All Questions
19 questions
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
53
votes
2
answers
3k
views
Silver's approach to the inconsistency of $\mathrm{ZFC}$
As all probably know, Jack Silver passed away about one month ago. The announcement released, with delay, by European Set Theory Society includes a quote by Solovay about his belief on inconsistency ...
- 2,381
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
26
votes
7
answers
6k
views
What "forces" us to accept large cardinal axioms?
Large cardinal axioms are not provable using usual mathematical tools (developed in $\text{ZFC}$).
Their non-existence is consistent with axioms of usual mathematics.
It is provable that some of ...
user47697
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 ...
18
votes
2
answers
2k
views
A “paradox” about the inner model problem
As stated in Woodin, Davis, and Rodriguez - The HOD dichotomy, a longstanding open problem in set theory is to construct a canonical inner model for supercompactness. In general there are various ...
- 18.6k
18
votes
3
answers
5k
views
Is there a compendium of the consistency strength between the most important formal theories?
Preliminar Notions:
A formal system is a tuple $(\Sigma,G,A,R)$ where $\Sigma$ is an alphabet (set of symbols), $G$ is a formal grammar on $\Sigma$ that generates a formal language $L$ (set of well ...
- 283
16
votes
2
answers
2k
views
Why should I believe the Singular Cardinal Hypothesis?
The Singular Cardinal Hypothesis (SCH) is the statement that $\kappa^{cf(\kappa)} = \kappa^+ \cdot 2^{cf(\kappa)}$ for every singular cardinal $\kappa$ (or various equivalent statements).
It is ...
- 1,793
11
votes
1
answer
1k
views
Belief in consistency of extremely large cardinals
One of the most common justifications for the consistency of large cardinals is the development of a coherent inner model theory for many large cardinal axioms. While the strength of this argument can ...
- 3,816
9
votes
2
answers
1k
views
The impact of large cardinals in mathematics [closed]
What are the main applications of large cardinals in ordinary mathematics, and what is the philosophy behind using them. In particular:
Question 1. What is the philosophy behind accepting large ...
- 32.1k
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
2
answers
1k
views
Large cardinal near inconsistencies
I am looking for examples of results about large cardinals, large cardinal axioms, or other objects of high (or seemingly high) consistency strength that are almost inconsistencies. I am looking for ...
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
5
votes
0
answers
947
views
Why are real-valued measurable cardinals never explicitly mentioned in Gödel's "What is Cantor's Continuum Problem"?
It is a matter of mathematical folklore that Gödel "entertained the idea of so called stronger axioms of infinity deciding $CH$...." (this quote from Radek Honzik's paper, "Large ...
- 6,132
1
vote
2
answers
580
views
A question regarding the relation between Freiling's Axiom of Symmetry and real-valued measurable cardnals
A major argument against Freiling's Axiom of Symmetry is the following (this from the wikipedia article of the same name):
"The naive probabalistic notion used by Freiling tacitly assumes that there ...
- 6,132
1
vote
1
answer
446
views
A question regarding extendible cardinals and a result of M. Magidor
The following definitions and Theorems come from M. Magidor's paper "On the Role of Supercompact and Extendible Cardinals in Logic" (Israel J. Math., Vol. 10, 1971):
"Definition: Logic is called $\...
- 6,132
1
vote
1
answer
345
views
Proving independence with large cardinals?
Suppose I want to prove some statement S is independent of ZFC.
Now instead of the usual approach of making models, I do the following:
- Take two large cardinal axioms L1 and L2
- Prove that ZFC + L1 ...
- 675
0
votes
1
answer
887
views
Forcing the existence of a weakly inaccessible cardinal in some strong set theory
Does the fact that, assuming the consistency of $ZFC$, no proof that the consistency of "$ZFC$ implies the consistency of '$ZFC$ + There exists a weakly inaccessible cardinal'" can be formulated in $...
- 6,132
0
votes
0
answers
78
views
'Maximising interpretative power entails maximising consistency strength'?
I'm hoping there is a clear mathematical answer to this question (hence asking it here) rather than anything more exegetical (in which case it's presumably not appropriate for this site).
In his paper ...
- 161