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Results tagged with lo.logic
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user 20597
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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Can the omega-rule rescue Hilbert's program?
Note: In Buldt's paper "The Scope of Goedel's First Incompletness Theorem" (find it under title on the web), he has the following statement (in Section 3.3):
"When we enter the (small) transfinite, w …
- 6,132
4
votes
Are Separation and Types the only way to avoid paradoxes?
In point of fact, there are (at least) a couple more ways to circumvent the paradoxes brought about by the use of the unrestricted comprehension axiom
$\exists$$y$$\forall$$x$($x$$\in$$y$$\leftrighta …
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1
answer
282
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A question regarding a fragment of Robinson Arithmetic
In his answer to the following mathoverflow question, The (un)decidability of Robinson Arithmetic without multiplication, Emil Jerabek proved that the following fragment:
$\forall$x(Sx$\neq$0)
$\for …
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1
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0
answers
142
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Can Basic Law $V$ be derived from Leibniz's Law in Second-Order Logic without comprehension ...
Consider Basic Law $V$:
$\hat x$$F$($x$)=$\hat x$$G$($x$)$\equiv$($\forall$$x$)($F$$x$$\equiv$$G$$x$)
At first glance, it seems to have the same form as Leibniz's law
$x$=$y$$\equiv$($\forall$$F$)( …
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2
votes
Literature on Kripke models
I can't say that this is the 'best' introduction to Kripke models (as 'best' is always a relative term), but John P. Burgess's survey article "Kripke Models" presumes only knowledge of propositional a …
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0
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1
answer
255
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Real-valued measurability vs. Two-valued measurability in determining whether $CH$ holds or not
The following fact is known:
If there is a measurable cardinal, then there are only countably many constructible reals.
It is also known that if $ZFC$ + "There is a (two-valued) mesurable cardin …
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5
votes
Is a paraconsistent and provably non-trivial foundation for math possible?
One should note that as regards your question 1, systems of relevant arithmetic with inconsistent models such as $R{\sharp}$, $R{\sharp}{\sharp}$, and the systems $RM3^{i}$ can prove (with finitary pr …
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3
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0
answers
137
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A question regarding forcing in $NGBC^{-f}$+$BAFA$
Suppose one has a model $M$$\vDash$$NGBC^{-f}$+BAFA. Does there exist a (class) forcing extension $M[G]$$\vDash$$NGBC^{-f}$+$BAFA$ that has a submodel $N$$\vDash$$NGB^{-f}$+$BAFA$+$\lnot$$AC$? Can o …
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1
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1
answer
386
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What restriction(s) of Goedel's primitive recursive functionals is (are) necessary and suffi...
It is well known that one can use Goedel's primitive recursive functionals of finite type to prove the consistency of $PA$ (Peano Arithmetic). As such, one can certainly use them to prove the consist …
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What are the sense and reference of the propositions $R \notin R$, $R \in R$, where $R=\{x \...
Consider the following passage from Frege's lecture "Funktion und Begriff" (English translation by Peter Geach with the title "Function and Concept" [Note: this is the lecture which introduced the no …
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0
votes
Analogues of Primitive Recursive Functions
Actually, for the case $\mathbb A$=$\mathbb H$$Y$$P_{\mathfrak M}$ (which seems, if I am understanding your question correctly, what you are referring to), since $\mathbb H$$F_{\mathfrak M}$$\subseteq …
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2
votes
Can we add set complements on top of ZF?
I believe that an answer to your question [1] is the system that Dana Scott developed in his paper, "Axiomatizing Set Theory" found in Proceedings of Symposia in Pure Mathematics, Volume 13, Part II, …
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4
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Set theoretical multiverse and truths
Since the Fundamental Theorem of Arithmetic is a theorem of $PA$, it holds for both standard and nonstandard models of $PA$. Since one can interpret $PA$ in both $ZFC$ and $GBC$ (e.g., for $ZFC$, it …
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0
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259
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A question regarding Koepke' s Ordinal Computability in HOD
Consider the following theorem of Koepke-Koerwien-Siders:
"A set x of ordinals is ordinal computable [either by ordinal Turing machines or ordinal register machines--my comment] if and only if it is …
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0
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1
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269
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What is the smallest countable limit ordinal in which 'lost melodies' occur
The question is in the title. This question is in response to the following paragraph found at the end of Prof. Hamkins' answer to my MathOverflow question, Are ITTM's necessary to compute Turing's " …
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