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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.
4
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Why should we believe in the axiom of regularity?
Consider first the Unrestricted Axiom of Comprehension
($\exists$y)($\forall$x)(x$\in$y $\leftrightarrow$ $\phi$(x))
and the resulting Russell paradox
y$\in$y $\leftrightarrow$ y$\notin$y
One can cert …
- 4,713
69
votes
5
answers
9k
views
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 …
- 2,858
1
vote
1
answer
360
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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 …
- 3,065
5
votes
1
answer
475
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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 …
- 2,031
7
votes
Why hasn't mereology succeeded as an alternative to set theory?
Considering the nature of your question, you might be interested in the following paper by Geoffrey Hellman and Stewart Shapiro:
"The Classical Continuum without Points", The Review of Symbolic Logic …
- 4,713
0
votes
1
answer
878
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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 …
- 2,912
-4
votes
1
answer
596
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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 …
- 13k
0
votes
1
answer
307
views
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 …
- 73.2k
2
votes
2
answers
554
views
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 …
- 237k
1
vote
2
answers
831
views
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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2
votes
1
answer
452
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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 $y$ …
- 19.4k
0
votes
Regarding Gentzen's note regarding 'Godel-points' (i.e., "Where is the Godel-point hiding?")
In his answer to David Roberts' mathoverflow question, "$Z_2$ versus second-order $PA$" (question 97077), Prof. Ali Enayat writes (Under the subheading, "Regarding the second question)":
One way t …
- 6,132
0
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3
answers
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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 …
- 6,132
6
votes
3
answers
3k
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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 …
- 167
-4
votes
2
answers
454
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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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