All Questions
Tagged with set-theory mathematical-philosophy
117 questions
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0
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There is a typo in Stall's textbook on Set Theory: unable to prove the trichotomy of sets (m ∈ n or m = n, or n ∈ m) [migrated]
Here is the textbook, chapter 7, page 300. This lemma seems very of important, and I've spend about 8 hours trying to figure it out, but I'm unable to prove even the weaker version of the lemma (only ...
- 105
43
votes
16
answers
9k
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Essential reads in the philosophy of mathematics and set theory
I am graduate student and have a decent understanding of logic and set theory.
Recently I have got interested in the philosophy of mathematics and set theory. I have read a number of papers by ...
- 236k
19
votes
3
answers
2k
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Is platonism regarding arithmetic consistent with the multiverse view in set theory?
A "truth" platonist for arithmetic believes, given a statement in the language of arithmetic, that the problem whether the statement is true has a definite answer.
Prof. Hamkins has argued for a ...
- 4,713
0
votes
0
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197
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Axiom of Choice : Reasons behind using it or avoiding it? [duplicate]
I am a 1st year PhD student in Mathematics.
During my masters course in Ring theory, the proof of the result : Every commutative unital ring has a maximal ideal was done in the class. Every proof of ...
- 18.6k
67
votes
10
answers
14k
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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 ...
- 4,985
16
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1
answer
1k
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Proving that ZF is Artemov-consistent
As discussed in another MO question, Sergei Artemov has proposed that the standard formalization Con(PA) of "PA is consistent" is flawed, and has proposed a different way to formalize "...
- 82.7k
60
votes
8
answers
10k
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Why should we believe in the axiom of regularity?
Today I started reading Maddy's Believing the axioms. As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of foundation/...
- 4,713
13
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0
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Is there any correspondence between Gödel and Kreisel that supports Kreisel's observation that Gödel changed his mind about his 1938 set theory note?
At a conference in 1965 there were some interesting comments made by Kreisel and Mostowski asserting that Gödel later changed his mind regarding his1938 note on his set theory results (see Problems in ...
- 481
6
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1
answer
447
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Why should I believe Martin's Maximum++?
$\sf MM^{++}$ is a 'good' set-theoretic axiom because it is 'Maximum'. Of course, bigger is better. But I'd like to know exactly how the argument works.
Let me be clear about the question posed:
What ...
- 82.7k
12
votes
5
answers
5k
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Proper classes and their consequences
I have two main questions:
What is a proper class? I've read that it's collection of objects that's "too big" to be a set, but in what sense is such a collection "too big"? Since I'd like this post ...
- 4,713
11
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3
answers
2k
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What governs our "perception?" about the platonic realm of sets?
Here, I want to delve into what do we exactly feel about what constitutes a platonic existence of a set? Or what makes us think or actually a kind of feel or sense the existence of a set in the ...
- 571
7
votes
9
answers
7k
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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 ...
- 2,031
7
votes
2
answers
1k
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Explaining the consistency of PRA and ZF from predicative foundations
Recently I got interested in predicative foundations, mostly because of Laura Crosilla's work and because Agda employs a predicative type theory.
From the point of view of a predicative foundation to ...
- 3,312
15
votes
5
answers
2k
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In what sense does the sentence $\operatorname{con}(\mathsf{PA})$ "say" that $\mathsf{PA}$ is consistent?
It seems common amongst logicians to think of "truth" as being relative to a particular structure. Consider, for instance, the first-order theory of groups. The sentence $\forall x\forall y(...
- 63.9k
31
votes
3
answers
5k
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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 ...
- 4,713
1
vote
1
answer
362
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How are Koepke's ordinal computability and E-recursion related?
In Koepke's paper, "Turing Computations On Ordinals", one has the following (well-known) result:
A set $x$ is ordinal computable from a finite set of ordinal parameters if and only if it is ...
- 3,065
8
votes
2
answers
1k
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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 ...
53
votes
2
answers
3k
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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 ...
- 236k
26
votes
6
answers
3k
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Is there a metamathematical $V$?
As with many of you, I've been following Peter Scholze's recent question about universes with great interest. In ring theory, we don't often have to deal with proper classes, but they occasionally ...
- 2,031
5
votes
1
answer
483
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Extensions of the Ackermann interpretation to nonstandard theories of arithmetic
In their paper, " On Interpretations of Arithmetic and Set Theory" (Notre Dame Journal of Formal Logic, Vol. 8, No. 4 (2007), pp. 497-510) in section 7, "Fragments of Arithmetic and Set ...
- 2,031
40
votes
5
answers
7k
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Were Bourbaki committed to set-theoretical reductionism?
A set-theoretical reductionist holds that sets are the only abstract objects, and that (e.g.) numbers are identical to sets. (Which sets? A reductionist is a relativist if she is (e.g.) indifferent ...
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74
votes
11
answers
12k
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Why hasn't mereology succeeded as an alternative to set theory?
I have recently run into this Wikipedia article on mereology. I was surprised I had never heard of it before and indeed it seems to be seldom mentioned in the mathematical literature. Unlike set ...
- 4,713
0
votes
1
answer
887
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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 $...
- 2,912
4
votes
0
answers
409
views
Do you know any deep paradoxes or controversial hypothesis in category theory similar to those we have in set theory?
