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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.
17
votes
CH in non-set theoretic foundations
One consideration will be that the CH statement can become ambiguous in weaker foundations, since statements that are equivalent in ZFC are not always equivalent in weaker theories. …
- 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
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
12
votes
Accepted
The "first-order theory of the second-order theory of $\mathrm{ZFC}$"
I understand your theory to be the set of all sentences in the first-order language of set theory that are true in every model of $\newcommand{\ZFC}{\text{ZFC}}\ZFC_2$. This is a perfectly sensible no …
- 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
8
votes
Accepted
What is the proof of consistency of anterior reflection?
This is a consequence of ZF as follows.
Assume that all quantifiers of $\varphi$ appear at the front as in the normal forms. Suppose that $\varphi(\vec v)$ has complexity at most $\Sigma_n$ for some l …
- 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
8
votes
Accepted
Bounded alternatives to powerset that interpret ZFC
The answer is Yes. The simple fact is that it is much easier to interpret ZFC from low-complexity assertions than one might expect. For example, even PA+Con(ZFC) can already interpret ZFC, since one c …
- 236k
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
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
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
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 …
- 4,713
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 …
- 4,713
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 …
- 236k
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