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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.
43
votes
Accepted
Why hasn't mereology succeeded as an alternative to set theory?
I have long found this question interesting, and in some recent joint work with Makoto Kikuchi, now available, we consider various aspects of the question of whether a set-theoretic version of mereolo …
- 236k
39
votes
Do set-theorists use informal set theory as their meta-theory when talking about models of ZFC?
The main fact is that a very weak meta-theory typically suffices, for theorems about models of set theory. Indeed, for almost all of the meta-mathematical results in set theory with which I am familia …
- 236k
34
votes
Accepted
Nontrivial circular arguments?
Perhaps an example of the kind of circularity you mention
arises with the self-reference phenomenon that arises in
connection with the incompleteness
theorems
and related applications. Specifically, G …
- 236k
33
votes
Set theory and alternative foundations
If we adopt a historical attitude, then there is an extremely good answer, namely, Geometry. For approximately two thousand years, (Euclidean) geometry was taken to be the foundation of all mathematic …
- 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
28
votes
Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic
If you are really satisfied with a model only of the theory $Q$, then you should be prepared for a bad situation, for this is an extremely weak theory. In fact one can make a computable model simply b …
- 236k
28
votes
Ultrainfinitism, or a step beyond the transfinite
My view is that the large cardinal hierarchy already has all the
principal features of the project you are proposing.
Each of the large cardinal concepts can be regarded as
corresponding to a certain …
- 236k
24
votes
Accepted
Large categories vs. $\mathrm{U}$-categories: why is the loss of category-theoretic informat...
Let me try to answer as a set theorist, rather than as a category
theorist, since I think that your question concerns at bottom a
matter often considered in set theory.
Namely, the essence of your qu …
- 236k
23
votes
Accepted
Are the categories of sets, abelian groups, and commutative rings unique?
Introduction to pluralism
A version of this question lies at the heart of the ongoing dispute on pluralism in the philosophy of mathematics. Is there at bottom just one mathematical reality? Does ever …
21
votes
Lists as a foundation of mathematics
Peter Koepke and Martin Koerwien developed the theory of sets of ordinals as a foundation of mathematics, showing senses in which it is equivalent to ZFC as a foundation.
Peter Koeopke and Martin Koe …
- 236k
19
votes
Accepted
What governs our "perception?" about the platonic realm of sets?
I see NF as arising by a process of formal manipulation of the axioms, rather than a mathematical idea, and in my view this is not a sound method of finding robust meaningful principles in the foundations …
- 236k
18
votes
Accepted
Finite versions of Godel' s incompleteness
Nice questions!
For the first question, I claim that every theory $T$ is $k$-incomplete in your sense for every finite $k$. This is because every statement appears explicitly as a part of any proof o …
- 236k
17
votes
Accepted
What set theoretical questions could never be answered by Turing machines of arbitrary cardi...
Peter Koepke and his numerous collaborators have studied the ordinal-length tape version of infinite time Turing machines, where one has a tape stretching the length of the ordinals, and one imagines …
- 236k
17
votes
Accepted
Does foundation/regularity have any categorical/structural consequences, in ZF?
Yes, the axiom of foundation has structuralist consequences.
Let $\phi$ be the assertion, "if every well-founded set is well-orderable, then every set is well-orderable."
This statement, I claim, …
- 236k
17
votes
Are there substantive differences between the different approaches to "size issues" in categ...
Although people often talk as though it just doesn't matter which approach you use — perhaps all universes are alike? Let me prove that in a strict sense this is not true. The nature of the mathematic …
- 236k