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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.
10
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1
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450
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Extending models of topological set theory
$\mathsf{GPK_\infty^+}$ is an alternative set theory in which we have comprehension for formulas which are positive in a certain sense; see the SEP article for more detail (or this MO post, which cont …
- 13.2k
5
votes
0
answers
232
views
Is there a ${\bf 0'}$-computable linear order with "all intervals wild"?
Say that a linear order $L$ is a thicket iff $L$ is infinite, and for all elements $a,b,c_1,...,c_n\in L$ with $a<_Lb$ and $[a,b]_L$ infinite the following are equivalent:
$\{a,b\}\subseteq \bigcup_{ …
- 20.7k
8
votes
2
answers
519
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A continuous notion of realizability
I have been interested in non-classical logics, off and on, for quite a while. This question is probably very basic, and I hope it is not too low-level for MO. My question stems from an attempt to def …
CommunityBot
- 1
12
votes
0
answers
241
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Is there a decidable theory of arithmetic with a non-collapsing quantifier hierarchy?
This question is very close to this old MSE question of mine, which is still unanswered.
Is there an (ideally reasonably-natural!) expansion of the structure $(\mathbb{N};+)$ in a finite language whos …
- 20.7k
4
votes
0
answers
149
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Are the natural powers of two conservatively embedded in $\mathbb{C}$?
This is a followup to this question.
Consider $\mathbb{C}$ as a structure - in the sense of first-order logic - with the graphs of addition and multiplication. Let $\mathcal{X}$ be the substructure wi …
- 10.1k
9
votes
0
answers
211
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Can $\mathbb{C}$ have a "doppelganger" in $L(\mathbb{R})$ with countable automorphism group?
Working in $\mathsf{ZFC}$ + large cardinals (a proper class of Woodins, to be precise), is there a field $F\in L(\mathbb{R})$ such that $V\models F\cong\mathbb{C}$ and $L(\mathbb{R})\models\vert\mathi …
- 3,065
8
votes
1
answer
176
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Weakly compact cardinals in $L$: how long do branches take to appear?
Throughout, we work in $\mathsf{ZFC+V=L+}$ "There is a weakly compact cardinal," $\kappa$ is the first weakly compact cardinal and "tree" means "subtree of $2^{<\kappa}$ of height $\kappa$"
Despite t …
- 10k
7
votes
1
answer
182
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Is Presburger arithmetic in stronger logics still complete?
Originally asked at MSE:
Let $\Sigma=\{+,<,0,1\}$ be the usual language of Presburger arithmetic. Given a "reasonable" logic $\mathcal{L}$, let $\mathbb{Pres}(\mathcal{L})$ be the $\mathcal{L}$-theory …
- 16.6k
46
votes
8
answers
12k
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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 …
- 6,377
5
votes
1
answer
252
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What oracles make finding isomorphism (of finite structures) easy?
Below, all structures are finite, in a finite language, with underlying set an initial segment of the natural numbers. This has been edited to fix errors pointed out by Emil Jerabek in his answer belo …
- 47.9k
6
votes
1
answer
207
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Can we computably escape infinitely many functions (allowing partiality)?
Let $(p_i)_{i\in\omega}$ be a uniformly computable sequence of partial functions (i.e. the partial function $q(i,x)=p_i(x)$ is computable) such that infinitely many $p_i$s are total. Must there be a …
- 20.7k
6
votes
0
answers
217
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Decidably clarifying ordinals
For a computable$^*$ ordinal $\alpha$ (viewed as an $\{<\}$-structure), say that a first-order sentence $\sigma$ in a relational language $\Sigma\supseteq \{<\}$ decidably clarifies $\alpha$ iff $\alp …
- 20.7k
6
votes
0
answers
248
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Whence compactness of automorphism quantifiers?
The following question arose while trying to read Shelah's papers Models with second-order properties I-V. For simplicity, I'm assuming a much stronger theory here than Shelah: throughout, we work in …
- 20.7k
3
votes
0
answers
89
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Can the differential field of d.c.e. reals be nicely construed as a field of functions?
This question is basically a special case of this older question of mine, which is still unanswered.
Let $\mathcal{D}$ be the field of d.c.e. reals; these turn out to be exactly the reals $\alpha$ for …
- 20.7k
21
votes
0
answers
912
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"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 …
- 20.7k