All Questions
15 questions
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 “...
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
0
votes
2
answers
582
views
Why are model theorists free to use GCH and other semi-axioms? [closed]
Looking into the open problem section of the book Model theory by Chang and Keisler, I noticed that many problems assumed semi-axioms like GCH. I talk about 'semi-axioms' because these "axioms" are ...
- 87
-1
votes
1
answer
319
views
Tarski-Grothendieck in the cumulative hierarchy
How can the Grothendieck-Tarski axiom seen to be true in the cumulative hierarchy of sets?
What are intuitions that would convince us that this axiom is true?
0
votes
1
answer
678
views
Why do we try to encode every mathematical object as a set? [closed]
Probably everyone of us has seen set-theoretic encodings of mathematical objects which we wouldn't naturally consider to be sets. May it be the "definition" of a function from $A$ to $B$ as a relation ...
- 27
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
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 ...
3
votes
0
answers
342
views
A Question Regarding Boolean-valued Models
What were the intuitions motivating the creation (or discovery, if you will) of Boolean-valued models? I have searched for the Scott-Solovay paper on the subject, but to no avail. There also seems to ...
- 6,132
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
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
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
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 ...
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
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
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