There is a lot of non-obvious and controversial topics and questions in set theory. From its begining in the first half of 20th century it have generated many paradoxes. For example there are ...
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122
votes
4
answers
39k
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Is the analysis as taught in universities in fact the analysis of definable numbers?
Ten years ago, when I studied in university, I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows:
All numbers are divided into two classes: those ...
- 4,713
1
vote
0
answers
194
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Does ${\sf ZC + Universes + ZFC}^V$ meet Muller's criteria for a founding theory of Mathematics?
I was re-thinking Muller's criteria in Sets, Classes and Categories: page 14 for a theory that founds Mathematics. To him, it should be able to be a foundation of Category Theory. He lays down six ...
- 11.3k
35
votes
8
answers
4k
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Interpretation of the Second Incompleteness Theorem
For simplicity, let me pick a particular instance of Gödel's Second Incompleteness
Theorem:
ZFC (Zermelo-Fraenkel Set Theory plus the Axiom of Choice, the usual foundation of mathematics) does not ...
- 4,713
23
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3
answers
2k
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Why would the category of sets be intuitionistic?
This question is probably really naive. And, I hope the title doesn't come off as too combative. I think that topoi of $\mathbf{Set}$-valued sheaves provide an excellent motivation for higher-order ...
- 4,713
0
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0
answers
78
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'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
70
votes
6
answers
8k
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The logic of Buddha: a formal approach
Buddhist logic is a branch of Indian logic (see also Nyaya), one of the three original traditions of logic, alongside the Greek and the Chinese logic. It seems Buddha himself used some of the features ...
CommunityBot
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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 ...
72
votes
13
answers
19k
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Logic in mathematics and philosophy
What are the relations between logic as an area of (modern) philosophy and mathematical logic.
The world "modern" refers to 20th century and later, and I am curious mainly about the second ...
- 82.7k
21
votes
2
answers
2k
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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 ...
- 101
6
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1
answer
333
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Does simple theory of types + ambiguity prove axiom of infinity?
Does simple theory of types + ambiguity prove axiom of infinity?
The simple theory of types known as $\sf TST$ is a multi-sorted first order theory, syntactical restrictions include $\in$ being a ...
- 19.6k
18
votes
2
answers
2k
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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 ...
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0
votes
0
answers
182
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Erotetic inference and extrinsic justification?
Gödel introduced his notion of what has come to be called extrinsic justification in the following terms:
Furthermore, however, even disregarding the intrinsic necessity of some new axiom, and even ...
16
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2
answers
2k
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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 ...
- 236k
1
vote
0
answers
261
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Is this theory using defined notions of classes, sets, and membership interpretable in ZFC?
The main difference with this formal theory is that it depends in an essential manner on a defined notion of class, set, and set membership $\in$, rather than the usual appraoch of leaving them ...
- 11.3k
8
votes
1
answer
330
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Impact of applying LEM to non-definite statements on definite statements
Solomon Feferman (1928 – 2016) hold that statements of arithmetic are definite, while "higher-order" notions (such as the set of all subsets of $\mathbb N$) are vague, and questions about ...
- 2,912
15
votes
1
answer
985
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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 ...
- 924
0
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1
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463
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Is a function needed here?
This question is related to my question Can we choose an element from a class?.
However, I decided to create a separate question.
Let $H$ be a complex Hilbert space and $H_1,\dotsc,H_n$ be closed ...
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19
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2
answers
2k
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Can we take a supremum over all Hilbert spaces?
In my paper On the optimal error bound for the first step in the method of cyclic alternating projections, I defined functions $f_n:[0,1]\to\mathbb{R}$,
$n\geqslant 2$, by
$$
f_n(c)=\sup\{\|P_n\dotsm ...
- 12.9k
2
votes
1
answer
452
views
Can we choose an element from a class?
Let $H$ be a complex Hilbert space and $H_1,...,H_n$ be closed subspaces of $H$.
Set $H_0:=H_1\cap H_2\cap...\cap H_n$ and let
$P_i$ be the orthogonal projection onto $H_i$, $i=0,1,2,...,n$.
I study ...
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18
votes
3
answers
5k
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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 ...
21
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1
answer
3k
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Philosophical consistency proof for set theory
In his ASL Gödel lecture (Las Vegas, Nevada, 2002), Harvey Friedman asked the following question:
Are there fundamental principles of a general philosophical nature which can be used to give ...
- 4,729
11
votes
1
answer
679
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Conceptual structuralism and continuum hypothesis
In Ferefman's paper 'Is the Continuum Hypothesis a definite mathematical problem?', he argues that within the philosophy of conceptual structuralism, the continuum hypothesis is not a definite ...
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124
votes
17
answers
18k
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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 “...
CommunityBot
- 1
1
vote
0
answers
163
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Can "description" of models revive formalism?
A model of a theory is a structure (e.g. an interpretation) that satisfies the sentences of that theory. Wikipedia
Let $A$ be a set of sentences in some language that has only one extra-logical ...
CommunityBot
- 1
1
vote
1
answer
345
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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 ...
- 1,133
0
votes
1
answer
295
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Formalizing ontological optimism
Inform speaking ontological optimisms means that everything that possibly exists in the abstract reality actually exists. From this principle we (again informally) get the Axiom of infinity, the Power ...
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