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Results tagged with lo.logic
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user 7206
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.
8
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1
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269
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What is the consistency strength of "Singular worldly that is inaccessible in an inner model"?
In short, what can we say about the consistency strength of "$\kappa$ is a singular worldly and inaccessible in an inner model"?
Clearly, $0^\#$ exists since we have a singular cardinal which is regul …
- 39.8k
6
votes
0
answers
259
views
Forcing Martin's Axiom without cardinal arithmetic
We know that if $\kappa>2^{\aleph_0}$ and $\kappa^{<\kappa}=\kappa$, then there is a c.c.c. forcing which forces $\sf MA+2^{\aleph_0}=\kappa$. Traditionally, we even start with $\sf GCH$, which simpli …
- 39.8k
13
votes
1
answer
708
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Who introduced the notation for $\beth$ numbers and when?
Georg Cantor, when developing the basics of set theory, noted that there are two ways to increase cardinality: power sets and successors (or, in modern terms, the Hartogs operation).1
Eventually the n …
- 39.8k
8
votes
1
answer
259
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Example of a distributive forcing which is entirely improper
One of the standard examples of a forcing which is distributive but not proper (equiv. not closed) is a club shooting into a stationary co-stationary set.
But that forcing is $S$-proper for the statio …
- 39.8k
14
votes
1
answer
799
views
What is the "Prikry–Silver collapse" when CH fails?
We all know and love Cohen reals, and we can (and often do) define the Cohen forcing as partial functions $p\colon\omega\to 2$ with finite domain. The Prikry–Silver forcing is defined as partial funct …
- 39.8k
11
votes
1
answer
417
views
Coding the universe into a real over better core models
One of the most incredible results in modern set theory, due to Jensen, is that given any model of $\sf ZFC$, there is a class forcing which adds a real number $r$ and in the extension $V=L[r]$. Moreo …
- 39.8k
4
votes
0
answers
174
views
Sequences of sequences of sequences and elementary embeddings
Suppose that $\kappa$ is the critical point of $j\colon V\to M$, and suppose that $\mathcal F=(F_\alpha\mid\alpha\leq\kappa)$ is a sequence such that for every limit ordinal $\alpha$, $F_\alpha$ is a …
- 39.8k
10
votes
1
answer
671
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Is Collection really stronger than Replacement?
The two powerhouse schemata of set theory are Replacement and Collection:
Replacement. For every definable function $f$ and every set $x$, $f"x$ is a set.
Collection. For every definable relation $R$ …
- 39.8k
6
votes
0
answers
317
views
Temporary destruction of measures in intermediate models
It is a well-known theorem that if $\kappa$ is measurable, then there is a generic extension in which $\kappa$ is no longer weakly compact, but we can force its weak compactness back and recover the f …
- 39.8k
10
votes
0
answers
324
views
What kind of objects can code a universe?
Jensen proved that given $V\models\sf ZFC+GCH$, there is a class generic real $r$, such that $V[r]=L[r]$, and no cardinals are collapsed.
We know that this can be modified such that $r$ is minimal, i. …
- 39.8k
10
votes
1
answer
431
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What is first-order logic with Dedekind-finite sets of variables?
The usual set up of first-order logic is with an infinite reservoir of variables which we can use in formulas. This is one of the annoying reasons why we need to put $\aleph_0$ into the cardinal equat …
- 39.8k
11
votes
1
answer
421
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If all transitive models have the same height, are they all "simple"?
Suppose that $\alpha$ is the unique ordinal for which $L_\alpha$ is a model of $\sf ZFC$. In other words, there is no transitive model of $\sf ZFC$ in which there is a transitive model of $\sf ZFC$.
W …
- 39.8k
14
votes
1
answer
824
views
Is there a minimal inner model for determinacy?
Assume $\sf ZF+AD$. Is there some inner model $M$ containing all the ordinals such that $M\models\sf ZF+AD$ as well?
What if we require $\omega_1$ and/or $\omega_2$ to be computed correctly?
Can we sa …
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10
votes
0
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283
views
How wealthy are canonical inner models?
One of the way a person shows their wealth is by having many diamonds. The same can be said about models of $\sf ZFC$. We can add generic diamond sequences, while preserving the old ones, so in some s …
- 39.8k
6
votes
0
answers
401
views
General theory of the reals in Solovay-like models
Solovay's model is a famous model of $\sf ZF$ where we start in $L$ with $\kappa$ inaccessible, and we collapse all the ordinals below $\kappa$ to be countable, without collapsing $\kappa$ itself, and …
- 39.8k