All Questions
25 questions
37
votes
6
answers
6k
views
Who needs Replacement anyway?
The set theory ETCS famously comes without the Replacement axiom schema (or an equivalent) that is part of ZFC. One (to me, not apparently useful) set that one cannot build in ETCS is $\coprod_{n\in \...
- 35.5k
21
votes
6
answers
3k
views
Where in ordinary math do we need unbounded separation and replacement?
[I have updated the question after initial comments in the hope of clarifying it.]
I do quite a bit of reasoning, typically about topology and metric spaces, in "non-standard" foundations, such as ...
38
votes
4
answers
6k
views
Could groups be used instead of sets as a foundation of mathematics?
Sets are the only fundamental objects in the theory $\sf ZFC$. But we can use $\sf ZFC$ as a foundation for all of mathematics by encoding the various other objects we care about in terms of sets. The ...
- 3,137
15
votes
2
answers
1k
views
Does foundation/regularity have any categorical/structural consequences, in ZF?
(Prompted by reflection on this old answer, and its suggestion of the “harmlessness” of the axiom of regularity.)
In ZFC, one may justify the axiom of foundation (AF, aka the axiom of regularity) as ...
- 21.3k
50
votes
4
answers
6k
views
Do set-theorists use informal set theory as their meta-theory when talking about models of ZFC?
Here, Noah Schweber writes the following:
Most mathematics is not done in ZFC. Most mathematics, in fact, isn't done axiomatically at all: rather, we simply use propositions which seem "intuitively ...
- 509
19
votes
3
answers
1k
views
Large categories vs. $\mathrm{U}$-categories: why is the loss of category-theoretic information inessential?
I've asked a related question about nine months ago here, however, apparently, I lacked expertise to ask the precise question I want to ask here, as I wish to revisit the matter of universes. I hope ...
- 1,115
74
votes
8
answers
14k
views
Category theory and set theory: just a different language, or different foundation of mathematics?
This is a question to research mathematicians, as well as to those concerned with the history and philosophy of mathematics.
I am asking for a reference. In order to make the reference request as ...
- 6,937
39
votes
7
answers
6k
views
Is V, the Universe of Sets, a fixed object?
When I first read Set Theory by Jech, I came under the impression that the Universe of Sets, $V$ was a fixed, well defined object like $\pi$ or the Klein four group. However as I have read on, I am ...
- 750
5
votes
1
answer
344
views
What is the proof of consistency of anterior reflection?
Let Anterior Reflection be the following principle: $$\forall \vec{v}~ \exists X: \operatorname {transitive} (X) \land \, (\varphi \to \varphi^{X"}) $$
where $\varphi$ is a formula in $\sf FOL(=,\in)$ ...
- 11.3k
43
votes
4
answers
5k
views
Lists as a foundation of mathematics
I am wondering if there is a foundation of mathematics where not sets or "set-like objects" (such as objects of a suitable topos as in ETCS) are the primitive notion, but rather lists. These ...
- 63.1k
23
votes
2
answers
1k
views
Statements in differential geometry independent from ZFC
It is well known that some problems in functional analysis and in general topology are independent from ZFC: to name a few, Kaplansky's conjecture, the existence of outer automorphisms of the Calkin ...
19
votes
1
answer
940
views
Positive set theory and the "co-Russell" set
This is a more focused version of a question which was asked at MSE a couple years ago, but is still unanswered there. That question asks about a broad range of theories, whereas this version focuses ...
- 20.5k
19
votes
2
answers
2k
views
Which kind of foundation are mathematicians using when proving metatheorems?
Zermelo-Fraenkel set theory (with choice) is commonly accepted as the standard foundation of mathematics. It is a material set theory. For every two objects/sets $a,b$ one can ask whether $a=b$ or not....
user103598
14
votes
0
answers
390
views
Can the axiom of choice be expressed in 4 quantifiers?
This 2007 paper presents a 5-quantifier $(\in, =)$-expression that is ZF-equivalent to the axiom of choice, but leaves open the 4-quantifier case:
Thus the gap is reduced to the undecided case of a 4 ...
- 2,203
14
votes
1
answer
2k
views
Martin's "Philosophical Issues about the Hierarchy of Sets"
Some months ago (October 2010), in the context of the Workshop on Set Theory and the Philosophy of Mathematics, Professor Donald A. Martin gave a talk entitled "Philosophical issues about the ...
- 1,465
10
votes
4
answers
1k
views
Direct axiomatization of ordinal and cardinal numbers
Again, this question is related (**) to a previous one:
in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: ...
- 7,853
8
votes
1
answer
1k
views
Ill-founded models of set theory with well-founded ordinals
Let $(\mathcal{M},E)$ be an internally non-well-founded model of set theory i.e of $ZFC^{\neg f}=ZFC\setminus \mathrm{foundation}+\neg \mathrm{foundation}$, then $\mathcal{M}$ includes an infinite ...
- 2,381
8
votes
1
answer
3k
views
Foundations: Existence of uncountable ordinals.
This isn't really a research question, but at least it's research-level mathematics. I'm talking with some other people about the first uncountable ordinal, and I want some facts to inform this ...
- 2,754
7
votes
9
answers
7k
views
Ultrainfinitism, or a step beyond the transfinite
Cantor has, in the immortal words of D. Hilbert, given all of us a paradise (or perhaps, I would rather say, a great vacation spot), the TRANSFINITE.
$\aleph_0, \aleph_1,\aleph_2\dots$
the lists ...
- 7,853
6
votes
1
answer
1k
views
Some questions about Ackermann set theory
In a comment on this site Andreas Blass stated:
"To fit this situation into my philosophical point of view, I'd say that what Ackermann's theory
calls proper classes are really certain sets. That ...
- 897
5
votes
2
answers
1k
views
Large cardinals without the ambient set theory?
In an attempt to understand a bit better large cardinals, I have been thinking along the following lines, which could be summarized under the slogan
Talk about cardinals without the
(ambient) ...
- 7,853
5
votes
4
answers
2k
views
Subsystems of Peano arithmetic and incompleteness theorem
I think everyone is familiar with Goedel's incompleteness theorems. In particular they imply that PA (Peano arithmetic) can not prove its own consistency. Now my question is what is the largest ...
- 741
3
votes
2
answers
720
views
Shortest axiom of infinity for foundationless set theory
Let $T$ be the theory with a binary symbol $\in$, an unary symbol $S$, and the following axioms:
Axiom of extension:
\begin{equation}
\forall x \forall y (\forall z (z \in x \leftrightarrow z \in ...
- 2,203
-3
votes
1
answer
296
views
Can this form of reflection be consistent?
Is this form of reflection consistent?
First I'll begin by clarifying the notation I'm using here:
By a quantifier being relativized or bounded it means that the first occurrence of the quantified ...
- 11.3k
-4
votes
1
answer
198
views
Is Bounding Reflection consistent?
Working in the first order language of set theory.
Let $\varphi^{*B}$ be the formula obtained from $\varphi$ by merely bounding all open quantifiers in $\varphi$ by the symbol "$B$".
Here a ...
- 11.3k