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Results tagged with lo.logic
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user 10384
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
Does the fixed point lemma / diagonalization require capturing or not?
this trivially leads to inconsistency: We can show that if T ⊢ 𝝍 then T ⊢ ⊥, and likewise if T ⊢ ¬𝝍 then T ⊢ ⊥.
In the case of $\neg\mathrm{Prov}(n)$, applying the diagonal lemma gives one (whe …
- 1,094
5
votes
Is a paraconsistent and provably non-trivial foundation for math possible?
Zach Weber has been working on developing a paraconsistent set theory that can serve as a foundation for mathematics.
- 1,094
14
votes
Are key theorems finitistically reducible?
There are various problems with finitistic reducibility as Simpson develops it; for a survey, see §5.3 of my Stanford Encyclopedia of Philosophy entry on reverse mathematics. I tend to agree that the …
- 1,094
2
votes
Reference Request: Non-Standard Models of PA
Peter Smith has a pretty good handout on Tennenbaum's theorem that I found useful when learning that material. As others have mentioned, Richard Kaye's Models of Peano Arithmetic is the go-to referenc …
- 1,094
5
votes
1
answer
231
views
Attribution of an equivalence of the existence of omega-models of RCA0
There are many well-known equivalences in reverse mathematics between statements of the form "Every set is contained a countable coded $\omega$-model of $T$" and $S$, where $S, T$ are subsystems of se …
- 1,094
5
votes
Equivalences between statements of (seemingly) different order
When we say "with set parameters" I take it that what we really mean is something of the form: the axioms of (e.g.) the $\Sigma^0_1$ induction scheme are the universal closures of all formulas of the …
- 1,094
10
votes
3
answers
1k
views
New research on coding in reverse mathematics?
Coding is obviously a fundamental tool in reverse mathematics, and practitioners take care to both demonstrate the correctness of their coding mechanisms and point out their limitations. Harvey Friedm …
- 1,094
10
votes
Book recommendation introduction to model theory
I quite like Wilfrid Hodges's A Shorter Model Theory (Cambridge University Press, 1997). It covers all the topics you mention, while also tackling a few more advanced ones in the final chapter. The bo …
- 1,094
10
votes
How much choice is needed to show that formally real fields can be ordered?
Let me give an answer from a different perspective. Konrad Swanepoel's accepted answer shows what happens in the general case, for formally real fields of any cardinality. However, it is possible to c …
- 1,094
10
votes
Ultrainfinitism, or a step beyond the transfinite
Perhaps you could take a look at William Reinhardt's paper 'Remarks on reflection principles, large cardinals, and elementary embeddings' (1974). Reinhardt suggests extending the set-theoretic univers …
- 1,094
4
votes
Accepted
Further research on relevant realizability etc
I thought this was an interesting question and so I asked some relevant logicians on Mastodon. Here's a quick summary of the answers, although the short version seems to be "No", with Shawn Standefer …
- 1,094