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Results tagged with lo.logic
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user 125703
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.
7
votes
Accepted
Preservation of projective stationarity
It is possible that $\sigma$-closed forcing destroys projective stationarity:
Suppose $\mathcal A$ is a maximal antichain in $\mathrm{NS}_{\omega_1}^+$. Feng-Jech have shown that
$$\mathcal S=\{N\in [ …
- 2,163
3
votes
Accepted
Closure properties of elementary embeddings resulting from generic iterations
It is impossible that $M^{\omega}\subseteq M$ in $V[G]$. The map $j$ is continuous at $\omega_2^V$, i.e. $\omega_2^M=\sup_{\alpha<\omega_2^V} j(\alpha)$ (this can be seen by induction along the length …
- 2,163
6
votes
Accepted
Countable closure of quotient forcing
This does (unfortunately?) not hold in general.
Consider the following (trivial) forcings $P$ and $Q$:
The only thing $Q$ does is generically pick out some $N\leq\omega$, it does this in the following …
- 2,163
7
votes
Accepted
What is the "Prikry–Silver collapse" when CH fails?
If $\mathrm{CH}$ fails then $\mathrm{Col}(\omega, \omega_1)$ does not add a generic for the "Prikry-Silver collapse" $\mathbb P$: Let $\mathbb U$ be $(\mathcal{P}(\omega)/I)^+$ where $I$ is the ideal …
- 2,163
5
votes
Amoeba collapse
This is basically a long comment to your answer saying a bit more about the structure of $\mathbb A(\kappa)$. The forcing $\mathbb A(\kappa)$ is equivalent to $\mathrm{Add}(\kappa, 1)\ast\dot{\mathbb …
- 2,163
4
votes
Accepted
Are the following two "tree properties" equivalent?
The $(\kappa,\lambda)$-STP and $(\kappa,\lambda)$-SSTP are equivalent for any uncountable cardinals $\kappa\leq\lambda$: Let $\mu<\kappa$ and $(L_\gamma)_{\gamma<\mu}$ a sequence of thin $(\kappa,\lam …
- 2,163
4
votes
Accepted
Highly improper forcings
If we allow only for separative forcings $\hat{\mathbb{P}}$ in the definition of rudeness then all non-trivial separative $\sigma$-closed forcings are rude. This does not answer your question literall …
- 2,163
15
votes
A Löwenheim–Skolem–Tarski-like property
Here is a lower bound that improves Joel's by a bit. If the reflection property of the post holds at $\kappa$ then there is some measurable $\lambda<\kappa$ with
$$V_\lambda\models``\text{there is a p …
- 2,163
1
vote
closure of separative quotients
Let me give a concrete example for the first question and a consistent example for the second question.
First, consider the partial order $\mathbb P$ consisting of atoms $a_n$ for $n<\omega$ and in ad …
- 2,163
3
votes
Accepted
Relative null-ness
$\newcommand{\scope}{\mathrm{Scope}}$
$\newcommand{\res}{\upharpoonright}$
The edit makes it seem like already the question of whether incompatible null sets can exists is of interest to you, I hope t …
- 2,163
3
votes
Accepted
Strategic vs. tactical closure
Consistently, there is a $\sigma$-strategically closed forcing which is not $\sigma$-tactically closed. Such a forcing is constructed (through forcing) by Jech-Shelah in
Jech, Thomas; Shelah, Saharon, …
- 2,163
9
votes
Iteration of $\aleph_2$-properness
No. The construction of Mitchell which is described in the paper by Laver and Shelah as indicated in the comments by Golshani can be modified to produce a counterexample to your question. I had to ans …
- 2,163
15
votes
Accepted
Why do we need the comparison lemma?
If you take a step back and squint your eyes, inner model theory is basically the theory of a big measuring stick that measures the vague notion of "strength of natural set theoretical theories". The …
- 2,163