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Results tagged with mathematical-philosophy
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user 8133
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
votes
Accepted
Critical points and the Foundation Axiom
As far as I understand your question, this is the answer:
Suppose $V$ is a model of $ZF$ minus Foundation (call this theory "$ZF^-$"). Then the following are equivalent:
$V$ satisfies Foundation - …
- 20.7k
3
votes
Accepted
Is the notion of measurable cardinal definable from the perspective of set-theoretical poten...
Paring the philosophy away (which I think just obscures things) this seems to just be a question of definability - the key point being that Hamkins' recursively-defined translation works for any formu …
- 20.7k
10
votes
Does Zorn's Lemma imply a physical prediction?
Here's a set-theoretic answer, paralleling Simon Henry's use of Barr's theorem. Shoenfield's absoluteness theorem implies (among other things) that no $\Pi^1_2$ property can depend on choice: if $\var …
- 20.7k
6
votes
Accepted
Extensions of the Ackermann interpretation to nonstandard theories of arithmetic
Based on the comments, I think it's worth clarifying some points about the Ackermann interpretation.
The Ackermann interpretation is definable: it is a formula $\varphi(x, y)$, and - given a model $M …
- 20.7k
6
votes
Ontological status of some "sets" in ZFC
Consider the following descriptions of sets:
(1) $X=\lbrace i\in\omega: i=0\rbrace=\lbrace0\rbrace$,
(2) $Y=\lbrace i\in\omega: i=0\iff CH, i=1\iff \neg CH\rbrace=\lbrace 0\rbrace$ or $\lbrace 1\rb …
- 20.7k
10
votes
Can only the constructible sets be proven to exist in $ZF$ without benefit of extra assumpti...
There are several issues with this question; Andres has pointed out one glaring one, which is the conflation of theories and models. There is another issue, however, around what you are asking in the …
- 20.7k
4
votes
Formalizing ontological optimism
As remarked above, claims from "optimism" are not very clear-cut: e.g. do we think of choice as asserting the existence of well-orderings of arbitrary sets, or the non-existence of cool stuff like amo …
- 20.7k
4
votes
Are ITTM's necessary to compute Turing's "computable numbers" and what does that mean for or...
It sounds like you're asking whether the following are equivalent:
$x$ is a computable infinite binary sequence in the usual sense, that is, the function $i\mapsto x(i)$ is computable.
There is an I …
- 20.7k
19
votes
Are there 'finitistic' nonrecursive functions (assuming Church's Thesis is false)?
Kalmar's argument is indeed wrong. The problem, of course, lies in his justification of our ability to compute $\psi(x)=0$, where he writes
"We can prove, not in the frame of some fixed postulate …
- 20.7k
2
votes
Accepted
A question regarding extendible cardinals and a result of M. Magidor
For your second question, the answer is "no." Since, with Henkin semantics, second-order logic is just re-syntacted first-order logic, of course theorem 4 fails for Henkin semantics: $L_\kappa^2$ with …
- 20.7k
8
votes
An example of a proof that is explanatory but not beautiful? (or vice versa)
I'm not sure this question is appropriate for MO, but: I find the usual proof of the Recursion Theorem beautiful but not explanatory. (See my answer to another MO question: Are there proofs that you f …
13
votes
Accepted
Regarding Gentzen's note regarding 'Godel-points' (i.e., "Where is the Godel-point hiding?")
Your question seems to boil down to (after fixing an error) the following:
Any model $\mathfrak{M}$ of ACA$_0$ has a first-order part $Num(\mathfrak{M})$, which satisfies PA; why doesn't this mean …
- 20.7k
2
votes
Accepted
How are Koepke's ordinal computability and E-recursion related?
I don't quite know what your main question is asking, or what (3) means, but (1) and (2) seem to have straightforward answers:
Any nontrivial forcing makes $L$ non-E-r.e.: if $M\models ZFC$ and $M[G] …
- 20.7k
49
votes
Why not adopt the constructibility axiom $V=L$?
Let me add to the existing answers a point which may seem "vulgar" at first, but I think is actually important:
V=L is complicated.
And whether or not this ought to be a reason to not raise it to ZF …
5
votes
Accepted
Compactness of existential second order logic and definability of certain quantifiers
Existential second-order logic is indeed compact for arbitrary languages. The proof I know is via ultraproducts, which I'll sketch here. (I once saw a Henkinization-style argument, but it was quite me …
- 20.7k