All Questions
Tagged with set-theory soft-question
81 questions
157
votes
5
answers
28k
views
What makes dependent type theory more suitable than set theory for proof assistants?
In his talk, The Future of Mathematics, Dr. Kevin Buzzard states that Lean is the only existing proof assistant suitable for formalizing all of math. In the Q&A part of the talk (at 1:00:00) he ...
- 1,667
124
votes
17
answers
18k
views
Pressure to defend the relevance of one's area of mathematics
I am a set theorist. Since I began to study this subject, I became increasingly aware of negative attitudes about it. These were expressed both from an internal and an external perspective. By the “...
86
votes
10
answers
11k
views
What's wrong with the surreals?
Of all the constructions of the reals, the construction via the surreals seems the most elegant to me.
It seems to immediately capture the total ordering and precision of Dedekind cuts at a ...
- 1,843
67
votes
10
answers
14k
views
Arguments against large cardinals
I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally ...
- 825
63
votes
4
answers
7k
views
When size matters in category theory for the working mathematician
I think a related question might be this (Set-Theoretic Issues/Categories).
There are many ways in which you can avoid set theoretical paradoxes in dealing with category theory (see for instance ...
- 3,318
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
39
votes
5
answers
6k
views
Why do categorical foundationalists want to escape set theory?
This is a question that I have seen asked passively in comments relating to the separation of category theory from set theory, but I haven't seen it addressed in full.
I know that it's possible to ...
- 855
33
votes
3
answers
2k
views
Wiki for consequences of axiom of choice?
I raised the following question as part of another MO question, but I am following the suggestion of Nate Eldredge to make it a question in its own right.
For many years, there has a been a valuable ...
- 82.7k
31
votes
4
answers
4k
views
Is "all categorical reasoning formally contradictory"?
In the December 2009 issue of the newsletter of the European Mathematical Society there is a very interesting interview with Pierre Cartier. In page 33, to the question
What was the ontological ...
- 33.3k
31
votes
3
answers
5k
views
Should there be a true model of set theory?
As I understand it, there is a program in set theory to produce an ultimate, canonical model of set theory which, among other things, positively answers the Continuum Hypothesis and various questions ...
- 3,982
31
votes
5
answers
3k
views
Why should we care about "higher infinities" outside of set theory?
Let's say you are a prospective mathematician with some addled ideas about cardinality.
If you assumed that the natural numbers were finite, you'd quickly vanish in a puff of logic. :)
If you ...
- 801
26
votes
9
answers
8k
views
Why are proofs so valuable, although we do not know that our axiom system is consistent? [closed]
As a person who has been spending significant time to learn mathematics, I have to admit that I sometimes find the fact uncovered by Godel very upsetting: we never can know that our axiom system is ...
26
votes
4
answers
3k
views
What is the definition of a large cardinal axiom?
In different books one can find different implicit definitions for a large cardinal axiom.
My question is that which one of these definitions are more popular or standard amongst set theorists?
Any ...
user45939
24
votes
1
answer
2k
views
Why do we need "canonical" well orders?
(I asked this question on Math.SE earlier but received no response and am therefore moving it here, please note that I realise this question is probably incredibly naïve for the experienced set-...
- 343
22
votes
4
answers
4k
views
Are proper classes objects?
Many of us presume that mathematics studies objects. In agreement with this, set theorists often say that they study the well founded hereditarily extensional objects generated ex nihilo by the "...
- 1,081
22
votes
2
answers
3k
views
Where are Georg Cantor's Original Manuscripts?
Georg Cantor is famous for introducing transfinite numbers and set theory.
A main part of his mathematical point of view about this new type of "numbers" and this new "realm of mathematics" cannot be ...
- 229
21
votes
2
answers
2k
views
Philosophical arguments in defense (or against) large cardinals
The question is essentially what is asked in the title. I split it into two parts
(A) (Arguments supporting the existence of large cardinals) What are the main philosophical arguments in defense of ...
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
19
votes
0
answers
905
views
What examples of existence forcing proofs are there?
Forcing proofs tend to be fairly constructive, in the sense that if I claim that there is a forcing that does something, I usually prove this by constructing that forcing.
