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Results tagged with mathematical-philosophy
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user 20597
Philosophical aspects of logic and set theory; truth status of mathematical axioms; Philosophy of Mathematics; philosophical aspects of mathematics in general; relation of mathematics to philosophy; etc. Consider also posting at http://philosophy.stackexchange.com/, where philosophy-of-mathematics is one of the most popular tags.
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The universe of sets, existential quantification in set theory
Considering what you wrote in your slide presentation "On the definitional character of axioms.", you might be interested in the following preprint by John L. Bell (found on his Homepage) titled "SETS …
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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$$F$)( …
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Can Dedekind's 'proof' of the existence of infinite sets be properly formulated and carried ...
This question is related to Mikhail Katz's recent mathoverflow question, "Has Dedekind's proof of the existence of existence of infinite sets been analyzed by historians?". Dedekind's 'proof' seems ( …
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Critical points and the Foundation Axiom
(Note: This question is related to my previous mathoverflow question, "Critical Points in $ZF$ without Choice".)
In the Stanford Encyclopedia of Philosophy entry "Non-Wellfounded Set Theory" (Sectio …
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Can the Kunen inconsistency (or the existence of Reinhardt cardinals) be 'properly formulate...
In their paper "Generalizations of the Kunen Inconsistency" (arXiv:1106.1951v1 [math.LO]10 Jun. 2011), Hamkins, Kirmayer, and Perlmutter write the following:
The first [metamathematical issue--my …
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Second-order characterizability and forcing
In their paper, "On Second-Order Characterizability", Hyttinen, Kangas, and V$\ddot a$$\ddot a$n$\ddot a$nen define that notion as follows:
Let us call a structure $\mathfrak U$ second order char …
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Is the notion of measurable cardinal definable from the perspective of set-theoretical poten...
Consider the definition of measurable cardinal (this definition was found in Neil Barton's paper, "Large cardinals and the iterative conception of set"):
Definition 8. A cardinal $\kappa$ is measura …
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Is $\mathit{GPK}^{+}_{ \infty}+\mathit{BAFA}$ inconsistent (and why does it matter)?
Consider Olivier Esser’s alternative axiomatic set theory $\mathit{GPK}^{+}_{\infty}$. Esser defines it as follows (this from his paper "Inconsistency of The Axiom of Choice with The Positive Theory …
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What is an oracle, really? [closed]
Regarding oracles, might this be a reasonable description of their inner workings (this from Hartley Rogers, Jr.'s text, Theory of Recursive Functions and Effective Computability)?
Why should I ask s …
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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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A Question Regarding Productive Sets in the Koepke-Koerwien System SO (Sets of Ordinals)
In their paper "The Theory of Sets of Ordinals" (arXiv), Koepke and Koerwien propose a theory SO axiomatizing the class of sets of ordinals in a model of ZFC and show that SO and ZFC are bi-interpreta …
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Can only the constructible sets be proven to exist in $ZF$ without benefit of extra assumpti... [closed]
I am interested in asking the following question:
What sets can be proven to exist in $ZF$ without the benefit of extra assumptions? (Thanks to Toshiyasu Arai for inspiring me to ask this variation o …
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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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Are ITTM's necessary to compute Turing's "computable numbers" and what does that mean for or...
In his celebrated paper, "On Computable Numbers, With An Application To the Entscheidungsproblem", Turing defines a "computable number" as follows:
The "computable" numbers may be described briefly a …
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