All Questions
Tagged with set-theory mathematical-philosophy
117 questions
11
votes
1
answer
1k
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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
55
votes
10
answers
11k
views
How should a "working mathematician" think about sets? (ZFC, category theory, urelements)
Note that "a working mathematician" is probably not the best choice of words, it's supposed to mean "someone who needs the theory for applications rather than for its own sake". Think about it as a ...
- 1,115
3
votes
1
answer
182
views
What is the weakest large cardinal property which is equiconsistent to weak compact cardinal?
Accoding to wiki, weak compact cardinal is a very weak property in the large cardinal ladder.
But, like ZFC+CH to ZFC, weak compact has some "useless part", so that even the first Woodin cardinal may ...
- 475
1
vote
1
answer
362
views
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 ...
- 6,132
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
1
vote
0
answers
260
views
Is $\mathit{GPK}^{+}_{ \infty}+\mathit{BAFA}$ inconsistent (and why does it matter)?
Consider Olivier Esser’s alternative axiomatic set theory $\mathit{GPK}^{+}_{\infty}$. Esser defines it as follows (this from his paper "Inconsistency of The Axiom of Choice with The Positive Theory $...
- 6,132
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
8
votes
1
answer
330
views
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 ...
- 81
5
votes
0
answers
323
views
Can the Kunen inconsistency (or the existence of Reinhardt cardinals) be 'properly formulated' in Ackermann set theory?
In their paper "Generalizations of the Kunen Inconsistency" (arXiv:1106.1951v1 [math.LO]10 Jun. 2011), Hamkins, Kirmayer, and Perlmutter write the following:
The first [metamathematical issue--my ...
- 6,132
-2
votes
1
answer
281
views
Critical points and the Foundation Axiom
(Note: This question is related to my previous mathoverflow question, "Critical Points in $ZF$ without Choice".)
In the Stanford Encyclopedia of Philosophy entry "Non-Wellfounded Set Theory" (...
- 6,132
1
vote
0
answers
265
views
Can Dedekind's 'proof' of the existence of infinite sets be properly formulated and carried out in positive set theory?
This question is related to Mikhail Katz's recent mathoverflow question, "Has Dedekind's proof of the existence of existence of infinite sets been analyzed by historians?". Dedekind's 'proof' seems (...
- 6,132
12
votes
3
answers
3k
views
Has Dedekind's proof of existence of infinite sets been analyzed by historians?
This pdf by David Joyce notes that in paragraph 66 of his famous essay, Dedekind claims to prove the existence of an infinite set.
The proof exploits the assumption that there exists a set $S$ of all ...
- 16.6k
13
votes
1
answer
751
views
Is there a class of mathematical structures with non-isomorphic natural representations as a standard Borel space?
Background. The field of Borel equivalence relation theory
provides a robust, unifying theory that organizes most of the
classification problems of classical mathematics into a hierarchy,
allowing us ...
- 236k
9
votes
3
answers
1k
views
The universe of sets, existential quantification in set theory
Yesterday, I posted a question that was received in a different way than I intended it. I would like to ask it again by adding some context.
In ZF one can prove $\not\exists x (\forall y (y\in x)).$ ...
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
19
votes
3
answers
2k
views
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 ...
- 261
9
votes
3
answers
3k
views
What does the axiom of replacement mean and why should I believe it?
Here Professor Blass describes the following cumulative hierarchy of sets:
Begin with some non-set entities called atoms ("some" could be "none" if you want a world consisting exclusively of sets), ...
- 101
13
votes
0
answers
882
views
Arguments against Freiling's argument against Continuum Hypothesis
Freiling's axiom of symmetry ($\sf AS$) is known as a justification for falsity of Continuum Hypothesis. Freiling in his 1986 paper, Axioms of symmetry: throwing darts at the real number line, ...
2
votes
0
answers
134
views
A question regarding an analogue of the Kleene $T$-predicate for Koepke's ordinal computability
Does Koepke's notion of ordinal computability admit an analogue of the Kleene $T$-predicate? If so, is the existence of such a $T$-predicate independent of $ZFC$? Also, if one assumes the existence ...
- 6,132
60
votes
8
answers
10k
views
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/...
- 28.2k
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
4
votes
1
answer
386
views
Plausibility argument for a measurable cardinal
The following question is not mathematically precise but perhaps of some philosophical interest.
A typical plausibility argument for assuming the existence of inaccessible cardinals goes as follows: ...
- 41
2
votes
1
answer
880
views
Is second-order ZFC categorical with regard to its proper class models
Second-order ZFC offers partial categoricity in the sense that, given any two models, one of them must be isomorphic to an initial segment of the other [1]. However, this leaves questions regarding ...
- 95
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
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
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
13
votes
2
answers
1k
views
Can we define an "empirically generic" real number?
Summary: My question, in a nutshell, is how we should intuitively imagine a generic real number (as opposed to a random one), and whether we can construct numbers which empirically behave like generic ...
- 32.5k
11
votes
2
answers
1k
views
Last Status of Feferman's Conjecture on Indefinite Value of Continuum
The "true" value of $2^{\aleph_0}$ is one of the most fundamental open questions of mathematics and its philosophy. Hundreds of set theoretic results during the last century don't say anything more ...
user45939
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
4
votes
2
answers
749
views
What is the impact on Godels theorem of Paraconsistency?
Russells paradox forced a restriction of the natural abstraction principle (that every predicate determines a set) so that Set Theory could be consistent. The standard one being ZF.
However ...
- 2,083
1
vote
0
answers
134
views
A Question Regarding Productive Sets in the Koepke-Koerwien System SO (Sets of Ordinals)
In their paper "The Theory of Sets of Ordinals" (arXiv), Koepke and Koerwien propose a theory SO axiomatizing the class of sets of ordinals in a model of ZFC and show that SO and ZFC are bi-...
- 6,132
1
vote
2
answers
819
views
Ontological status of some "sets" in ZFC [closed]
Let $\phi$ be an undecidable statement of ZFC set theory, for example let's take continuum hypothesis.
What is the ontological status of the "set" $X=\bigl\{x\in\{1,2\}:x=1\text{ or }(x=2\text{ and }\...
- 91
11
votes
1
answer
679
views
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 ...
- 111
1
vote
0
answers
260
views
A question regarding Koepke' s Ordinal Computability in HOD
Consider the following theorem of Koepke-Koerwien-Siders:
"A set x of ordinals is ordinal computable [either by ordinal Turing machines or ordinal register machines--my comment] if and only if it is ...
- 6,132
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
1
vote
3
answers
1k
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Sets = structured sets without structure
Motivation
There is presumably no single and widely accepted formal definition of structured sets = sets plus structure based on sets as primitive objects, but several approaches are around. See e.g. ...
- 9,720
9
votes
4
answers
1k
views
Does there exist a non-trivial Ultrafinitist set theory?
Does there exist a set theory T-which has not yet been proved to be inconsistent-and in which
one can prove the existence of (1) the empty set (2) sets that are singletons and (3) sets which
have non-...
- 6,215
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
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
43
votes
16
answers
9k
views
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 ...
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
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
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
72
votes
13
answers
19k
views
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 ...
- 24.7k
74
votes
11
answers
12k
views
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 ...
- 5,902
11
votes
3
answers
2k
views
Kunen's use of Countable Transitive Models
Hi,
I have a doubt concerning Kunen's exposition of forcing in his classical book (arguably $the$ book on forcing). When dealing with Countable Transitive Models to set up the forcing machinery, ...
