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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.

122 votes

Axiom of choice, Banach-Tarski and reality

There are two ingredients in the Banach-Tarski decomposition theorem: The notion of space, together with derived notions of part and decomposition. The axiom of choice. Most discussion about the the …
Andrej Bauer's user avatar
  • 48.8k
104 votes

Au revoir, law of excluded middle?

You make a couple of basic mistakes in your question. Perhaps you should correct them and ask again because I am not entirely sure what it is you are asking: Topos theory does not "freely use $P \lo …
Andrej Bauer's user avatar
  • 48.8k
81 votes

Set theory and Model Theory

Your worries arise from asymmetry between how you view ordinary mathematics and how you view logic and model theory. If it is the business of logic and model theory to provide foundations for the res …
Andrej Bauer's user avatar
  • 48.8k
60 votes

Consequences of technically proving anything in Coq (on at least Linux) exploiting a bug?

For the innocent observers, let me explain what joro did. He has tricked Coq into thinking that True = False is a theorem by providing an external piece of code (i.e., something that Coq does not chec …
Andrej Bauer's user avatar
  • 48.8k
59 votes
Accepted

Function extensionality: does it make a difference? why would one keep it out of the axioms?

I am going to draw heavily from Github discussion on HoTT book pull request 617. There are different kinds of equality. Let us say that equality is "intensional" if it distinguishes objects based on …
Andrej Bauer's user avatar
  • 48.8k
54 votes
Accepted

What is... a grossone?

I do not understand what the bounty on this question is for, as it seems to me that the other answers were already rather devastating. Here is a semi-reasoned technical answer. According to G. Lolli …
Andrej Bauer's user avatar
  • 48.8k
46 votes
Accepted

Choice vs. countable choice

Here is one explanation of why countable choice is not problematic in constructive mathematics. For this discussion it is useful to formulate the axiom of choice as follows: $(\forall x \in X . \ …
Andrej Bauer's user avatar
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40 votes

A Model where Dedekind Reals and Cauchy Reals are Different

With classical logic or countable choice Cauchy and Dedekind reals coincide. Therefore we must look at a model of intuitionistic mathematics without countable choice, such as a topos of sheaves over a …
Andrej Bauer's user avatar
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39 votes
Accepted

What is some current research going on in foundations about?

It is quite difficult to answer this question comprehensively. It's a bit like asking "so what's been going on in analysis lately?" It is probably best if logicians who work in various areas each answ …
39 votes
Accepted

Applications of nonconstructive mathematics

In general it is very difficult to be sure that a theorem cannot be constructivised in some form that preserves its applicability. As you will notice most of the answers offered have comments attestin …
38 votes

Logic in mathematics and philosophy

It is easier to list the differences than similarities between the two kinds of logic. Mathematical logic is the branch of mathematics that studies mathematical activity. It has all the usual propert …
Andrej Bauer's user avatar
  • 48.8k
36 votes
Accepted

Is Bauer–Hanson’s result “there is a topos where the Dedekind reals are countable” novel?

[Update 2024-04-15: The preprint The countable reals is now available.] Please allow me to list some basic observations that might clear up things. I work constructively (without excluded middle) and …
Andrej Bauer's user avatar
  • 48.8k
36 votes

What do we gain with higher order logics?

The debate about first-order and higher-order logics is a bit of a religious issue. There are many ways to argue one way or the other: first-order logic has very nice meta-theoretical properties, hig …
Andrej Bauer's user avatar
  • 48.8k
33 votes
Accepted

What can be expressed in and proved with the internal logic of a topos?

In general the internal language of a topos can only express those statements that make sense in every topos. In essence, this limits you to something like bounded Zermelo set theory, without global m …
Andrej Bauer's user avatar
  • 48.8k
32 votes

In What Way are Set Theorists' 'Experiences' in the CH Worlds Flawed, if Any?

I find it useful to make an analogy between what set theorists and geometers do. A geometer has mastered the study of models of a certain theory known as "affine geometry". He eats models of geometry …
Andrej Bauer's user avatar
  • 48.8k

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