All Questions
Tagged with set-theory mathematical-philosophy
117 questions
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, ...
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
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
11
votes
2
answers
721
views
Inconsistency and workaday independence.
Set-theoretic topologists, for example, encounter many propositions that turn out independent from set theory. Sometimes these results require novel forcing arguments, but often they simply rely on ...
- 17.6k
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
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
9
votes
2
answers
2k
views
Using the multiverse approach to decide the law of the exluded middle?
Recently, in response to deciding the Continuum Hypothesis $CH$, Hamkins and Gitman have proposed consider a multiverse of set-theoretic universes, some in which $CH$ is true, some in which $\neg CH$ ...
user2529
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
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
9
votes
1
answer
1k
views
How are material set theory and structural set theory related from the point of view of category theory?
In his comments to both cody and Nik Weaver regarding his answer to user7280899's mathoverflow question "What kind of foundation are mathematicians using when proving metatheorems?", Mike Shulman ...
- 6,132
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 $\not\exists x (\forall y (y\in x)).$ ...
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 ...
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
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
7
votes
2
answers
1k
views
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
6
votes
1
answer
209
views
Is $PRA$ + $TI({\epsilon_0})$ mutually interpretable with some theory in the language of set theory?
As is well known, the following theory is equiconsistent with $PA$:
$ZFC$ with the axiom of infinity replaced by its negation.
Since this theory is equiconsistent with $PA$, it would seem ...
- 6,132
6
votes
1
answer
447
views
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 ...
- 695
6
votes
1
answer
333
views
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 ...
- 11.3k
5
votes
3
answers
488
views
Counting without one-to-one correspondence? [closed]
Ash and Gross in their wonderful book Fearless Symmetry found it worth mentioning (and thus suggesting) another way of counting for which "we do not even need to know how to count" (in the sense of ...
- 9,720
5
votes
1
answer
483
views
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 ...
- 6,132
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
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
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
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
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
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 ...
- 141
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
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
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 ...
- 935
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
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
2
votes
0
answers
2k
views
Compatible and incompatible sets [closed]
Definition of the compatibility relation
I have defined a relation $\mathsf{C}$ for sets that captures (to some extent) the notion of compatibility.
In order to do this, we need an operation $': \...
- 121
2
votes
0
answers
305
views
Does this axiomatic system satisfy requirements for founding mathematics?
In this article, the author, F.A.Muller, suggests criteria for a founding theory of mathematics (pp:14-16). The author proposes $ARC$ Class Theory to embody these requirements. The motivation is ...
- 11.3k
2
votes
0
answers
325
views
The universe and multiverse views of set theory from the perspective of $ZFC^2$
(Note: the 'Second-order $ZFC$' ($ZFC^2$) I am talking about is the theory [in the second order language of set theory consisting of a single non-logical symbol $\in$ ] consisting of the axioms ...
- 6,132
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
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
1
vote
3
answers
1k
views
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
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
2
answers
436
views
What is against having distinct membership relations on sets in the Platonic realm?
This question is in connection with the question that I've asked at:
Where do models of false theories exist?
The answer to that question was that any consistent theory can have its primitives be re-...
- 11.3k
1
vote
1
answer
719
views
Where do models of false theories exist?
I have some difficulties in understanding the [uni]verse platonic view. How are we to understand the existence of a model of a false theory? what is the relationship of this model to the platonic ...
- 11.3k
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
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
1
vote
0
answers
194
views
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
1
vote
0
answers
261
views
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
1
vote
0
answers
163
views
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 ...
- 11.3k
1
vote
0
answers
125
views
Can $GPK^{+}_{\infty}$ + $AC_{WF}$ prove "$ZFC$ proves that the class of ordinals is not weakly compact for definable classes?
Consider the topological set theory $GPK^{+}_{\infty}$ +$AC_{WF}$, where $GPK^{+}_{\infty}$ is defined as follows (this from Olivier Esser's papers, "An Interpretation of the Zermelo-Fraenkel Set ...
- 6,132
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
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