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Results tagged with lo.logic
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user 17064
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.
43
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Do the analogies between metamathematics of set theory and arithmetic have some deeper meaning?
By "formal analogies" between the metamathematics of $\mathsf{ZFC}$/set theory and $\mathsf{PA}$(=Peano Arithmetic)/first order arithmetic, I mean facts such as the following:
We are considering a f …
- 32.5k
29
votes
2
answers
3k
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Does Taranovsky's system of ordinal notations make sense?
Dmytro Taranovsky has a Web page on which he claims to define a system of ordinal notations strong enough to provide an ordinal analysis of full second-order arithmetic. I think (perhaps unjustly) th …
- 32.5k
25
votes
1
answer
2k
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In what ways is ZF (without Choice) "somewhat constructive"
Let me summarize what I think I understand about constructivism:
"Constructive mathematics" is generally understood to mean a variety of theories formulated in intuitionist logic (i.e., not assuming …
- 32.5k
22
votes
1
answer
498
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What is known about the consistency of $2^{\aleph_\alpha} = \aleph_{\alpha+\gamma}$ for all ...
For $\gamma$ an ordinal, let “$H_\gamma$” be the statement:
For all ordinals $\alpha$, we have $2^{\aleph_\alpha} = \aleph_{\alpha+\gamma}$.
So clearly $H_0$ is false, and so is $H_\omega$; in fact, …
- 32.5k
20
votes
1
answer
1k
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Axiom of Choice versus V=L in opposition to large cardinals
Consider the following two observations:
The axiom $V=L$ is incompatible with large cardinal axioms that are somehow "too large", like measurable cardinals.
The axiom of Choice is incompatible with …
- 32.5k
18
votes
1
answer
366
views
Proof as a Σ₁ approximation to truth: what about higher degrees?
Let us posit that the goal of mathematics is to study mathematical truths, and let us stick to arithmetic for simplicity: so let $\mathscr{T} \subseteq \mathbb{N}$ be the set of (Gödel codes of) true …
- 32.5k
17
votes
0
answers
362
views
Joyal's topos in which $[0,1]$ fails to be compact
Some time around 1977, André Joyal constructed a topos (actually a locale, i.e., a localic topos, necessarily non-spatial) in which the closed unit interval $[0,1]$ fails to be compact. There are onl …
- 32.5k
17
votes
1
answer
1k
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Looking for a copy of Leo Harrington's unpublished notes on the first nonprojectible ordinal
Sometime around 1975, Leo Harrington wrote a set of notes, apparently 13 pages long, entitled Kolmogorov's $R$-operator and the first nonprojectible ordinal. I do not know how widely they were circul …
- 32.5k
17
votes
1
answer
442
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Examples of statements that are valid in every spatial topos
I am looking for statements¹ that, when interpreted in the internal language of a topos, are valid in all spatial toposes (i.e., the topos of sheaves of any topological space) that are not valid in al …
- 32.5k
14
votes
1
answer
545
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How exactly are realizability and the Curry-Howard correspondence related?
Consider, on the one hand:
the Curry-Howard correspondence between, on the one hand, types and terms (programs) in various flavors of typed $\lambda$-calculus, and on the other, propositions and proo …
- 32.5k
14
votes
3
answers
1k
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Examples of concrete games to apply Borel determinacy to
I'm teaching a course on various mathematical aspects of games, and I'd like to find some examples to illustrate Borel determinacy. Open or closed determinacy is easy to motivate because it proves th …
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14
votes
0
answers
291
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Ordinal-valued sheaves as internal ordinals
Let $X$ be a topological space (feel free to add some separation axioms like “completely regular” if they help in answering the questions). Let $\alpha$ be an ordinal, identified as usual with $\{\be …
- 32.5k
13
votes
1
answer
623
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About primitively recursively recognizable ordinals
Preliminary: I believe the notion of primitive recursive functions on ordinals is standard and unproblematic (the main difference with the finite case is that one needs to introduce a $\sup$ or $\lims …
- 32.5k
13
votes
1
answer
642
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Kleene realizability in Peano arithmetic
For completeness of MathOverflow and for clarity of the question, I will first recall a few things, including the the definition of Kleene realizability: experts can jump directly to the question belo …
- 32.5k
13
votes
0
answers
251
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Intuitionistic proofs of propositional formulae versus natural transformations between finit...
The setup: Given a formula $\varphi$ of intuitionistic propositional logic (i.e., made from the connectors $\Rightarrow$, $\land$, $\lor$, $\top$ and $\bot$ from propositional variables $A,B,C,\ldots$ …
- 32.5k