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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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What was Hilbert's view of Gödel's Incompleteness Theorems?
According to Solomon Feferman, in his slide presentation "Three Problems for Mathematics", Hilbert wrote (in regards to Gödel's second incompleteness theorem):
...the end goal [is] to establish as …
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How are Koepke's ordinal computability and E-recursion related?
In Koepke's paper, "Turing Computations On Ordinals", one has the following (well-known) result:
A set $x$ is ordinal computable from a finite set of ordinal parameters if and only if it is an elemen …
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Extensions of the Ackermann interpretation to nonstandard theories of arithmetic
In their paper, " On Interpretations of Arithmetic and Set Theory" (Notre Dame Journal of Formal Logic, Vol. 8, No. 4 (2007), pp. 497-510) in section 7, "Fragments of Arithmetic and Set Theory", Kaye …
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Forcing the existence of a weakly inaccessible cardinal in some strong set theory
Does the fact that, assuming the consistency of $ZFC$, no proof that the consistency of "$ZFC$ implies the consistency of '$ZFC$ + There exists a weakly inaccessible cardinal'" can be formulated in $Z …
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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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Can finite sets be non-c.e. depending on how they are presented?
I ask the question because of the following statement found in Mark Burgin's paper, "Algorithmic complexity of recursive and inductive algorithms", Theoretical Computer Science 317 (2004) 31-60 (pg. 3 …
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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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Is there an analogue of the Lost Melody Theorem in ordinary recursion theory and if not, why...
In their arXiv preprint, "Infinite Time Turing Machines" (arXiv:math/9808093v1 [math.LO] 21 Aug 1998) Hamkins and Lewis state the Lost Melody Theorem for ITTM's as follows:
Lost Melody Theorem 4.9 [p …
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Is the statement "All numbers are counting numbers" independent of $PA$?
In his paper, "Completed versus Incomplete Infinity in Arithmetic" (which can be found here), the late Edward Nelson defines the notion of 'counting number' as follows:
0 is a counting number
if $y$ …
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Regarding Gentzen's note regarding 'Godel-points' (i.e., "Where is the Godel-point hiding?")
Consider the following note written by Gerhard Gentzen in early 1932, on the onset of his work on a consistency proof for arithmetic:
The axioms of arithmetic are obviously correct, and the princi …
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The Lucas argument vs the theorem-provers -- who wins and why?
In his paper, "Minds, Machines and Gödel", J.R. Lucas writes the following:
Gödel's theorem [First Incompleteness Theorem, that is—my comment] must apply to cybernetic machines, because it is of t …
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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 …
CommunityBot
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The universe and multiverse views of set theory from the perspective of $ZFC^2$
(Note: the 'Second-order $ZFC$' ($ZFC^2$) I am talking about is the theory [in the second order language of set theory consisting of a single non-logical symbol $\in$ ] consisting of the axioms Exte …
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Why are real-valued measurable cardinals never explicitly mentioned in Gödel's "What is Cant...
It is a matter of mathematical folklore that Gödel "entertained the idea of so called stronger axioms of infinity deciding $CH$...." (this quote from Radek Honzik's paper, "Large cardinals and the Co …
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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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