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Results tagged with lo.logic
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user 11145
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
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 …
- 18.7k
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
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 …
- 18.7k
9
votes
1
answer
417
views
Natural set-theoretic principles implying the Ground Axiom
The Ground Axiom states that the set-theoretic universe is not a set-forcing extension of an inner model. By
Reitz, it is first-order expressible and easy to force over any given ZFC model with class …
- 18.7k
7
votes
0
answers
183
views
Interest in the size of ultrapowers in model theory
It seems that in the 60s (at least), there was interest in computing the size of ultrapowers by countably incomplete ultrafilters. For example, given an ultrafilter $U$ on some relatively small set l …
- 18.7k
5
votes
1
answer
154
views
Countable closure of quotient forcing
Let us say that a partial order is "countably closed with infima" if every descending $\omega$-sequence has an infimum.
Suppose $P$ and $Q$ are posets that are countably closed with infima, and for so …
- 18.7k
13
votes
2
answers
593
views
Amoeba collapse
Here is a naive idea for a forcing $\mathbb A(\kappa)$, for an inaccessible cardinal $\kappa$. Conditions are pairs $(P,p)$, where $P \in V_\kappa$ is a partial order and $p \in P$. We define the or …
- 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 …
- 18.7k
5
votes
1
answer
157
views
Preservation of stationary sets by Mitchell forcing quotients
It is well-known that Mitchell's forcing $\mathbb M$ for the tree property at $\omega_2$, which turns a weakly compact $\kappa$ into $\omega_2$ while adding many reals, is a projection of adding $\kap …
- 18.7k
7
votes
1
answer
219
views
Preservation of projective stationarity
A set $S \subseteq [\kappa]^\omega$ is called projective stationary if for every stationary $A \subseteq \omega_1$, and every algebra $F : \kappa^{<\omega}\to\kappa$, there is $z\in S$ such that $z$ i …
- 18.7k
9
votes
0
answers
305
views
Is there a relationship between the $\Omega$-conjecture and choiceless large cardinals?
Woodin’s $\Omega$-conjecture is absolute under set-forcing. One proposal to decide it is that some large cardinal hypothesis might refute it. On the other hand, Woodin is seeking extensions of the inn …
- 18.7k
3
votes
1
answer
239
views
Veličković's model game
This question is about the argument for Lemma 3.7 in Forcing axioms and stationary sets (MSN) by Boban Veličković.
He defines a game $G_\alpha$ between two players, playing objects in $H_\kappa$, depe …
- 18.7k
9
votes
0
answers
314
views
Non-closed Neeman forcing
This question is something of a follow-up to this one:
Iterating Neeman's forcing
It regards the work of Itay Neeman, MR3201836.
Neeman formulates his two-type models forcing seemingly in greater gene …
- 18.7k
9
votes
1
answer
286
views
Countably closed end-extensions of elementary submodels
The following is well-known. If $\kappa$ is measurable, $\theta > \kappa$, and $M \prec V_\theta$ has size $<\kappa$, then there is $N\prec V_\theta$ such that $N \supseteq M$, $M \cap \kappa \not= N …
- 18.7k
6
votes
1
answer
227
views
Extension of a sequence of complete embeddings
Suppose for $\langle P_i : i < \omega \rangle$ is a sequence of countably closed partial orders. Suppose for each $n$, there is a complete embedding
$$e_n : \prod_{i<n} P_i \to B(Q),$$
where $Q$ is co …
- 18.7k