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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.
7
votes
1
answer
399
views
continuity points of elementary embeddings from $0^\sharp$
Suppose $0^\sharp$ exists, and let $\langle \alpha_i : i \in \text{Ord}\rangle$ be the Silver indiscernibles for $L$. Let $j : L \to L$ be the embedding generated by mapping $\alpha_n$ to $\alpha_{n+ …
- 18.7k
13
votes
2
answers
555
views
Is categoricity retained when reducing the language?
Suppose $\mathcal L \subseteq \mathcal L’$ are first-order languages, $\kappa$ is a cardinal, and $T’$ is a theory in $\mathcal L’$ that is $\kappa$-categorical. Let $T = T’ \restriction \mathcal L$. …
- 18.7k
5
votes
Accepted
Assuming a transitive set model of ZF
Most likely the author is referring to the following fact:
The Reflection Scheme proves each instance of the following scheme: Let $\Delta$ be a finite subset of the axioms of ZF. ZF proves that th …
- 18.7k
8
votes
Accepted
Can there be an upper bound on definability of cardinal numbers in ZF?
Yes. This is an easy consequence of the reflection theorem. The point is that we can reason about definability with respect to a small collection of formulas within a model.
Put all formulas of len …
- 18.7k
4
votes
1
answer
303
views
bijections and order types
Suppose $\kappa$ is an infinite cardinal and $\alpha$ is an ordinal of cardinality $\kappa$. Is it possible to find a bijection $f : \kappa \to \alpha$ such that for all $x \subseteq\kappa$, $\mathrm …
- 18.7k
6
votes
1
answer
302
views
Internal vs. external definability of inner models
Suppose $\kappa$ is an inaccessible cardinal. Is the following situation consistent?
There is $p \in V_\kappa$ and a formula $\phi(x)$ such that there is exactly one $M \subseteq V_\kappa$ such tha …
- 18.7k
14
votes
3
answers
483
views
For ideals, does normal imply countably complete?
The following little question has bugged me for a while.
Suppose $Z \subseteq \mathcal P(X)$. We say an ideal $I$ on $Z$ is normal when it is closed under diagonal unions, which means that if $\{ A_x …
- 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
6
votes
0
answers
273
views
A strengthening of Chang's conjectures
In the Handbook of Set Theory, Foreman has (essentially) the following proposition (3.9):
Suppose $\kappa_n>...>\kappa_0$ and $\lambda_n>...>\lambda_0$ are regular cardinals and $(\kappa_n,...,\ka …
- 18.7k
2
votes
Accepted
Two-cardinal diamond principles and saturation of the nonstationary ideal
I just came across this, but probably you resolved this question long ago. In any case, to address your questions in order as asked,
1) Shioya has an article on this. Let me know if you'd like me to …
- 18.7k
7
votes
1
answer
420
views
Skolem Hulls in $H_{\omega_2}$
I put this on stack exchange over a week ago with no answer, so let's try here.
Consider a model of the form $\mathfrak{A} = (H_{\omega_2}, \in, \lhd, f_0, f_1, ...)$, some expansion of $H_{\omega_2} …
- 18.7k
8
votes
Accepted
Forcings that are not equivalent to Levy collapse
First note that in the special case $\kappa = \omega_1$, countably closed implies countably directed-closed.
Answer 1: No. In any model of CH, there is a countably closed, $\omega_2$-c.c. forcing t …
- 18.7k
5
votes
1
answer
246
views
Amalgamation via elementary embeddings
Can there exist three transitive models of ZFC with the same ordinals, $M_0,M_1,N$, such that there are elementary embeddings $j_i : M_i \to N$ for $i<2$, but there is no elementary embedding from $M_ …
- 18.7k
13
votes
1
answer
572
views
End-extending cardinals
Let us say a cardinal $\kappa$ end-extending if there is a function $F : V_\kappa^{<\omega} \to V_\kappa$ such that:
(a) If $M \subseteq V_\kappa$ is closed under $F$, then $M \prec V_\kappa$.
(b) If …
- 18.7k
9
votes
1
answer
500
views
Solovay’s model
Solovay proved that if $\kappa$ is inaccessible, then if we adjoin a generic $G \subseteq \mathrm{Col}(\omega,{<}\kappa)$, then in the extension, every set of reals in $L(\mathbb R)$ is Baire- and Leb …
- 18.7k