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0 answers
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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 ...
Iaroslav Baranov's user avatar
1 vote
0 answers
80 views

How strong is separation + reflection without transitivity?

Consider a theory $T$ with a binary relation $\in$ and the following axiom schemas: $\exists u \forall x (x \in u \leftrightarrow x \in a \land \phi)$ where $u$ is not free in $\phi$. This is the ...
user76284's user avatar
  • 2,203
14 votes
0 answers
389 views

Can the axiom of choice be expressed in 4 quantifiers?

This 2007 paper presents a 5-quantifier $(\in, =)$-expression that is ZF-equivalent to the axiom of choice, but leaves open the 4-quantifier case: Thus the gap is reduced to the undecided case of a 4 ...
user76284's user avatar
  • 2,203
12 votes
1 answer
227 views

Is there a $\Pi_2$ sentence $A$ such that $\text{ZFC}^- + A$ proves powerset?

This is a follow-up to this question. Let $\text{ZFC}^-$ be ZFC without powerset and with collection rather than replacement, as described here. Is there a $\Pi_2$ (or perhaps $\Sigma_2$) sentence $A$ ...
user76284's user avatar
  • 2,203
16 votes
2 answers
1k views

CH in non-set theoretic foundations

I asked this question on stack exchange and got little attention, barring a nice example I intend to look into. The original post can be found here: https://math.stackexchange.com/q/4941233/1053681 I ...
Joseph_Kopp's user avatar
13 votes
1 answer
933 views

Consistency strength of HoTT

What is the consistency strength of Homotopy type theory (HoTT) relative to various set theories (e.g., are there any known set theories that it can interpret)? Does this question even make sense?
Jesse Elliott's user avatar
3 votes
1 answer
256 views

Can these short set-building expressions of the finite set world extend to the infinite set world?

A formula of the form $\forall \vec{p}\, \exists x \, \forall y\, (y \in x \leftrightarrow \phi(y,\vec{p}))$ is to be named a "set-building" formula. Now, when $\vec{p}$ includes a predicate ...
Zuhair Al-Johar's user avatar
14 votes
2 answers
1k views

Type vs. Set Theory: Expressive Ability

In the modern mathematical arena, the two primary contenders for the ‘correct’ foundation of mathematics are set and type theory. Set theory, very roughly, captures the intuition that we frequently ...
Alec Rhea's user avatar
  • 10.1k
29 votes
3 answers
3k views

Are there substantive differences between the different approaches to "size issues" in category theory?

In category theory, there are different ways to approach the "size issues" that crop up when we try to formalise the subject in axiomatic set theory. As far as I can tell, there are two main ...
Joe Lamond's user avatar
2 votes
0 answers
220 views

Which is richer Set or Graph Theory?

This theory about structures, defined as abstractions over isomorphic graphs, can interpret Set Theory in a rather creepy manner. Though the theory is largely technical, yet it is not far from being ...
Zuhair Al-Johar's user avatar
3 votes
0 answers
165 views

Suitability of formal type theory for mathematical thinking (vs. traditional set theory)

Type theory has advantages over set theory for the (computer) formalisation of mathematics, but has anybody who does mathematics with pen and paper found proof assistants or automated theorem provers, ...
Troubled Shallows's user avatar
11 votes
3 answers
2k views

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 ...
Zuhair Al-Johar's user avatar
5 votes
1 answer
597 views

The "first-order theory of the second-order theory of $\mathrm{ZFC}$"

$\newcommand\ZFC{\mathrm{ZFC}}\DeclareMathOperator\Con{Con}$It is often interesting to look at the theory of all first-order statements that are true in some second-order theory, giving us things like ...
Mike Battaglia's user avatar
2 votes
1 answer
200 views

Does inductive definitions must be supported by the set theoretical definition of natural numbers?

In page 4 of Gödel's book The Consistency Of The Axiom Of Choice and Of The Generalized Continuum Hypothesis With The Axioms Of Set Theory, Gödel defined the $n$-tuple as $\langle x \rangle = x$; $\...
Wenchuan Zhao's user avatar
-4 votes
1 answer
198 views

Is Bounding Reflection consistent?

Working in the first order language of set theory. Let $\varphi^{*B}$ be the formula obtained from $\varphi$ by merely bounding all open quantifiers in $\varphi$ by the symbol "$B$". Here a ...
Zuhair Al-Johar's user avatar
5 votes
1 answer
344 views

What is the proof of consistency of anterior reflection?

Let Anterior Reflection be the following principle: $$\forall \vec{v}~ \exists X: \operatorname {transitive} (X) \land \, (\varphi \to \varphi^{X"}) $$ where $\varphi$ is a formula in $\sf FOL(=,\in)$ ...
Zuhair Al-Johar's user avatar
-3 votes
1 answer
296 views

Can this form of reflection be consistent?

