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Results tagged with lo.logic
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user 9833
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.
0
votes
Is there any physical or computational justification for non-constructive axioms such as AC ...
Not an answer, but an opinion. The question whether it would be
difficult to model physical phenomena using constructive mathematics would make some sense if those physical phenomena were modeled by …
- 6,891
4
votes
1
answer
282
views
A weak fragment of analysis?
Question: Is there a decidable theory sufficient to formulate and prove (many) theorems of classical analysis?
What I have in mind is a theory with two kinds of objects, reals (which are introduced a …
- 6,891
6
votes
Gödel's speed-up from constructive to classical logic?
I am no expert in logic, but let me give it a try. There is an old (1970) theorem of M. Waldschmidt that at least one of the numbers $e^e$ and $e^{e^2}$ is transcendental.
So, the statement is a d …
- 6,891
18
votes
0
answers
486
views
What is the logical complexity of the Hodge conjecture?
The Hodge conjecture seems to me the most mysterious among the Millennium problems
(and many others). In particular, I am not sure about its logical complexity. It is not difficult to see that the …
- 6,891
24
votes
2
answers
2k
views
Is the Invariant Subspace Problem arithmetic?
Invariant Subspace Conjecture: A bounded operator on a separable Hilbert space has a non-trivial closed invariant subspace.
Can this conjecture be reformulated as an arithmetic statement, that is, $ …
- 6,891
17
votes
Accepted
For which Millennium Problems does undecidable -> true?
As Harry Altman pointed out, for a conjecture undecidable -> true
means that it can be formulated as a $\Pi_1^0$ statement. To put it simply, if the conjecture
is false, one can prove this by an e …
6
votes
0
answers
580
views
How to prove a $\Pi_2$ statement properly?
Consider the following situation. In a parallel world (let's hope not in this one),
in 2020 a clever guy proved $P\neq NP$ in a theory $T_1=\{ZFC+some\, reasonable\, new\, axioms\}$.
Then, in 2021 a c …
- 6,891
7
votes
2
answers
1k
views
Are all the theorems true?
The title sounds a bit philosophical, but it is
about mathematics. Let me explain.
Consider a first order theory $T$, which is an extension
of Peano Arithmetic. Call this theory "good" if it is
co …
- 6,891
6
votes
2
answers
602
views
Are undecidable consequences of Con recursively enumerable?
Let $X\subset\Pi_1^0$ be the set of statements which are
provable in PA$+$Con(PA) but independent of PA.
Is $X$ recursively enumerable?
- 6,891
0
votes
The disjunction property in Peano Arithmetic?
This is not actually an answer but rather a comment to Joel's answer.
I am not very good in models, so here is an idea how to do without them.
There is a theorem of Kreisel: if a $\Pi_1^0$ statement i …
- 6,891
14
votes
4
answers
1k
views
The disjunction property in Peano Arithmetic?
Let $\phi,\psi\in\Pi_1^0$ be independent of PA.
Is the disjunction $\phi\vee\psi$ independent of PA?
- 6,891