V), x\subseteq L(\beta)$ for some $\beta …
- 3,038
5
votes
Accepted
How many disjoint compact sets are needed to form a connected compactum?
It is indeed axiom independent. In the Solovay random model (adding $\aleph_2$ random reals over a model of ZFC+GCH) you have $2^\omega=\aleph_2$ and there exists a $\aleph_1$ partition of $[0,1]$ int …
- 3,038
7
votes
0
answers
288
views
Higher dimensional $\Delta$-system lemma
Consider the following statement (called $\Delta^d(\kappa^+, \lambda)$ where $d\in \omega$ and $\kappa<\lambda$ are cardinals): For every $A': [\lambda]^d\to [\lambda]^{\leq \kappa}$, there exist $E\i …
- 3,038
10
votes
1
answer
432
views
Countable support iteration of proper forcings and the tree property
I'm mainly concerned with countable support iterations of proper forcings that add reals of some large cardinal length. It is known that countable support iteration of Sacks forcing/Cohen forcing of w …
- 3,038
2
votes
Accepted
Strongly non-Ramsey order type in polarized partition problems
We can't find such pair of bad order types, aka there indeed is some Erdös-Rado phenomenon in the polarized partitions with respect to linear orderings. Indeed given $\gamma$ number of colors and $\th …
- 3,038
2
votes
Is the fusion argument on trees of uncountable height consistent?
The answer is yes. In fact, more general statements are true. See https://arxiv.org/abs/1704.06827 for more detail (in particular Theorem 3.1).
- 3,038
7
votes
1
answer
305
views
Very weak square and good points
This is probably well known but I'll appreciate pointers to references: Is there any model where for a singular cardinal $\kappa$ of cofinality $\omega$, Very Weak Square holds at $\kappa$ but every …
- 3,038
8
votes
1
answer
334
views
Is the fusion argument on trees of uncountable height consistent?
In the countable context where we are given a perfect subtree $T$ of $2^{<\omega}$ and a sequence of colorings $f_i: T\to 2, i\in \omega$, it is possible to obtain a perfect subtree $T'\subset T$ and …
- 3,038
2
votes
Can there be an almost-special not-fully-special Aronszajn tree?
To supplement this with another example, it is also possible to construct a tree that is $\omega$-distributive and $S$-st-special (in Shelah's terminology) from $\Diamond^*(S^c)$ for some $S$ bi-stati …
- 3,038
3
votes
Accepted
Indecomposable ordinals and pseudointersection
I believe the claim is wrong:
If the claim is right I claim I can show $\alpha\to (\alpha)^2_2$ which is obviously wrong for countable ordinal $\alpha\geq \omega+2$.
Given a coloring $f: [\alpha]^ …
- 3,038
4
votes
1
answer
235
views
Strong partition property + DC + existence of non-principal ultrafilter on $\omega$
It was mentioned after Theorem 30.27 in Kanamori's Higher Infinite that Woodin constructed a model of $DC$ + there exists unboundedly many many $\kappa<\Theta$ such that $\kappa \to (\kappa)^\kappa_{\ …
- 3,038
6
votes
Accepted
End-extending cardinals
Suppose $\kappa$ carries an $\omega_1$-saturated $\kappa$-complete ideal $I$, given $M\prec (V_{\kappa+2},\in , <)$ ($<$ well orders $V_{\kappa+2}$) of size $<\kappa$ containing $I$, we show how to fi …
- 3,038