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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.

5 votes

Does type theory help us avoid the "defining postulate"?

Indeed, type theory does not need to add new axioms to represent definitions. The basic idea is that the formal language of type theory contains a binding form for terms -- something like: $\mathsf{le …
Neel Krishnaswami's user avatar
17 votes

Is there any relationship between Bourbaki's Epsilon Calculus and Lambda Calculus? Is $\lamb...

Bourbaki's tau-box notation is somewhat insane (e.g., see Adrian Mathias's A Term of Length 4,523,659,424,929), so I'll eventually answer in terms of Hilbert's epsilon-calculus. But first, the laws o …
Neel Krishnaswami's user avatar
5 votes
Accepted

Formal verification of simple equational proofs (as in Universal Algebra...)?

SMT (Satisfaction Modulo Theories) solving is pretty much the go-to technology for this these days, and works shockingly well in practice, often even on undecidable theories. Here are links to a few s …
Neel Krishnaswami's user avatar
20 votes

Proof strength of Calculus of (Inductive) Constructions

IIRC, the calculus of inductive constructions is equi-interpretable with ZFC plus countably many inaccessibles -- see Benjamin Werner's "Sets in Types, Types in Sets". (This is because of the presence …
Neel Krishnaswami's user avatar
16 votes

Logic in mathematics and philosophy

I agree with Timothy and Andrej's answers, and will complement them by suggesting a few books by philosophers and philosophically-inclined logicians which I have found very interesting. I am sure the …
Neel Krishnaswami's user avatar
65 votes
Accepted

Is there any formal foundation to ultrafinitism?

Wikipedia also says that Troelstra said in 1988 that there were no satisfactory foundations for ultrafinitism. Is this still true? Even if so, are there any aspects of ultrafinitism that you can get …
Neel Krishnaswami's user avatar
9 votes
Accepted

Can a typing judgment admit essentially different derivations?

This property is called "coherence", and no, it doesn't always hold. Establishing this property holds for a given semantics of proofs is a proof obligation. An example of when it doesn't arises with …
Neel Krishnaswami's user avatar
12 votes

The use of the word "model" in Mathematical Logic vs the same word in Natural Sciences

No, it's not strange. Understand why it's not strange, and you understand the essence of one of Frege's great innovations in logic: the so-called "linguistic turn", in which he taught us to shift from …
Neel Krishnaswami's user avatar
3 votes

What is the proper name for "compact closed" multiplicative intuitionistic linear logic?

Compact closed categories are models of classical linear logic when tensor and par collapse. As an aside, I'm not sure that the particular resource interpretation you're suggesting genuinely works, s …
Neel Krishnaswami's user avatar
22 votes

Community experiences writing Lamport's structured proofs

From a proof-theoretic point of view, Lamport essentially suggests is writing proofs in natural deduction style, along with a system of conventions to structure proofs by the relevant level of detail. …
31 votes

What is Realistic Mathematics?

At the other side, existence of large cardinals, non-measurable subsets of the reals, etc. are not (immediately) useful for such a study. I don't know about non-measurable subsets, but large card …
Neel Krishnaswami's user avatar
23 votes
Accepted

How do proof verifiers work?

What exactly is the role of type theory in creating higher-order logics? Same goes with category theory/model theory, which I believe is an alternative. Don't think of type theory, categori …
Neel Krishnaswami's user avatar
21 votes

Are real numbers countable in constructive mathematics?

It depends on what you mean. If you are working in classical mathematics, and regard the computable reals to be those real numbers for which a program exists to generate their digits, then they are …
Neel Krishnaswami's user avatar
21 votes

[solved] sequent calculus as programming language

There is no single answer to this question, because of the high degree of nondeterminism inherent in the sequent calculus -- to get a computational interpretation, you need to resolve the ambiguity, a …
Neel Krishnaswami's user avatar
33 votes

The unification of Mathematics via Topos Theory

This statement is true, but there's substantially less than meets the eye to it. Topoi are gadgets which are both models of both a fairly large fragment of logic (typed higher-order logic), and are …
Neel Krishnaswami's user avatar

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