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Results tagged with lo.logic
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user 33505
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.
19
votes
Accepted
Can you do math without knowing how to count?
A "philosophy of math" tag would have been a good idea.
To answer your question, take a look at Hatry Field's "Science without Numbers". Also, Edward Nelson has developed a theory of "proto-integers" …
- 4,359
16
votes
Accepted
Reverse mathematics of Cousin's lemma
Sam Sanders here, one of the authors of the paper you mention. Thanks for the nice words.
I will answer your questions based on my personal opinion.
You write:
[...] would like to know if it ha …
- 4,359
13
votes
Accepted
Can nonstandard analysis be used to prove results in constructive or computable analysis?
Nonstandard Analysis (NSA) can be used to prove results in computable/constructive analysis; The central notion is $\Omega$-invariance, defined as follows.
[As usual, the set $N$ consists of the st …
- 4,359
10
votes
Set-theoretical reverse mathematics of the reals
TL;DR: A most basic property of $\mathbb{R}$ is that it is not countable, which is surprisingly hard to prove (namely far beyond the Big Five you
mention), as explored in [1, 2, 3].
The longer version …
- 4,359
10
votes
Uniqueness results that follow from CH
Assuming $\textsf{CH}$, a lot of natural fourth-order functionals are computationally equivalent (Kleene's S1-S9) to $\exists^3$. These equivalences do not seem to go through without the former. ([ …
9
votes
Simpler proofs using the axiom of choice
Many examples from set theory are known, but here is a very basic (third-order) theorem from most ordinary mathematics:
"A regulated$f:[0,1]\rightarrow \mathbb{R}$ is bounded", (&)
where 'regulated' m …
9
votes
What can be preserved in mathematics if all constructions are carried out in ZF?
ZF cannot prove the local equivalence between "epsilon delta" continuity and "sequential" continuity for functions on the reals. A small fragment of (countable) AC does suffice for proving the equiva …
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9
votes
Church–Turing thesis for higher order functions
Dag Normann (and myself) have written in many places that:
there is no Church-Turing thesis for computability of (even just) type 2 objects.
As noted here by others, John Longley has explored this the …
- 4,359
9
votes
Interview of Connes, Caramello, and L. Lafforgue about topos theory
My answer to this question would be: perhaps the religious zeal of some of its followers, based on the following experience.
About 12 years ago, the John Templeton Foundation organised a nice meeting …
- 4,359
9
votes
Accepted
Is there a constructive version of internal set theory?
As you have proven (by a well-known construction), one cannot expect to have full Transfer in constructive NSA. For different but related reasons, full Standardisation is off the table, though its re …
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8
votes
Is there a semantics for intuitionistic logic that is meta-theoretically "self-hosting"?
Comment environment was acting funny, so I am writing an "answer". Here is a good place to start:
Harry de Swart's PhD from the University of Nijmegen (the Netherlands) was about this kind of topic:
…
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8
votes
How much of mathematical General Relativity depends on the Axiom of Choice?
My (attempt at an) answer goes in the direction of "2. natural restrictions".
First of all, as noted in the comments, provable in ZF are restrictions of AC to the language of second-order arithmetic. …
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8
votes
Lists as a foundation of mathematics
The original question was:
Can we use lists as the primary notion and build mathematics around it?
My answer is that any such approach runs into trouble very quickly, as can be gleaned from the follow …
- 4,359
8
votes
Did Edward Nelson accept the incompleteness theorems?
Your second question has been properly answered by Emil Jerabek, I would say. Reading some of the comments, I feel I should write the following about your first question:
From talking to Ed Nelson an …
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7
votes
Is there a completeness proof of intuitionistic predicate calculus using Heyting algebra sem...
Harry de Swart's PhD from the University of Nijmegen (the Netherlands) was (explicitly) about this kind of topic. He establishes the completeness of IPC using search trees, in an intuitionistic meta …
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