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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.
5
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1
answer
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Mathematical strength of the statement "Heyting Arithmetic admits Markov's rule"
Consider the following theorem about Heyting arithmetic (HA):
For every arithmetical formula $\phi$ whose only free variable is $n$, if $\text{HA} \vdash \forall n. \phi \lor \lnot \phi$ and $\text{H …
- 12.9k
8
votes
3
answers
491
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Is there a constructive version of internal set theory?
Is there a theory T such that:
T includes all the axioms of CZF.
T includes the Idealization, Standardization, and Transfer schemas from IST.
Every axiom of T is a theorem of IST.
T has Church's rule …
- 16.6k
8
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0
answers
153
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How to define Dedekind reals and Eudoxus reals such that they are equivalent to unmodulated ...
In constructive mathematics without choice, we have three different versions of the real numbers (each embedding into the next).
Regular Cauchy reals (functions $f : \mathbb N \to \mathbb Q$ such tha …
- 6,403
6
votes
0
answers
295
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What are these non-classical versions of ZFC defined by realizability?
See Kleene realizability in Peano arithmetic for a similar question, but about PA instead of ZFC. (In particular, an answer as specific as Emil Jeřábek's answer would be great!)
In the context of cons …
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9
votes
2
answers
448
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Completing half of Hilbert's program: Foundations that are conservative over Peano Arithmetic
The goal of the Hilbert program was to find a complete and consistent formalization of mathematics. Gödel's first incompleteness theorem establishes that completeness is impossible with first-order lo …
- 6,403
2
votes
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 …
- 6,403
11
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6
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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 …
- 6,403
3
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0
answers
160
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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 …
- 6,403
5
votes
1
answer
204
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What is the theory of computably saturated models of ZFC with an *externally well-founded* p...
For any model of $M$ of ZFC, we can extend it to a model $M_{ew}$ with an "externally well-founded" predicate $ew$. For $x \in M$, We say that $M_{ew} \vDash ew(x)$ when there is no infinite sequence …
- 47.8k
7
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0
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253
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Is it decidable whether a statement about reals (in the language of ordered rings) is constr...
The language of ordered rings is a first-order language with operators for $+$, $-$, and $\cdot$, constants for $0$ and $1$, and relations for $<$, $=$ and $>$.
To decide whether such a statement can …
- 6,403
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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18
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3
answers
3k
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What's the earliest result (outside of logic) that cannot be proven constructively?
Although mathematicians usually do not work in constructive mathematics per se, their results often are constructively valid (even if the original proof isn't).
An obvious counter-example is the law o …
- 82.7k
6
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1
answer
308
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Is univalence equivalent to every type function being a functor over equivalence?
Introduce a rule in type theory that if $\Gamma \vdash f : \text{Type} \to \text{Type}$ and $\Gamma \vdash e : A \simeq B$ then $\Gamma \vdash f[e] : f(A) \simeq f(B)$.
It may seem like such a rule is …
- 48.8k
3
votes
0
answers
94
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What does the computation of irrationality and transcendentality via a fancy implementation ...
Proofs that various real numbers are not rational or not algebraic tend to be constructively valid as is. Examples include the proofs that $\sqrt 2$ and $\log_2(3)$ are not rational and that $e$ is no …
- 6,403
14
votes
1
answer
515
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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 …
- 47.8k