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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.
2
votes
When is $\mathbb{L}$-rank definable in inner models of $\mathbb{V} = \mathbb{L}$?
The following isn't an answer to your question, as it's only one example, but I'm not able to make comments here.
It can be definable over $M$, and quite simple. (However, note that for the model I'm …
- 10k
7
votes
The relation between $\Pi_1$-Foundation and $\Sigma_1$-Foundation over Kripke-Platek set theory
The original version of this answer contained a mistake, as pointed out by Hanul Jeon.
The main issue was that I was assuming too much of the $\omega$ of the model. This version makes a weaker claim.
…
- 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
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
5
votes
Accepted
Are there outer models $V \subset W$ of $L$ such that $V$ is "far" from $L$ but $W$ is "not ...
One example: do some class forcing over $L$ to produce a proper class $V\models$ ZFC + "there is no set-forcing which forces that $V[G]=L[x]$ for a set $x$". Now do Jensen coding forcing over $V$ to p …
- 10k
7
votes
Accepted
Can there exist such a sequence of elementary embeddings of the universe to itself?
It was pointed out by Monroe that if $\lambda$ is a limit and $j:V_\lambda\to V_\lambda$ is elementary and $j_n$ is the $n$th iterate, i.e. $j_0=j$ and $j_{n+1}=j_n(j_n)$, and $\kappa_n=\mathrm{crit}( …
- 10k
4
votes
Accepted
Is this sequence of embeddings possible?
Work in ZF+AC$_\omega$. Suppose $V_\theta$ is inaccessible. Suppose $j,k:V_{\theta+1}\to V_{\theta+1}$ are elementary and $\mathrm{crit}(j)=\mathrm{crit}(k)=\kappa$ and $\kappa_\omega(j)<k(\kappa)$. H …
- 10k
5
votes
Accepted
Would automorphisms cause nested subset-hood?
Regarding the first question, no:
Let $\beta=j(\alpha)$ and $\beta+\gamma=\alpha$. Define a version of the cumulative hierarchy with the empty set replaced with $\beta$. (That is, let $V_0’=\beta$, $V …
- 10k
5
votes
Accepted
Does the axiom schema of collection imply schematic dependent choice in ZFCU?
(Remark: Sam Roberts pointed out to me an error in the first version of the proof I posted earlier -- there was no reason that the desired automorphisms exist, as there was no specification of them on …
- 10k
5
votes
Accepted
Can a model of $V\neq L$ contain a class giving the $L$-ordering on all its sets?
This answer just considers the version of the question for transitive models $M$.
Under a reasonable interpretation of the order $<_L$ of constructibility, there is no such transitive model $M$. Howev …
- 10k
7
votes
Accepted
Can Z + Ranks + Successor cardinals + Ordinal inaccessibility be equal to ZF?
This theory doesn't prove Replacement (assuming the consistency of an inaccessible, at least).
Assume ZFC + $\kappa$ is inaccessible and force over $V$ to add $\kappa$-many Cohen reals (i.e. the forc …
- 10k
6
votes
Accepted
Is this compactness property for "satisfiability on $\mathbb{R}$" consistent?
It looks to me like under ZFC, $\mathbb{R}$-satisfiability is not (consistently) $(\omega_2,\omega_3)$-compact. To see this, we'll emulate your argument above for $\mathbb{R}_{\mathbb{Z}}$. So basical …
- 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
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
8
votes
Accepted
Can Deep Choice entail Axiom of Choice?
It proves AC. For this, recall it's enough to see that for every ordinal $\alpha$, $\mathcal{P}(\alpha)$ is wellorderable, and for that it's enough to see that $\mathcal{P}(\mathcal{P}(\alpha))\backsl …
- 10k