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Results tagged with mathematical-philosophy
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user 8991
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.
7
votes
Interpretation of the Second Incompleteness Theorem
While it's not directly a philosophical benefit, the Second Incompleteness Theorem is quite useful for giving concrete unprovability results: if we want to prove that theory T does not prove theorem X …
- 7,145
9
votes
Accepted
Reasoning Using Countable Subsets of Real Numbers
Suppose that during some argument (involving ℕ) one switches to real numbers and then back to discrete domain (before completing the argument). My question is, can one give examples where a switch …
- 7,145
6
votes
Accepted
Is $PRA$ + $TI({\epsilon_0})$ mutually interpretable with some theory in the language of set...
Yes, the consistency of "ZFC with the axiom of infinity replaced by its negation" is provable in "PRA + TI($\epsilon_0$)". Technically one has to also show that "PRA + TI($\epsilon_0$)" can prove the …
- 7,145
76
votes
Accepted
How should a "working mathematician" think about sets? (ZFC, category theory, urelements)
Set theory provides a foundation for mathematics in roughly the same way that Turing machines provide a foundation for computer science. A computer program written in Java or assembly language isn't …
- 7,145