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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.
3
votes
Translate ZFC statements into systems of Diophantine equations compatibly with geometry
Andrey Bovykin used to work on this kind of topic until ten (?) years ago:
formulate unprovable statements in terms of some kind of polynomial equations, possibly involving quantifiers.
Here you can f …
- 4,359
0
votes
What is the status of Cantor-Schroder-Bernstein in Reverse Math?
The papers [1b, 2] answer your question for the Cantor-Bernstein theorem (CBN) for $\mathbb{N}$, working in Kohlenbach's higher-order Reverse Mathematics ([0]). As is customary, sets are defined via …
- 4,359
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 …
- 4,359
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
4
votes
Accepted
"At most one" versus "at most finitely many"
The answer is positive, assuming extra induction, and a sketch is as follows.
Let $\varphi(X,n)$ be as in (*).
Define an analytic code $A_n$ as follows $X\in A_n\leftrightarrow \varphi(X, n)$.
Use i …
- 4,359
3
votes
Axiomatic system made just for playing
I guess you should take a look at the work of John Conway. In particular, his game of life seems to be exactly what you are looking for, a pioneering part of recreational mathematics.
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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 …
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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
4
votes
Relationship between provable in $RCA_0$ and effectively true
Three (related) approaches that have not been mentioned are as follows:
the meta-theorems from proof-mining (see U. Kohlenbach's "Applied Proof Theory") provide the kind of results you are looking fo …
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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 …
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7
votes
Does anyone still seriously doubt the consistency of $ZFC$?
Some more detail about Nelson's work:
For most of his life, Ed Nelson tried to show that Peano Arithmetic (and weaker systems) was somehow inconsistent.
This was (initially) based on the idea that one …
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 …
6
votes
Comprehension axiom that helps in the opposite direction
There are many such examples in higher-order Reverse Mathematics. I discuss a couple and indicate a relevant paper of mine related to David's. As usual, let RCA$_0^\omega$ be Kohlenbach's base theory …
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3
votes
What subsystem of third order arithmetic proves the real numbers are Dedekind complete?
The answer to your question (unsurprisingly) depends on the formalisation of "being a subset of $\mathbb{R}$". Alex Kreuzer [1] has used characteristic functions to represent subsets of Cantor space …
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4
votes
A conservative extension of Peano Arithmetic
I believe Feferman's theory S (under a different name) is used here by Feferman to formalise mathematics in a predicative setting.
Erik Palmgren used to have a list on his webpage with papers on (cons …
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