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Results tagged with lo.logic
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user 65915
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.
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3
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Is true arithmetic + $\lnot Con (TA)$ consistent?
Is the theory $TA+\lnot Con(TA)$ consistent?
In particular, for every $TA \vdash \phi$, we take as an axiom $\phi$, and $TA \vdash \phi$. We also assert $TA \vdash 0 = 1$. We call this theory $TA + \ …
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4
votes
1
answer
220
views
Instead of classes, can't we just use sets of elements of a model?
When working in set theory, we often want to work with large collections of sets, so large that they themselves do not form a set. This requires us to have a notion of classes. But then we might want …
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4
votes
1
answer
539
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Can adding Hilbert's epsilon to a theory, and then expanding that theories axiom schema to i...
Epsilon Calculus is a formalism developed by Hilbert adding his $\epsilon$ operator to predicate logic. $\epsilon x. A(x)$ is a term such that $\exists x.A(x) \implies A(\epsilon x.A(x))$. In can actu …
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2
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0
answers
138
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Constructively, when do functions that agree on $[a, b] \cup [b, c]$ also agree on $[a,c]$?
Let $a, b, c \in \mathbb R$ such that $a \le b \le c$. Let $S$ be some set and $f, g : [a, c] \to S$ be functions. As a follow up to When can a function defined on $[a, b] \cup [b, c]$ be constructive …
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1
vote
0
answers
204
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Lowest Turing degree that allows a Turing machine to tell whether $\operatorname{Con}(PA)$?
Let $T$ be a given turing machine. We say that $T$ decides $\operatorname{Con}(PA)$ if $PA + \operatorname{Con}(PA) \vdash T \text { accepts}$ and $PA + \lnot \operatorname{Con}(PA) \vdash T \text { r …
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3
votes
0
answers
274
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Can we prove the epsilon theorems without the axiom of choice?
Hilbert's epsilon $\epsilon$ is a quantifier. It follows the rule that if $\exists x. p(x)$, for some predicate $p$, we can infer $p(\epsilon x. p(x))$. Semantically, it represents picking some elemen …
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2
votes
1
answer
682
views
Is ZFC plus a truth predicate capable of variable substitution consistent?
Let us define a truth predicate that allows one to substitute in variables. For example, for a 3-ary predicate $p$ and sets $A$, $B$, and $C$, then $\text{true}(\ulcorner p \urcorner, A, B, C)$ (where …
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3
votes
0
answers
255
views
A theory which denies the existence of a truth predicate
Is there a paper or other reference that explores the implications of a theory that denying the existence of a truth predicate in its own language (perhaps based on ZFC or PA)?
I know that a theory t …
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33
votes
1
answer
3k
views
Is there a position in infinite Go for which the life of a particular stone has transfinite ...
As follow up to Checkmate in $\omega$ moves?, we can ask the same question about go. Is there a position on a $\mathbb Z \times \mathbb Z$ goban such that either black can kill a white group, but whit …
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9
votes
1
answer
245
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What are these generalizations of the principles of omniscience called?
I will give some principles that are slightly stronger versions of the principles of omniscience. Despite being about the natural numbers, they imply their analytic versions! Under countable choice (o …
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14
votes
1
answer
515
views
Is there a theory between HA and PA that doesn't have Markov's rule?
A theory $T$ admits Markov's rule when
For every formula $\phi(n)$, if $$T \vdash \forall n \in \mathbb N. \phi(n) \lor \lnot \phi(n)$$ and $$T \vdash \lnot \lnot \exists n \in \mathbb N. \phi(n)$$ t …
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1
vote
1
answer
149
views
Is there a three valued logic whose game semantics corresponds to potentially infinite games?
Consider game trees with the following properties:
Each node in the tree is one of the following:
Verifier Choice: Has one or more children
Falsifier Choice: Has one or more children
No Choice: Ha …
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11
votes
6
answers
1k
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When can a function defined on $[a, b] \cup [b, c]$ be constructively extended to a function...
Let $a, b, c \in \mathbb R$ such that $a \le b \le c$. Let $S$ be some set and $f : [a, b] \cup [b, c] \to S$ be a function. When can we find a function $g : [a, c] \to S$ that meets the following cri …
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3
votes
0
answers
160
views
Is the Tarski–Seidenberg theorem constructively provable?
The Tarski–Seidenberg theorem asserts that the projection of a semialgebraic set is also a semialgebraic set. My question is whether this is provable in constructive mathematics.
First, let me formali …
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8
votes
2
answers
719
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Is every true statement independent of $PA$ equivalent to some consistency statement?
Most true statements independent of PA that I know of is equivalent to some consistency statement. For example
Con(PA), Con(PA + Con(PA)), Con(PA + Con(PA) + Con (PA + Con(PA)), $\dots$
Goodstein's …
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