There are only a handful of ...
- 39.7k
18
votes
3
answers
7k
views
What is so special about set theory anyway? [closed]
(Later edit - tried to clarify a couple of vague places concerning interpretations of theories that became evident in comments (thanks to Andrej Bauer, Mauro ALLEGRANZA and Emil Jeřábek). (To closers ...
- 18.4k
17
votes
8
answers
2k
views
The Importance of ZF
It seems as though many consider ZF to be the foundational set of axioms for all of mathematics (or at least, a crucial part of the foundations); when a theorem is found to be independent of ZF, it's ...
- 397
17
votes
2
answers
2k
views
Why did Gödel name his constructible universe $L$?
It seems like Gödel didn't use the letter $L$ for his model before his book "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory", which is ...
- 1,170
16
votes
1
answer
991
views
What are the current views on consistency of Reinhardt cardinals without AC?
It's well known that Reinhardt cardinals are inconsistent, provided that we have access to axiom of choice, but, as far as I know, we are clueless about this when we don't assume choice. For me, the ...
15
votes
3
answers
1k
views
What did Paul Cohen mean by saying that generic sets of natural numbers have "no asymptotic density?"
In Paul Cohen's original 1963 paper on forcing, The independence of the Continuum Hypothesis, published in PNAS, he gives his general proof sketch of how he intends to create a model of ZFC that doesn'...
- 4,897
15
votes
2
answers
1k
views
Why is inner model theory evidence for consistency of large cardinals?
I want to understand the viewpoint that existence of canonical inner model for a large cardinal notion is strong evidence for its consistency. For example, below is Trevor Wilson's answer to What &...
- 667
15
votes
1
answer
1k
views
Where did Zermelo first model the natural numbers by iterates of the singleton operator, and have the definitions been compared by himself?
E. Zermelo is widely said to have modelled the (axioms of the) natural numbers by iterating the singleton operation $\{\cdot\}\colon \mathsf{Set}\rightarrow\mathsf{Set}$, $S\mapsto\{S\}$, whence the ...
- 6,051
15
votes
1
answer
977
views
What is the motivation behind inner model theory?
Inner model theory aims to construct canonical inner models which captures as much of V as possible, which now is formulated more concretely as to build (fine structural) mice that contain many large ...
- 1,170
15
votes
1
answer
985
views
Does inner model theory seek canonical models for large cardinals?
Like the author of this question, I have heard that a main goal of inner model theory is building canonical inner models for large cardinals. My questions are: (a) Is this accurate? (b) If so, in ...
- 18.6k
14
votes
2
answers
678
views
Are there interesting examples of theorems proved using ‘height’ extensions?
It's well known that forcing is more than a tool for proving independence: We can prove theorems and formulate axioms in theories like $\mathsf{ZFC}$ by moving to forcing extensions (e.g. $\mathfrak{p}...
- 613
13
votes
1
answer
2k
views
Hausdorff and Naive Set Theory
Erhard Scholz, in his article "Felix Hausdorff and the Hausdorff edition" writes the following:
"Hausdorff considered the contemporary attempts to secure axiomatic foundations for set theory as ...
- 6,132
12
votes
8
answers
5k
views
Is there a ground between Set Theory and Group Theory/Algebra?
It is well known that there are strong links between Set Theory and Topology/Real Analysis.
For instance, the study of Suslin's Problem turns out to be a set theoretic problem, even though it started ...
11
votes
5
answers
9k
views
Models of ZFC Set Theory - Getting Started
For just any first-order theory: What are the sets I am supposed/allowed to think of when thinking of models as sets (of something + additional structure)?
Provided:
I can think of models of any ...
- 9,720
11
votes
3
answers
1k
views
Why isn't there more interest in "large powerset axioms"?
By a large powerset axiom, let us mean informally an axiom that says that for some cardinal numbers $\kappa$, we have that $2^\kappa$ is somehow "large" or "difficult to access from below." For ...
- 3,793
11
votes
4
answers
3k
views
the delta system lemma outside set theory
The lemma:
Any uncountable set $S$ of finite sets has an uncountable subset $\Delta \subseteq S$ and an $x$ such that $\forall a,b \in \Delta$, if $a \neq b$ then $a \cap b = x$. $\Delta$ is called a ...
