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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.
2
votes
1
answer
110
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Does $\mathrm{L}_{s_{n+1}}$ contain a surjection from $\omega$ to $\mathrm{L}_{s_n}$?
Let $s_n$ be the least $\Sigma_n-$admissible ordinal, so that $\mathrm{L}_{s_n}$ is a model of Kripke-Platek set theory with $\Sigma_n-$collection and $\Sigma_n-$separation.
Does $\mathrm{L}_{s_{n+1}} …
- 3,574
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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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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2
votes
1
answer
597
views
"Potency set" for power set?
Cross-posted at HSM.
Has the term "potency set" been used in English language mathematics for power set, and, if so, what are good references?
It is relevant that for historical reasons, "power set" i …
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2
votes
1
answer
379
views
Impredicativity, definition, recursion and conservatism
Suppose we in an impredicative framework isolate the fixed point
$$Gx\leftrightarrow A(G,x)$$
from a $Gx$ obtained by $\Pi^1_1$-comprehension as equivalent to $\forall K((A(K,x)\to Kx)\to Kx)$, where …
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1
vote
0
answers
96
views
Is Jaskowski's paraconsistent system moderate if sparked?
Stanislaw Jaskowski published a non-adjunctive paraconsistent logic, which does not have the inference rule $\vdash A \ \& \ \vdash B\Rightarrow \ \vdash A\wedge B$. The paper first appeared in Poli …
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2
votes
0
answers
115
views
Will the least class satisfying Scott set theory interpret AC and CH?
I use ST for the set theory used by Dana Scott in More on the Axiom of Extensionality,
in Y. Bar Hillel et alia, Essays on the Foundations of Mathematics}, Hebrew University, Jerusalem: $115-131$. 1 …
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2
votes
1
answer
464
views
First use of corner quotes for Gödel numbers
Who first used the corner quotes, $\ulcorner$ and $\urcorner$, for the notion of Gödel number? They can also be written as\Godelnum with Sam Buss's macro.
They were used by Joseph R. Shoenfield, in Ma …
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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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2
votes
0
answers
115
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Does NBG set theory minus power set minus extensionality interpret NBG set theory minus powe...
In On the Axiom of Extensionality, Part II, JSL, Vol. 24, No. 4, 1959, 287-300, R. O. Gandy shows that NBG set theory minus extensionality interprets NBG set theory. His systems make use of set abstra …
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1
vote
0
answers
154
views
Countable $L_\alpha$ model for $S$ if $S$ has a countable well founded model?
Let $S$ be a proper fragment of $\mathit{ZFC}$, and let $S$ properly extend $\mathit{ZFC}^-$, i.e. $\mathit{ZFC}$ minus the power set axiom. Is there a countable ordinal $\alpha$ so that $L_\alpha$ is …
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2
votes
0
answers
114
views
Does $ZFC^-$ plus $\mathcal{P}^\gamma$ have a countable $L_\alpha$ model if it has a countab...
Suppose $ZFC^-$ is $ZFC$ minus the power set axiom, and that, for $\gamma$ a countable ordinal, $\mathcal{P}^\gamma$ is an axiom that allows less than $\gamma$ applications of the power set operation …
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1
vote
1
answer
242
views
Are countable models constructible?
Suppose a first order set theory $S$ has a countable model. Does it follow that there is a countable ordinal $\alpha$ so that $L_\alpha$ in the constructible hierarchy is a model of $S$?
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0
votes
0
answers
287
views
ZF plus class-choice?
$\DeclareMathOperator\Cls{Cls}\newcommand{\ZFC}{\mathrm{ZFC}}$Let ZF be extended with class and set predicates, and extend the theory with a class abstraction schema so that for all formulas $\phi$:
$ …
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-1
votes
1
answer
399
views
Ubiquity beyond infinity, transitive closure and the recursion theorem?
I am considering a Principle of Ubiquity, expressed as follows - for a class theory where precisely the elements are sets - with the aid of set abstracts:
For $\alpha(y,z)$ a first order condition so …
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