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Results tagged with lo.logic
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user 160347
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.
17
votes
A Löwenheim–Skolem–Tarski-like property
Here is an upper bound:
Suppose $\kappa$ is $2$-fold supercompact. Then the property holds at $\kappa$. (Recall that $2$-fold supcompactness means that for each ordinal $\lambda$, there is $j:V\to M$ …
- 10k
14
votes
Accepted
If $L_\alpha \vDash ZFC$, then do we have $L_{\alpha+1} \vDash \alpha\text{ is inaccessible}$?
Yes. The elements of $L_{\alpha+1}$ are exactly those subsets of $L_\alpha$ which are definable from parameters over $L_\alpha$. But $L_\alpha\models\mathrm{ZFC}$, so from here we can just use the usu …
- 10k
14
votes
Accepted
Detecting uncountable cardinals in $(\mathbb{R};+,\times,\mathbb{N})$
For the first question (distinct regular cardinals $>\aleph_1$): Force ZFC + MA + $2^{\aleph_0}=\aleph_3$ over $L$ in the usual way (see Jech, Theorem 16.13; note the forcing is ccc and it forces MA + …
- 10k
14
votes
Accepted
Undefinable inner model
Here is an example from a transitive set model of ZFC + "there is a proper class of measurable cardinals". If there is one, then there is a countable one $N$, so fix such an $N$. Fix a sequence $\left …
- 10k
13
votes
Accepted
Is $\mathsf{ZFC+V=L}$ consistently $\omega$-complete?
Claim: $T+$"$T$ is $\omega$-complete" is inconsistent. For suppose it's consistent and now work in a model $V$ of this theory. Let $T^+$ be the resulting completion of $T$ (i.e. the unique theory of …
- 10k
13
votes
Accepted
Can $\mathcal{L}_{\omega_1,\omega}$ detect $\mathcal{L}_{\omega_1,\omega}$-equivalence?
(Working in ZFC.)
No (re $\mathcal{L}_{\omega_1,\omega}$). Suppose it is. Consider the signature $\Sigma$ with just one binary relation symbol $<$. Let $\Sigma',\eta$ witness SED-ness for $\Sigma$.
Le …
- 10k
13
votes
Accepted
Projective well-ordered sets, higher up
The theory $T_0$ = ZFC + "there is an inaccessible $\kappa$ such that every wellorder of a subset of $V_{\kappa+1}$ which is definable over $V_{\kappa+1}$ from parameters has length $<\kappa^+$" is eq …
- 10k
13
votes
Accepted
Is "every infinite set of strictly subnumerous sets is supernumerous to its union" equivalen...
Yes, this is like Tarski's result that if $\mathfrak{c}^2=\mathfrak{c}$ for every infinite cardinal $\mathfrak{c}$, then AC holds.
In fact, it suffices to suppose that $\mathfrak{c}\mathfrak{d}=\mathf …
- 10k
12
votes
Accepted
Logics detecting their own equivalence notions, take two: $\mathcal{L}_{\omega_2,\omega}$
(Working in ZFC.)
$\omega_2$ is not Fraissean. In fact, it is not Fraissean with respect to $\Sigma$, where $\Sigma$ is the signature with a single binary relation $<$.
To see this we use a variant of …
- 10k
12
votes
Can local $0^\#$ exists in L?
(EDIT: Now edited to compute the precise value of $\alpha$.)
@AsafKaragila already answered the question, but this is answering the follow-up question od @Reflecting_Ordinal in the comments to Asaf's …
- 10k
11
votes
Do $X$ and $Y$ have the same cardinality if their families of finite subsets do?
If $\mathrm{ZF}$ is consistent, then it does not prove the statement.
Let $\left<x_n\right>_{n<\omega}$ be generic
for the finite support product of an $\omega$-sequence of Cohen forcings. Let $C=\{x_ …
- 10k
11
votes
Accepted
Is this relation about elementary embedding transitive?
EDIT: If the codomain $N$ is allowed to be illfounded, then the answer is yes.
(Here if $\kappa=\mathrm{crit}(j)$ then this will mean that $\kappa\subseteq N$, but it might be that $N$ is illfounded a …
- 10k
10
votes
Accepted
Can we have mutual elementary embeddability between distinct transitive sets?
There are examples of this where $M,N$ are also models of ZFC, in the following paper, which is joint with Monroe Eskew, Sy Friedman and Yair Hayut: https://arxiv.org/abs/2108.12355
Thus, you certainl …
- 10k
10
votes
Can there be an embedding j:V → L, from the set-theoretic universe V to the constructible un...
Theorem 1. If $V$ is a non-trivial set generic extension of $W\models\mathrm{ZFC}$ then there is no
$j:V\to W$ as described (i.e. with $x\in y\iff j(x)\in j(y)$ for all $x,y\in V$).
(So in particular, …
- 10k
10
votes
Accepted
Minimum transitive models and V=L
Yes, I claim you can in fact get one whose minimum model is a set forcing extension of a segment of $L$. Let $L_\alpha$ be least modelling ZFC. Let $\mathbb{P}=\mathbb{P}^{L_\alpha}$ be Jensen's forci …
- 10k