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Results tagged with lo.logic
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user 31233
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.
3
votes
Explaining the consistency of PRA and ZF from predicative foundations
Last week, I learned from Ulrik Buchholtz that there are proof-theoretic reductions from certain impredicative systems to certain predicative ones. While these fall short of predicatively explaining t …
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9
votes
When can we prove constructively that a ring with unity has a maximal ideal?
If the ring is countable (or the image of a linear well-ordering), then no choice of any kind (not even countable choice) and in fact not even the law of excluded middle is required: There is an expli …
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12
votes
Examples of statements that are valid in every spatial topos
Great question!
One example is Zorn's lemma. Assuming ZL holds in the metatheory, ZL also holds in toposes of sheaves over locales, so in particular in toposes of sheaves over topological spaces. Howe …
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7
votes
2
answers
1k
views
Explaining the consistency of PRA and ZF from predicative foundations
Recently I got interested in predicative foundations, mostly because of Laura Crosilla's work and because Agda employs a predicative type theory.
From the point of view of a predicative foundation to …
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9
votes
0
answers
374
views
Reflection principle for intuitionistic Zermelo–Fraenkel?
The well-known reflection principle for classical Zermelo–Fraenkel states:
For any formula $\varphi(x_1,\ldots,x_n)$ of the language of ZFC with free variables $x_1,\ldots,x_n$, ZFC proves
$$ \f …
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6
votes
Constructively, is the unit of the “free abelian group” monad on sets injective?
Yes! Here is a proof which is slightly different from both your proof and the proof in Mines–Richman–Ruitenberg.
First define the similarity relation on $\mathrm{List}(R \times X)$ as in Mines–Richma …
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11
votes
What does the Zariski topos of $\mathbb{P}^1$ classify?
First note that a morphism $\operatorname{Spec}(A) \to \mathbb{P}^1$ is just given by an element of the "classical projective space" $\mathbb{P}^1(A) = \{ [a:b] \,|\, \text{$a$ is invertible or $b$ is …
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11
votes
Accepted
Constructive proof that a kernel consists of nilpotent elements
This answer provides a scheme how to construct a constructive proof, though I'm still working to actually explicitly extract the constructive proof, so please don't accept the answer just yet. (Update …
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4
votes
constructive Serre classes
Any subclass $\mathcal{C}$ of an abelian category determines a smallest Serre class containing it, by iteratively adding (the zero object and) the object $Y$ for any exact sequence $X \to Y \to Z$ whe …
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41
votes
Accepted
Joyal's construction of the spectrum of a commutative ring
Since I don't know precisely which parts of Lawvere's article you have difficulties with, this answer is a bit a long and tries to give a bit of context. If you want me to be more specific at some poi …
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9
votes
1
answer
449
views
Base change in homotopy type theory
Recall that with the internal language of 1-toposes, we have the nice, basic, and useful result that geometric sequents are stable under base change along geometric morphisms: If $\varphi$ and $\psi$ …
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