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Results tagged with lo.logic
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user 23835
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
10
votes
0
answers
279
views
Martin's Maximum implies stationary/club Chang's conjecture?
Chang's Conjecture (CC) states: for any $f: [\omega_2]^{<\omega} \to \omega_1$, there exists a set $X\subset \omega_2$ of order type $\omega_1$ such that $|f''[X]^{<\omega}|\leq \aleph_0$.
Martin's Ma …
- 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
8
votes
Accepted
Poset dimension and width (Dilworth's theorem)
For your modified question, namely, does there exist a poset $P$ such that $\dim(P) > \sup \{|A|: A\subset P \text{ is an antichain}\}$, the answer is positive. Due to Laver (An Order Type Decompositi …
- 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
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
Ways to add Aronszajn trees which are neither Souslin nor special
Regarding your comment before Question 3, you mean all such trees with size $<\mathfrak{m}$ are special. In particular, it doesn't say anything about trees of size $\geq 2^{\omega}$. Indeed, $T(\mathb …
- 3,038
5
votes
Accepted
Iterated forcing and the super tree property at $\omega_2$
The answer to the supercompact case is yes. More specifically, in the forcing extension obtained by iterating Sacks forcing of supercompact length, the super tree property at $\omega_2$ holds. This fo …
- 3,038
8
votes
1
answer
338
views
Iterated forcing and the super tree property at $\omega_2$
It is a theorem of Baumgartner and Laver that iterating Sacks forcings of weakly compact length gives rise to the tree property at $\omega_2$. Natural questions (at least for me) are: do we get strong …
- 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
Infima in the Rudin-Keisler ordering
This part only shows the existence of lower bounds, which is not the point. See edit.
This is consistently true for example when the near coherence principle of ultrafilters holds. It says, for any tw …
- 3,038
5
votes
0
answers
239
views
A possible characterization of weakly compact cardinals
Aside from the well-known characterization of weakly compact cardinals in terms of the usual partition calculus, I've been wondering if there are other characterizations that are variants of the typic …
- 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
6
votes
3
answers
239
views
$(\kappa, \kappa, 2)$-saturated ideals?
Is it consistent to have a $(\kappa,\kappa,2)$-saturated ideal $I$ on $\kappa$ that is $\kappa$-complete and $\kappa$ is not weakly compact? Here $\kappa$ is inaccessible. An ideal is $(\kappa,\kappa, …
- 3,038
7
votes
Is it possible for countably closed forcing to collapse $\aleph_2$ to $\aleph_1$ without col...
Here is for the benefit of the curious (and it answers the original question): By a result of Baumgartner-Taylor (Saturation properties of ideals in generic extensions. I) we know $[\omega_2]^\omega$ …
- 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