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Results tagged with lo.logic
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user 26705
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.
16
votes
Is there a theorem whose only known proof uses "$A$ or not $A$" for undecidable $A$
Another example is: Mycielski proved in 1964 under ZF that there is some $A \subseteq \omega_1^{\omega}$ such that the two-player game with payoff set in $A$ is not determined. The proof uses that eit …
- 2,146
2
votes
1
answer
213
views
If M is an inner model containing all the reals, might every game in M be determined in V?
Let $M$ be an inner model (of height $\mathsf{Ord}$) containing all the reals. I am wondering about the consistency strength of the statement "Every game in $M$ is determined in $V$."
MOTIVATION
For …
- 2,146
5
votes
0
answers
147
views
If M is an inner model containing all the reals, might every game in M be "strongly" determi...
QUESTION
Let M be an inner model (of height Ord) containing all the reals. For each $X \in M$, define $S_X = \{x \in X^\omega : x_I \in M \land x_{II} \not \in M\}$. ($x_I$ is the set of plays in $x$ …
- 2,146
11
votes
3
answers
946
views
When is $\mathbb{L}$-rank definable in inner models of $\mathbb{V} = \mathbb{L}$?
Suppose $\mathbb{V} = \mathbb{L}$ and there is a countable transitive model $\mathbb{M}$ of $ZFC$.
Let $\rho$ be the $\mathbb{L}$-rank, i.e. for all $a \in \mathbb{V}$, $\rho(a) = $the least $\alpha …
- 2,146
10
votes
1
answer
365
views
limits of stable theories
Say that a complete theory $T$ is a limit of stable theories if for every $\phi \in T$ there is a stable completion of $\{\phi\}$. (Equivalently, $T$ is the ultraproduct of stable theories.)
Question …
- 2,146
4
votes
Accepted
How many elementary embeddings can there be?
It is a fact (following from the Ehrenfeucht–Mostowski theorem) that for every complete theory $T$ and for every $\lambda \geq |T|$, there is $M \models T$ with $|M| = \lambda$ and $M$ having $2^\lamb …
- 2,146
6
votes
2
answers
636
views
A "Completion" of $ZFC^-$
Let $T_0$ be the set theory axiomatized by $ZFC^-$ (that is $ZFC$ without powerset) + every set is countable + $\mathbb{V}=\mathbb{L}$.
Question 1: Suppose $\phi$ is a sentence of set theory. Must t …
- 2,146
1
vote
A "Completion" of $ZFC^-$
Overnight the following occurred to me...
The answer to Question 2 is negative (with an asterisk), and so the same is true of Question 1. Namely, let $T$ be a set of $\Pi_2$ sentence with $ZFC^- \cup …
- 2,146
2
votes
Accepted
Theories with the infinitary Vopenka property
The answer to the question is yes, assuming $VP$ holds and thus, in the terminology from your earlier question, that $VP(\mathcal{L}_{\omega_1 \omega})$ holds. Namely let $T$ be any unstable theory wi …
- 2,146
8
votes
Accepted
Elementary extensions of direct product
Yes. Let $T$ be the theory of two disjoint groups, in the language $(\cdot_1, \cdot_2, U_1, U_2)$. Note that if $(G_1, G_2) \models T$ then the group operation on $G_1 \times G_2$ is definable without …
- 2,146
4
votes
Accepted
Perfectly transversable theories
The property you are asking for is a very strong condition on $T$. Let met try to rephrase the question more carefully:
The set of countable $\Sigma$-structures with universe $\omega$ is naturally a …
- 2,146
6
votes
Accepted
Models with few types in infinitary logics
Here is a partial answer: consistently, the generalization can fail for all uncountable $\kappa$. Namely:
Suppose $\mathbb{V} = \mathbb{L}$ and let $\kappa$ be any uncountable cardinal. Let $\mathca …
- 2,146
19
votes
0
answers
932
views
What is the Cantor-Bendixson rank of the space of first order theories?
Let $L$ be the language $\{R\}$ containing a single binary relation symbol. Consider the space $S_0(L)$ of complete, first-order $L$-theories. This is a seperable, compact Hausdorff space; what is its …
- 2,146
3
votes
Accepted
Consistency of Weak Diamond with a Weak Version of Martin's Axiom
So I was looking through related questions on this site and the book "Proper and Improper Forcing" by Shelah kept popping up. So I checked it out and the appendix actually resolves the question. I rep …
- 2,146
6
votes
1
answer
387
views
Consistency of Weak Diamond with a Weak Version of Martin's Axiom
If $S \subset \omega_1$ is stationary, then the weak diamond principle $\Phi(S)$ states that for any $F: 2^{<\omega_1} \to 2$, there is a $g: \omega_1 \to 2$ such that for all $f: \omega_1 \to 2$, the …
- 2,146