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Results tagged with lo.logic
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user 8133
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.
46
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8
answers
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What are some proofs of Godel's Theorem which are *essentially different* from the original ...
I am looking for examples of proofs of Godel's (First) Incompleteness Theorem which are essentially different from (Rosser's improvement of) Godel's original proof.
This is partly inspired by question …
34
votes
2
answers
3k
views
Ur-elemental surprises
For most of my (mathematical) life, I believed that there was really no essential difference between set theory without urelements and set theory with urelements. However, while that may be true in ce …
- 20.7k
32
votes
1
answer
2k
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Godel on recursion-theoretic hierarchies
At the end of his excellent article, "The Emergence of Descriptive Set Theory" (http://math.bu.edu/people/aki/2.pdf), Kanamori writes:
"Another mathematical eternal return: Toward the end of his …
- 20.7k
27
votes
0
answers
2k
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Supercompact and Reinhardt cardinals without choice
A friend of mine and I ran into the following question while reading about proper forcing, and have been unable to resolve it:
Definition. A cardinal $\kappa$ is supercompact if for all ordinals $\la …
- 20.7k
26
votes
0
answers
1k
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Where do uncountable models collapse to?
Suppose $T$ is a complete first-order theory (in an finite, or at worst countable, language). Given any model $\mathcal{M}\models T$ of cardinality $\kappa$, we can ask whether $\mathcal{M}$ can be ma …
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25
votes
2
answers
1k
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Detecting uncountable cardinals in $(\mathbb{R};+,\times,\mathbb{N})$
For a structure $\mathcal{X}=(X;...)$, say that a cardinal $\kappa$ is $\mathcal{X}$-detectable iff there is some sentence $\varphi$ in the language of $\mathcal{X}$ together with a fresh unary predic …
- 20.7k
24
votes
9
answers
2k
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Self-containing structures
This question is partly inspired by this question: independently of the original context, I'm interested in the general claim* that an ill-founded set theory would represent certain mathematical objec …
22
votes
2
answers
1k
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Gently changing measure
This question was asked and bountied on MSE without answer, so I'm porting it here:
There's an easy way to change the measure of a set of reals by moving to a larger universe: simply make $\mathbb{ …
- 20.7k
22
votes
1
answer
746
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Undetermined Banach-Mazur games in ZF?
This question was previously asked and bountied on MSE, with no response. This MO question is related, but is also unanswered and the comments do not appear to address this question.
Given a topolo …
- 20.7k
21
votes
1
answer
833
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Relative null-ness
Here, "measure" always means Lebesgue measure on $\mathbb{R}$. This question is partly motivated by my answer https://math.stackexchange.com/questions/1444498/is-there-a-categorizaiton-system-for-null …
- 20.7k
21
votes
2
answers
1k
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Antirandom reals
This is a crossposting of https://math.stackexchange.com/questions/1446602/anti-random-reals, which has not gotten any answers; after thinking about the problem, I've become more convinced that it bel …
- 20.7k
21
votes
1
answer
1k
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Is "almost-solvability" of Diophantine equations decidable?
Say that a Diophantine equation is almost-satisfiable iff for each $n\in\mathbb{N}$ it has a solution mod $n$. Trivially genuine satisfiability over $\mathbb{N}$ implies almost-satisfiability, but the …
- 20.7k
21
votes
0
answers
912
views
"Compactness for computability" - does it ever happen?
Throughout, "computable structure" means "first-order structure in a computable language with domain $\omega$ whose atomic diagram is computable."
Say that a computable structure $A$ is compact for co …
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20
votes
1
answer
1k
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Does sine interact equationally with addition alone?
$\DeclareMathOperator\Eq{Eq}\DeclareMathOperator\Th{Th}$Originally asked at MSE without success:
For a structure $\mathcal{A}$ whose signature only contains function and constant symbols, let $\Eq(\ma …
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20
votes
3
answers
2k
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A limit to Shoenfield Absoluteness
Shoenfield's Absoluteness Theorem states that if $\phi$ is any $\Sigma^1_2$ sentence of second-order arithmetic, then $\phi$ is absolute between any two models of $ZF$ which share the same ordinals. T …
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