All Questions
85 questions
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
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
8
votes
1
answer
1k
views
Does equality between sets contradict the philosophy behind structural set theory?
Zermelo-Fraenkel set theory (with choice) is commonly accepted as the standard foundation of mathematics. It is a material set theory. This means that for every two objects/sets $a,b$ one can ask ...
- 89
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
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
4
votes
3
answers
915
views
Compactness of existential second order logic and definability of certain quantifiers
It is known (as a slogan) that the "existential fragment of second-order logic (ESO) is compact".
My first question is:
(1) Is ESO compact for:
(a) uncountable languages
(b) languages with ...
- 135
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
9
votes
1
answer
856
views
Taller models of ZFC
This question is somewhat related to a previous one, where I asked for new forms of infinite beyond the cardinal hierarchy.
Using forcing techniques, at least the ones I know of, one starts from a ...
- 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
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
-3
votes
1
answer
262
views
An axiomatic system with a set of constants that form a complete ordered field [closed]
I am developing a ZFC axiomatic system where together with the empty set, there is a singular (and huge) set of constants that are themselves sets and form a complete ordered field (cof) these ...
- 201
6
votes
2
answers
1k
views
Why can't mathematics be formalised in terms of classes rather than sets? [closed]
I've always been curious about the seeming compulsion to found mathematics upon sets, be it ZF(C) or some other system. Of course, there are other approaches these days like category theory and type ...
- 820
9
votes
3
answers
2k
views
What set theoretical questions could never be answered by Turing machines of arbitrary cardinality?
Let us assume that there are Turing machines of arbitrary cardinality, by that I mean they can have input tapes of any arbitrarily high cardinality and compute for a number of steps also of ...
- 297
12
votes
3
answers
649
views
Has the Ramified Theory of Types been applied to NBG?
Questions of predicativity are well-studied in the context of arithmetic. We have a base theory, first-order Peano arithmetic. Some people, like Edward Nelson (in chapter 1 of his book) and Charles ...
- 4,597
12
votes
2
answers
748
views
Ways to define "definability"
The notion of a definable set is not expressible in the language of set theory: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y)$ such that $x = \lbrace y : \phi(...
- 9,720
4
votes
2
answers
452
views
On wild behavior of $\omega_{1}$ in the absence of some essential axioms of $ZFC$
The regularity of $\omega_{1}$ is one of the most well known facts of set theory. But it seems that in order to prove this simple fact we need the "full power" of mathematics! For example by an ...
user36136
1
vote
0
answers
257
views
Is there a non-trivial consistency preserving transformation?
In set theory "equiconsistency" (and not "consistency") of the theories is the main part of researches. So we usually try to construct a new model using a given one. In the ...
user36136
15
votes
4
answers
2k
views
Where is the end of universe?
In some sense the empty set ($\emptyset$) and the global set of all sets ($G$) are the ends of the universe of mathematical objects. The world which $ZFC$ describes has an end from the bottom and is ...
user36136
-3
votes
3
answers
836
views
Can different extensions of ZF have contradictory consequences for first-order arithmetic?
My question is basically, does there exist a statement X independent of ZF such that ZF + X implies a statement P of first-order arithmetic, but ZF + not X implies not P?
Now X cannot be the axiom ...
- 4,597
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 ...
1
vote
1
answer
261
views
Finite level super classes over ZFC
My question is: "Is it possible to have a sound and rigorous legitimation of the following construction ?". This construction is:
0/ Let ZFC be the usuel set theory, and let us add to the language ...
- 2,655
3
votes
1
answer
422
views
Should functions be assumed to behave like the identity function when evaluated outside their domain?
Suppose we have a set $f$ of ordered pairs (so not a triple $(X,Y,f)$ but just the $f$) and suppose that $f$ has the appropriate property such that we can view $f$ as a function. Formally, we wish to ...
- 31
8
votes
2
answers
797
views
weakening naive comprehension to avoid the paradoxes
Weakening the axiom of naive comprehension has not been a popular way of escaping from the set-theoretic paradoxes because no consistent weakenings seem to be particularly well motivated or even to ...
- 81
1
vote
1
answer
535
views
Tarski-Grothendieck set theory, the axiom of pairing and the axiom of specification
I am building upon MO question 102846 concerning the Tarski-Grothendieck set theory (TG).
I have two questions;
1/ I think that it is possible that the axiom of pairing (axiom 4 of the TG theory ...
- 2,655
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
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
9
votes
1
answer
798
views
Ultimate Maximality Principle
I wonder if it's possible to formulate an "ultimate" maximality principle (UMP) and prove its consistency. I envision UMP to express the idea that no matter how we enlarge the universe of set theory V ...
- 93
2
votes
1
answer
275
views
comprehension and ideal elements
A not uncommon thought in philosophy is that we should distinguish (in philosophy, anyway) between "sparse" ("real", "serious") and "abundant" ("ideal", "superficial") properties/classes and relations....
- 313
26
votes
4
answers
1k
views
Are there lightweight foundations for arbitrarily extendable objects?
My experience with foundations is rather scant, but I've run into some types of objects that seem to resist the sort of set-theoretic encoding schemes via Kurowski tuples that are rather common for ...
- 45.7k
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
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
2
answers
736
views
Sets as Combinatorial Games
Just a few days ago my seemingly eternal and recurrent fascination for Conway's combinatorial game theory (CGT) & surreal numbers had a recrudescence, so I grabbed this excellent survey, and began ...
- 7,853
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
13
votes
7
answers
2k
views
(Non?)-linearity of the consistency strength ordering in ZF
Much of the research taking place in set theory, is related to the classification of sentences according to their consistency strength relative to ZF. In order to be more specific, we say that for all ...
- 360
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