Is this form of reflection consistent? First I'll begin by clarifying the notation I'm using here: By a quantifier being relativized or bounded it means that the first occurrence of the quantified ...
Zuhair Al-Johar's user avatar
3 votes
1 answer
96 views

Is this form of replacement suitable for ZF - Powerset + well-ordering principle?

The following scheme can be understood as a form of replacement. Axiomatizing $\sf ZF$ with it instead of the usual replacement schema renders it immune to removal of extensionality; see here. In an ...
Zuhair Al-Johar's user avatar
12 votes
0 answers
210 views

Are there times when replacement is "more natural" than collection?

There are a couple examples I'm aware of where choosing to axiomatize $\mathsf{ZF(C)}$ using collection instead of replacement results in a much nicer (or at least less surprising) picture: Let $\...
Noah Schweber's user avatar
4 votes
1 answer
368 views

Bounded alternatives to powerset that interpret ZFC

In set theory, many properties/relations of interest can be expressed as $\Delta_0$ formulas (formulas with only bounded quantifiers): \begin{align} \text{empty}(a) &\equiv \forall x \in a . \...
user76284's user avatar
  • 2,203
7 votes
3 answers
460 views

How much Dependent Choice is provable in $Z_2$? And what about Projective Determinacy?

So, second order arithmetic, $Z_2$, is capable of proving quite a few things. One thing which would be of use is dependent choice for $\mathbb{R}$. Basically, dependent choice on $\mathbb{R}$ says ...
Alex Appel's user avatar
11 votes
1 answer
1k views

Existence of skeletons in ZFC

Influenced by this question from a fellow lagomorph, I would like to get to the bottom of existence of a skeleton of a category. I want to stay in ZFC, so I do not assume the global axiom of choice. ...
Bugs Bunny's user avatar
  • 12.3k
43 votes
4 answers
5k views

Lists as a foundation of mathematics

I am wondering if there is a foundation of mathematics where not sets or "set-like objects" (such as objects of a suitable topos as in ETCS) are the primitive notion, but rather lists. These ...
Martin Brandenburg's user avatar
6 votes
0 answers
190 views

Is Vopěnka's principle inherited by Grothendieck topoi?

I call the Vopěnka's principle: Every subfunctor of an accessible functor is accessible but other formulations (which may lose equivalence in weak contexts?) are also interesting to me. If this is ...
Arshak Aivazian's user avatar
22 votes
4 answers
4k views

How much of the axiom of choice do you need in mathematics?

Say we have DC-λ where λ is some inaccessible cardinal. Is that enough to develop all of ordinary mathematics? If not, is there a strengthening that is but that nevertheless does not assume full ...
Someone211's user avatar
2 votes
0 answers
133 views

Higher-order oracle computation of reals and axiom of constructibility

Certain real numbers can be approximated arbitrarily well by computable functions. If we introduce halting oracles, then more real numbers can be "computed", like Chaitin's constant or the ...
GChromodynamics's user avatar
12 votes
0 answers
269 views

Freiling's question

(I've asked this question at Math StackExchange here but haven't gotten any response, so I decided to take a shot here as well.) In his paper "Axioms of Symmetry: Throwing Darts at the Real ...
Y.Z.'s user avatar
  • 231
2 votes
1 answer
69 views

Infinite decreasing sequence for class relation without minimal elements

Let us assume $<$ is some class relation without minimal elements, meaning $\forall a, \exists b, b< a$. This means that for every $n\in\omega$, one can build a decreasing function $f$ with ...
ViHdzP's user avatar
  • 447
23 votes
2 answers
1k views

Statements in differential geometry independent from ZFC

It is well known that some problems in functional analysis and in general topology are independent from ZFC: to name a few, Kaplansky's conjecture, the existence of outer automorphisms of the Calkin ...
3 votes
2 answers
333 views

On the definition of small categories in SGA4

We assume ZFC+U. A category is an ordered pair $(\operatorname{Ob} \mathcal{C},\operatorname{Mor} \mathcal{C},\operatorname{dom},\operatorname{codom},e,∘)$ of sets (not classes) and maps satifying ...
LOCOAS's user avatar
  • 405
3 votes
1 answer
510 views

Harvey Friedman: The expanding mind

In reference 1, Friedman writes: I discuss my efforts concerning 3 crucial issues in the foundations of mathematics that are deeply connected with the great work of Kurt Gödel. [...] B. Are there ...
user76284's user avatar
  • 2,203
6 votes
1 answer
892 views

Can the axiom of choice be proved with ZF+Tarski axiom?

Can choice be proved with ZF+Tarski axiom?
Carlos Freites's user avatar
16 votes
2 answers
2k views

Why do we care about small sets?

I have been a user of category theory for a long time. I recently started studying a rigorous treatment of categories within ZFC+U. Then I become suspecting the effect of the smallness of sets. We ...
LOCOAS's user avatar
  • 405
6 votes
1 answer
205 views

How strong is separation + reflection of unbounded quantifiers?

