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Results tagged with mathematical-philosophy
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user 5442
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.
33
votes
Interesting meta-meta-mathematical theorems?
In Reverse Mathematics, we can study what happens if we use weak systems of second-arithmetic as metatheories. For example, we can study the strength of the completeness theorem and prove results such …
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14
votes
Accepted
Notion of Truth and Axioms
The proof of the incompleteness theorem can already be done syntactically, ignoring truth, if we remove the conclusion that the Gödel sentence is true and leave only that it is neither provable nor di …
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7
votes
Intended interpretations of set theories
I have a few comments that I hope are useful even if they don't clarify things completely. As Michael Blackmon says, different people have different ways of resolving things to their own satisfaction. …
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6
votes
Interpretation of the Second Incompleteness Theorem
The fact that the second incompleteness theorem refers to consistency is important for several applications, both philosophical and mathematical.
Philosophically, the second incompleteness theorem i …
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20
votes
Proving Independence of Axioms by Exhibiting Models Which Don't Satisfy Our Intuition
You have run into one of the main themes of contemporary logic: the difference between "truth in the standard model" and "provability". This is an extremely deep issue, so I'm sure other people will a …
7
votes
Abstract thought vs calculation
In computability theory, it is often necessary to prove some particular function is a "computable function". Until the 1960s, this was most commonly done by actually demonstrating a formal algorithm f …
22
votes
Why are proofs so valuable, although we do not know that our axiom system is consistent?
Gödel's theorems do not say that we can never know our axiom systems are consistent. Not at all. What they say is that we can never prove that certain systems are consistent within those systems thems …
23
votes
Accepted
Clarification of Gödel's second incompleteness theorem
The key idea Feferman is exploiting is that there can be two different enumerations of the axioms of a theory, so that the theory does not prove that the two enumerations give the same theory.
Here i …
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