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Results tagged with lo.logic
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user 4613
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
votes
1
answer
363
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Are these theories of real and complex number biinterpretable?
Let $T_R$ be the first-order theory of real closed fields. This is precisely the theory over the language $\{0,1,+,\times\}$ such that the theorems are the formulas that hold in $\Bbb R$. It can be ef …
- 3,137
21
votes
Nuances Regarding Naturality
The issue here is that the inverses to $V\rightarrow V^{**}$ and $V^*\otimes V\rightarrow \mathrm{Hom}(V,V)$ don't exist in the infinite dimensional case. So in order to show that they exist one has t …
- 3,137
34
votes
1
answer
2k
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Does "every" first-order theory have a finitely axiomatizable conservative extension?
I originally asked this question on math.stackexchange.com here.
There's a famous theorem (due to Montague) that states that if $\sf ZFC$ is consistent then it cannot be finitely axiomatized. Howe …
- 3,137
16
votes
2
answers
1k
views
Is there a universal way to force the Axiom of Choice to be true?
Given a model of set theory $V$ there are various ways to construct a model in which the Axiom of Choice holds, such as Gödel's constructible universe $L^V$ or by using forcing*. I'm wondering if any …
- 3,137
38
votes
4
answers
6k
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Could groups be used instead of sets as a foundation of mathematics?
Sets are the only fundamental objects in the theory $\sf ZFC$. But we can use $\sf ZFC$ as a foundation for all of mathematics by encoding the various other objects we care about in terms of sets. The …
- 3,137