All Questions
64 questions
16
votes
1
answer
1k
views
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
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
15
votes
5
answers
2k
views
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(...
- 961
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
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
0
votes
0
answers
182
views
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 ...
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
0
votes
1
answer
463
views
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 ...
- 935
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
19
votes
2
answers
2k
views
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 ...
- 935
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
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
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
35
votes
8
answers
7k
views
Why not adopt the constructibility axiom $V=L$?
Gödelian incompleteness seems to ruin the idea of mathematics offering absolute certainty and objectivity. But Gödel‘s proof gives examples of independent statements that are often remarked as having ...
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 “...
14
votes
2
answers
994
views
Set-theoretical foundations of Mathematics with only bounded quantifiers
It seems that outside of researchers in Mathematical Logic, mathematicians use almost exclusively bounded quantifiers instead of unbounded quantifiers. In fact, I haven't observed any other practice ...
- 5,327
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
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
20
votes
1
answer
1k
views
Axiom of Choice versus V=L in opposition to large cardinals
Consider the following two observations:
The axiom $V=L$ is incompatible with large cardinal axioms that are somehow "too large", like measurable cardinals.
The axiom of Choice is incompatible with ...
- 32.5k
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
70
votes
6
answers
8k
views
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 ...
-3
votes
1
answer
341
views
What is the intuitive notion that ZF-Extensionality-Foundation+Collection can be said to capture? [closed]
This question has been moved to philosophy.stackexchange.com
I'll try to abbreviate it here: the question asks about the "informal notion" that the fragment of $\text{ZFC}$ that is axiomatized by ...
- 11.3k
15
votes
1
answer
986
views
Does every model of ZF-foundation have an extension, with no new well-founded sets, where every set is bijective with a well-founded set?
This question follows up on an issue arising in Peter LeFanu
Lumsdaine's nice question: Does foundation/regularity have any
categorical/structural consequences, in
ZF?
Let me mention first that my ...
- 236k
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 ...
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