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Results tagged with lo.logic
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user 1384
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.
7
votes
2
answers
1k
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candidate for rigorous _mathematical_ definition of "canonical"?
In this question: What is the definition of "canonical"?
, people gave interesting "philosophical" takes on what the word "canonical" means. Moreover I percieved an underlying opinion that there was n …
- 41.4k
59
votes
8
answers
8k
views
Cauchy reals and Dedekind reals satisfy "the same mathematical theorems"
The succinct question
The conjecture of Birch and Swinnerton-Dyer (to take a random example) mentions L-functions and hence the complex numbers and hence the real numbers (because the complexes are b …
- 41.4k
21
votes
When are two proofs of the same theorem really different proofs
My opinion, and it's only an opinion, is that it would be very difficult to formalise what it means for two proofs to be different. Here's an intuitive reason why. If I give you two proofs of theorem …
- 41.4k
54
votes
2
answers
5k
views
Automatically solving olympiad geometry problems
Warning: I am only an amateur in the foundations of mathematics.
My understanding of this Wikipedia page about Tarski's axiomatization of plane geometry (and especially the discussion about decidabil …
- 41.4k
37
votes
Does anyone still seriously doubt the consistency of $ZFC$?
For decades I was not particularly suspicious about the consistency of ZFC but I was rather surprised about how it had become the standard choice when it contained axioms such as Replacement, which se …
37
votes
Accepted
Is "all categorical reasoning formally contradictory"?
Note: I am not a historian. I'm just guessing as to what prompted the comments.
Here's my guess: if you do set theory naively, in the old-fashioned "anything is a set" way, then you run into Russell' …
- 41.4k
5
votes
Textbook recommendations for undergraduate proof-writing class
I am not so sure of the US system but one of the books we recommend at our university is
Martin Liebeck's "A concise introduction to pure mathematics".
http://www.amazon.co.uk/Concise-Introduction-Pu …
31
votes
Most 'unintuitive' application of the Axiom of Choice?
The fact that there exist non-measurable sets is highly counter-intuitive; the reason we don't find it so is that we've all been conditioned from day 1 to do measure theory very carefully, and define …