Consider a set theory with the following axioms: separation: $\exists y \forall x (x \in y \leftrightarrow \phi \land x \in a)$ where $y$ is not free in $\phi$ reflection: $\phi \to \exists u \phi^u$ ...
user76284's user avatar
  • 2,203
-6 votes
1 answer
403 views

Is ZFC set theory a satisfactory foundation for mathematics?

The conventional wisdom seems to be that it is, but there are problematic mismatches. Some are well-known: the use of first-order logic, and the many different implementations of the axioms, neither ...
Frank Quinn's user avatar
1 vote
0 answers
57 views

What is the consistency strength of this addition on simple type-set theory?

Language: multi-sorted first order logic with equality and membership, where for each natural $n$ there is a set $x^n$ of sort $n$. Equality "$=$" only occurs between variables of the same ...
Zuhair Al-Johar's user avatar
1 vote
0 answers
123 views

Is $\sf \Gamma_0$ the proof theoretic ordinal of this kind of predicative class theory?

Adopting the approach of Mono-sorted $\sf NBG$, define sets as elements of classes, then axiomatize: Extensionality, Predicative Class comprehension, emptyset, in the usual manner along mono-sorted $\...
Zuhair Al-Johar's user avatar
1 vote
0 answers
127 views

What is the proof theoretic ordinal of this kind of predicative type-set theory?

The following is a kind of Predicative Type Set Theory. The question is about what is exactly the proof theoretic ordinal of this theory? Is it lower than the one expected for predicative theories, i....
Zuhair Al-Johar's user avatar
1 vote
0 answers
117 views

Can this type theory interpret second order arithmetic?

Language: multi-sorted first order logic with equality and membership, where for each natural $t$ there is a set $x^t$ of sort $t$. Equality "$=$" only occurs between variables of the same ...
Zuhair Al-Johar's user avatar
1 vote
0 answers
94 views

Is definability in $V$ in $\sf Ack+MK$ expressible in its language?

Recall Ackermann set theory. If we extend Ackermann's set theory by adding all axioms of $\sf MK$ to it. We shall denote the universe of all elements by $W$, while $V$ is the primitive constant symbol ...
Zuhair Al-Johar's user avatar
7 votes
2 answers
588 views

Consistency strength of an attempt at higher order set theory

Work in a theory with (deep breath) a countable number of primitives denoted with capital letters from the end of the alphabet with numerical subscripts $\{X_n,Y_n,Z_n,\dots\}_{n<\omega}$ ...
Alec Rhea's user avatar
  • 10.1k
1 vote
0 answers
204 views

Can Frege set extensions of second order unary predicates serve as a foundation for mathematics?

To second order logic, add a primitive partial one place function symbol "$\epsilon$" from unary predicate symbols [upper cases] to object symbols [lower cases]. Define: $ x=y \equiv_{df} \...
Zuhair Al-Johar's user avatar
6 votes
1 answer
1k views

An axiomatic approach to the multiverse of sets

Work in a theory where the primitives are classes $X,Y,Z,\dots$, and class membership $X\in Y$, and add an individual constant $\mathcal{M}$ called 'the multiverse'. Classes $V$ which are members of ...
Alec Rhea's user avatar
  • 10.1k
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 ...
aidangallagher4's user avatar
17 votes
0 answers
509 views

The free complete lattice on three generators, beyond ZF

This was originally asked at MSE; although it is still under bounty it seems unlikely to be answered there. $\mathsf{ZF}$ proves that there is no free complete lattice on three generators since any ...
Noah Schweber's user avatar
12 votes
0 answers
574 views

Harvey Friedman's minimalist axioms for set theory

[This is a question on the FOM mailing list.] In 1997, Harvey Friedman introduced the following theory: Let $\in$ be a binary predicate and $U$ be a constant. Add the following axioms: Subworld ...
user76284's user avatar
  • 2,203
1 vote
0 answers
192 views

Does foundationless Ackermann set theory prove replacement?

From Ackermann's set theory equals ZF (1970) by William N. Reinhardt: Let A be the theory determined by the following axioms: Extensionality: $\forall z (z \in x \leftrightarrow z \in y) \to x = y$ ...
user76284's user avatar
  • 2,203
3 votes
0 answers
184 views

Can we interpret ZFC in GEM?

I have long held similar views as put forth here. As professor Hamkins points out though, $(M,\subseteq)$ is insufficient to model most of mathematics. However, this is unsatisfactory to me, as this ...
tox123's user avatar
  • 433
6 votes
1 answer
444 views

What do you call the generalisation of the direct image?

This question was posted on Math Stack Exchange, but did not attract an answer. Here is the question: Informal Description Let me start with an example. Let $X$ be the set $\{a, b, c, d, e\}$ and $E$ ...
Yes's user avatar
  • 115
5 votes
2 answers
629 views

Applications of ZFA-Set Theory

The set theory with atoms (ZFA), is a modified version of set theory, and is characterized by the fact that it admits objects other than sets, atoms. Atoms are objects which do not have any elements. ...
Yes's user avatar
  • 115