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Results tagged with mathematical-philosophy
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user 6794
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.
61
votes
Accepted
What is Realistic Mathematics?
When Solovay showed that ZF + DC + "all sets of reals are Lebesgue measurable" is consistent (assuming ZFC + "there is an inaccessible cardinal" is consistent), there was an expectation among set-theo …
- 73.2k
40
votes
Arguments against large cardinals
Most of the answers have addressed the "consistency" part of the original question, "Why is it so unreasonable to think that the existence of large cardinals contradicts ZFC?" There's another part, a …
- 73.2k
38
votes
Why should we believe in the axiom of regularity?
I think of the axiom of regularity along with the axiom of extensionality as formalizing what I mean by "set". Once upon a time, before paradoxes, one could think of sets as just any collection of th …
- 73.2k
34
votes
Accepted
Interpretation of the Second Incompleteness Theorem
For the philosophical point encapsulated in (*) in the question, it seems that corollaries of the second incompleteness theorem are more relevant than the theorem itself. If we had doubts about the c …
- 73.2k
29
votes
Is PA consistent? do we know it?
As Mirco Mannucci's answer suggests, the alternatives labeled 1) and 2) in the question are (for some people) not mutually exclusive. The consistency of PA indeed "has a proof as valid as any other t …
28
votes
Accepted
Are proper classes objects?
Proper classes are not objects. They do not exist. Talking about them is a convenient abbreviation for certain statements about sets. (For example, $V=L$ abbreviates "all sets are constructible.") …
- 73.2k
26
votes
Why hasn't mereology succeeded as an alternative to set theory?
It seems worthwhile to point out that Steve’s answer also essentially answers Carl Mummert’s question (in a comment) about why one can’t get set theory as a definitional extension of mereology by defi …
- 73.2k
26
votes
Proving Independence of Axioms by Exhibiting Models Which Don't Satisfy Our Intuition
I'm inclined to agree that "if you precisely identify the notion we're talking about (like the integers, set theory), then these pathological models don't exist." The problem is that it's not so easy …
- 73.2k
22
votes
Ultrainfinitism, or a step beyond the transfinite
In an (I hope) temporary bout of megalomania, I answer as follows. What you and Cantor and others regard as the absolute infinite, $V$, is really only a level $V_\kappa$ of the cumulative hierarchy, …
- 73.2k
20
votes
Why not adopt the constructibility axiom $V=L$?
When I explain the cumulative hierarchy at the beginning of a set theory course, I point out that it involves two rather vague ideas: (1) forming arbitrary subsets of a given set (which is what we do …
18
votes
Accepted
Proper classes and their consequences
A fairly general "definition" of "proper class" is that it means a collection of sets that is not itself a set.
In the usual picture of sets as constituting a transfinite cumulative hierarchy (in w …
- 73.2k
18
votes
How to tell a paradox from a "paradox"?
Both the Russell paradox and the Banach-Tarski "paradox" show that certain ideas are contradictory. It seems to me that the key difference between the two is that, in Russell's case, the ideas in que …
- 73.2k
17
votes
What is the status of irrational numbers within finitism/ultrafinitism?
Primitive recursive arithmetic (PRA), mentioned on the second Wikipedia page to which the question links, is, I believe, generally accepted as an appropriate formalization of finitist foundations. T …
- 73.2k
16
votes
nonstandard models and mathematical theorems
Well, let's compare the compactness example you cited with a non-standard models approach to the same result. Of course, since the result is about vertices of a graph, not just natural numbers, I'll …
- 73.2k
14
votes
Accepted
How can we know the well-foundedness of $\epsilon_0$?
Let me use the notation $\omega_n$ for an exponential tower of $\omega$'s of height $n$, so $\omega_{n+1}=\omega^{\omega_n}$. Then $\epsilon_0$ is the supremum of $\{\omega_n:n\in\omega\}$. PA proves …
- 73.2k