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Results tagged with mathematical-philosophy
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user 7743
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.
45
votes
Accepted
The sets in mathematical logic
I have been asked this question several times in my logic or set theory classes.
The conclusion that I have arrived at is that you need to assume that we know how to deal
with finite strings over a f …
- 16.2k
1
vote
Meaning of Kronecker's comment to Lindemann
There is some sense in socalled ultrafinitism (the believe or ideology that not only there is no infinite set of all integers, but that there are just finitely many integers). After all, there is an …
- 16.2k
9
votes
Accepted
When is a statement provable?
Not all statements that are not provable in ZFC are "strong", if by strong you mean that
ZFC + the statement in question is stronger than ZFC in the sense that it implies the consistency of ZFC.
The t …
- 16.2k
10
votes
Why is it OK to rely on the Fundamental Theorem of Arithmetic when using Gödel numbering?
I am not absolutely sure I understand your problems with the meta-language and such.
You take your favorite base theory, let's say PA or something stronger like ZFC. But I am quite sure that PA is en …
- 16.2k
5
votes
Is finitism an extreme form of constructivism?
Hilbert was looking for proofs by "finitistic methods", and I believe that any of the common
proof calculi of first order logic qualifies for this.
Finitism, on the other hand, rejects the existenc …
- 16.2k
10
votes
Are proper classes objects?
I think Andreas provided an excellent answer, and he pointed out that his answer expresses his philosophical opinion, not some absolute mathematical truth.
I wanted to add some things, though.
Russel …
- 16.2k
4
votes
Is there any formal foundation to ultrafinitism?
There is this argument against Nelson's predicative arithmetic which basically says that the assumption that exponentiation is not total, which is in some sense the whole reason to start predicative a …
- 16.2k
4
votes
Universe view vs. Multiverse view of Set Theory
There are two ways that come to my mind right now of how the multiverse view can be "simulated" in the universe:
One is to assume that there is an actual countable transitive model $M$ of ZFC.
(Not …
- 16.2k
16
votes
Can a problem be simultaneously polynomial time and undecidable?
Of course, every problem in P is decidable by definition of P. This was mentioned in the previous answers.
But there is another problem here that hasn't been addressed yet:
you apparently are look …
- 16.2k
31
votes
Arguments against large cardinals
First of all, why is it so unreasonable to think that ZFC itself is contradictory? Because
we have a good intuition about sets and we have a lot of experience with ZFC. The same basically applies to …
- 16.2k
20
votes
Should there be a true model of set theory?
As was pointed out in some answers to this question, since the large cardinal axiom are linearly ordered by consistency strength, there is a natural direction in which we can strengthen set theory.
Si …
- 16.2k