- 543
10
votes
3
answers
1k
views
Philosophy of forcing and ctm
I asked a similar question on SE before and received an answer. Not completely convinced, I decided to ask it here with some modifications. Note: I understand how forcing works and how it proves ...
- 237
10
votes
2
answers
763
views
measure theory for regular cardinals
Measure theory is somewhat focused on the cardinal $\aleph_0$: First of all we have the usual $\sigma$-additivity, Polish (separable!) spaces such as $\mathbb{R}^n$, countable sequences and limits, ...
- 63.1k
9
votes
1
answer
414
views
Are there some interesting propositions independent with ZF+V=L that do not increase consistency strength?
In some MO questions such as this and this, Hamkins gave some examples that is independent with ZF+V=L, however, all of them increase the consistency strength.
Are there some propositions P, which is ...
- 1,490
9
votes
1
answer
918
views
What is the etymology of zero-sharp?
I have wondered for a while what gave rise to the notation $0^\sharp$. According to wikipedia this is due to Solovay in 1967, but (perhaps unsurprisingly) there's no discussion of why that notation ...
- 437
8
votes
2
answers
1k
views
Road to Solovay's Land.
In the first semester of 2012 I took a course in General Topology and Set Theory, at undergraduate level. For topology, I was instructed to use Engelking's General Topology; albeit I had a great ...
- 405
8
votes
2
answers
789
views
Why is this theorem (about $L(P(\omega_1))^V$ and $L(P(\omega_1))^{V[G]}$) nice?
I was recently told that the following (due to M. Viale) is a nice theorem:
Suppose there are arbitrarily large supercompacts, and $\mathrm{MM}$ holds in $V$. Let $G$ be generic for a proper ...
8
votes
1
answer
689
views
Explicit uses of alephs above 'small ones'
In a paper placed on the arXiv today Shelah references theorem 0.9 from this paper (also Shelah) that uses $\aleph_{736}$ as an upper bound. This strikes me as analogous to Skewes' number. Are there ...
- 35.5k
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
8
votes
0
answers
187
views
Intuition for branch uniqueness in inner model theory
In inner model theory, what is the intuition behind the expectation that under appropriate conditions, we should have a single preferred branch to continue an iteration at a limit stage?
At the level ...
- 7,545
8
votes
0
answers
682
views
Is there any theorem achieving Conway's "Mathematician's Liberation Movement"
John Conway on in the appendix to part zero of ONAG describes a "Mathematician's Liberation Movement". The goal would be to give mathematicians the freedom to create mathematical theories with the ...
- 6,353
8
votes
0
answers
172
views
Sharply less regular cardinals in set theory
If $\lambda<\kappa$ are two regular cardinals, then every locally $\lambda$-presentable category is locally $\kappa$-presentable. The analogous statement does not hold for $\lambda$-accessible ...
- 5,482
8
votes
0
answers
1k
views
What's Reeb's take on naive integers?
Georges Reeb's "claim Q" is the statement that "naive integers don't fill up $\mathbb{N}$". To anyone familiar with model theory this could easily be interpreted as the existence of nonstandard models ...
- 16.6k
7
votes
5
answers
1k
views
the example of ccc but not separable
I am interested in the relation between the property of countable chain condition (ccc) and the property of separable. Could someone recommend some papers or books about this to me? thanks in advance.
- 654
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
6
votes
1
answer
994
views
Which branches of mathematics can be done just in terms of morphisms and composition?
Consider the first-order language $L_{\omega\omega}$ of the signature $L:=\{\mathrm{dom}, \mathrm{cod}, \mathrm{comp}\}$, where $\mathrm{dom}$ and $\mathrm{cod}$ are unary function symbols and $\...
user103598
6
votes
1
answer
662
views
Intuitive descriptions of some large cardinals
I was trying to formulate intuitive descriptions of some large cardinals.
Roughly something equivalent to "A manifold is an object which looks like patches of $R^n$ glued together". Not perfectly ...
- 675