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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.

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 …
Ingo Blechschmidt's user avatar
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 …
Ingo Blechschmidt's user avatar
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 …
Ingo Blechschmidt's user avatar
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 …
Ingo Blechschmidt's user avatar
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 …
Ingo Blechschmidt's user avatar
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 $\psi$ …
Ingo Blechschmidt's user avatar
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 …
Ingo Blechschmidt's user avatar
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 …
Ingo Blechschmidt's user avatar
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 …
Ingo Blechschmidt's user avatar
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 …
Ingo Blechschmidt's user avatar
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 …
Ingo Blechschmidt's user avatar