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Results tagged with foundations
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user 1946
Mathematical logic, Set theory, Peano arithmetic, Model theory, Proof theory, Recursion theory, Computability theory, Univalent foundations, Reverse mathematics, Frege foundation of arithmetic, Goedel's incompleteness and Mathematics, Structural set theory, Category theory, Type theory.
4
votes
Accepted
What do you call the generalisation of the direct image?
This idea is commonly used in set theory with atoms. I'm not sure whether it has a standard name, but I would be inclined to call it the natural extension of $f$ to sets.
There is no need to stop the …
- 236k
15
votes
Accepted
Meta-undecidability
One can never prove in ZFC itself that a given statement is independent of ZFC, because the assertion "$S$ is independent of ZFC" implies Con(ZFC), since every statement is settled by an inconsistent …
- 236k
10
votes
Accepted
Dedekind's theorem
Regarding the narrower interpretation of your question, the fact that the finite numbers are not equinumerous with
any proper subset is also expressed as the classical
pigeon hole
principle.
And for t …
- 236k
4
votes
Accepted
Original proof of Gödel's completeness theorem compared to Henkin's proof
With regard to the amount of set theory required to prove the completeness theorem, the wikipedia page on The completeness theorem asserts:
When considered over a countable language, the comp …
- 236k
9
votes
Accepted
Why not $\sf ZFC+[V=HOD]$?
What does it mean to be a "standard" theory?
By any account, the theory ZFC + V=HOD already is one of the "standard" theories. The axiom V=HOD is intensely studied by set theorists; it appears as a hy …
- 236k
12
votes
Do set theorists work in T?
You address your question to set theorists, and so let me answer as
a set theorist that, yes, when I think purely as a set theorist,
then indeed the idea that every object is a set just goes without
s …
- 236k
5
votes
weakening naive comprehension to avoid the paradoxes
I'm not clear on why you don't regard ZFC as an example. You say:
Weakening the axiom of naive comprehension has not been a popular way of escaping from the set-theoretic paradoxes because n …
- 236k
13
votes
Accepted
Concrete models of abstract structures
I take your question to be about what we might call the
structuralist perspective, the view that we specify
mathematical objects and structures by their defining
structural features, ignoring any inte …
- 236k
5
votes
Are inference laws consistent?
A somewhat stronger property than mere consistency, which is universally desired in a proof system, is the property of soundness.
A formal system is sound, if whenever it proves a statement $\varphi …
- 236k
13
votes
Accepted
Ultimate Maximality Principle
There are several maximality principles that already have
some of this flavor, with a growing literature surrounding them.
For example, the maximality principle MP as introduced in
my paper A simple …
- 236k
10
votes
Accepted
Set-theoretical multiverse and foundations
(It happens that I will be giving a talk this week on this topic at the Exploring the Frontiers of Incompleteness series at Harvard, which is focussing on the question of determinism in set theory. Th …
- 236k
4
votes
Resource request on "$\in$-homomorphisms" in Set Theory
I believe that you may have misstated the definition of what it means to be an $\in$-homomorphism. (I couldn't find your notion in Jech at your links — have I missed it?)
For example, with your defini …
- 236k
32
votes
Accepted
How much of the axiom of choice do you need in mathematics?
Your hypothesis is in a sense stronger than just assuming ZFC outright.
Namely, if we have $\lambda$-DC for some inaccessible cardinal $\lambda$, and ZF in the background, then in particular, we will …
- 236k
16
votes
Accepted
Propositional calculus, first order theories, models, completeness
Unfortunately I don't quite agree with your summary.
First, in the context of propositional logic, the relevant notion of model is simply a row of the truth table, a propositional world, a valuation a …
- 236k
5
votes
Accepted
Is this form of replacement suitable for ZF - Powerset + well-ordering principle?
The answer is no.
Your version of replacement is a weakening of ordinary replacement. To see this assume that replacement holds (but perhaps not power set or collection), and then observe that for any …
- 236k