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Results tagged with lo.logic
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user 37385
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.
23
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4
answers
20k
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Is there a natural bijection from $\mathbb{N}$ to $\mathbb{Q}$?
In a conversation where it came up that the Pythagoreans probably found an enumeration of the rational numbers I erroneously remarked that Georg Cantor found a natural bijection from $\mathbb{N}$ to $ …
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22
votes
1
answer
2k
views
"The boat is not longer than it is."
Bertrand Russell, I believe, somewhere presents a joke (if I remember correctly). Someone is shown the boat of another, and the first says: "I thought that your boat is longer than it is." The owner r …
- 3,574
13
votes
1
answer
2k
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Are some interesting mathematical statements minimal?
Gödel's set $\mathrm{L}$, of constructible sets, decides many interesting mathematical statements, as the Continuum hypothesis and the Axiom of Choice.
Are some interesting mathematical questions, whi …
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9
votes
1
answer
558
views
A question on an ordinal for ZFC-
ZFC-, which is ZFC minus power set, is modelled by $ L_{\delta}$ where $\delta$ is an admissible ordinal larger than
any least $\Sigma_{n}$-admissible ordinal for n a natural number. Can some provide …
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8
votes
4
answers
3k
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The Halting Problem and Church's Thesis
In the opening chapters of Hartley Rogers, Jr.'s book Theory of Recursive Functions and Effective Computability, the proofs of the unsolvability of the halting problem and related unsolvability result …
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7
votes
2
answers
297
views
At what ordinal $\chi$ does $\mathrm{L}_\chi$ contain a surjection from $\omega$ to $\mathrm...
Let $\mathrm{ZF^-}$ be $\mathrm{ZF}$ minus power set, and let $\beta_0$ be the ordinal for ramified analysis so that $\mathrm{L}_{\beta_0}$ is the least $\mathrm{L}$-model of $\mathrm{ZF}^-$. Clearly, …
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6
votes
1
answer
189
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Does $WKL_0$ provide more comprehension than $RCA_0$?
$WKL_0$ extends $RCA_0$ with the statement that any infinite subset of the infinite binary tree has an infinite branch. Does $WKL_0$ Prove that there are sets which are not proven to exist by the $\De …
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6
votes
0
answers
117
views
Ackermann set theory without extensionality?
Scott showed that ZF minus the axiom of regularity is interpreted by ZF minus the axioms of regularity and extensionality.
Is Ackermann set theory interpreted by Akermann set theory without extensiona …
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6
votes
1
answer
831
views
Did Brouwer evade uncountability?
I have the distinct memory of having often heard and read that intuitionism was inter alia geared to avoid Cantor's uncountable sets, and it may be that this was Brouwer's plan. But are there accounts …
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6
votes
2
answers
464
views
When do we get $CON(ZF)$ in transfinite progressions of consistency statements?
Given the work of Turing and Feferman all arithmetical truths can be isolated through a transfinite progression of theories like $T_0=PA$, $T_{\beta+1}=T_β \ plus \ CON(T_\beta)$ and $T\lambda=\cup T\ …
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6
votes
1
answer
306
views
Is a computer program for correspondence theory available?
In the 1990s I some times used a computer program with the Max Planck Institute which helped with calculating complicated correspondences for modal logical formulas. Is some program like that availabl …
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6
votes
4
answers
2k
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How short can we state the Axiom of Choice?
How short can we state a principle which is equivalent with the Axiom of Choice under $ZF$? The principle should be a sentence in the language of set theory with only $\in$ and$=$ as extralogical rela …
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5
votes
1
answer
194
views
Can extensions of $Q$ contradict Löb with recursive reflection?
It is an odd and arguably unacceptable situation that $PA$ does not have $\vdash_{PA}(Pr_{PA}\ulcorner A\urcorner\to A)$ for false recursive sentences $A$.
However, it is not clear to me that Löb …
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5
votes
1
answer
316
views
Is this set theory used by Gandy first-order with signature $(\in, \lambda)$?
In On the Axiom of Extensionality, Part II, The Journal of Symbolic Logic, Vol. 24, No. 4 (Dec., 1959), https://doi.org/10.2307/2963897, pp. 287-300, R. O. Gandy shows that a class theory X containin …
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5
votes
2
answers
355
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Transfinite recursion, collection and replacement in KP and KF
Kripke Platek set theory has collection instead of replacement, and it is a weakening of KP if one has replacement instead of collection. Call KP minus collection plus replacement KF for Kripke Fraenk …
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