All Questions
85 questions
1
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82
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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 ...
- 2,203
14
votes
0
answers
391
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 ...
- 2,203
12
votes
1
answer
227
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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 $A$ ...
- 2,203
13
votes
1
answer
933
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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?
- 4,985
3
votes
1
answer
256
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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 ...
- 11.3k
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 ...
- 11.3k
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 ...
- 4,907
-4
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1
answer
198
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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 ...
- 11.3k
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 $\sf FOL(=,\in)$ ...
- 11.3k
-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 ...
- 11.3k
3
votes
1
answer
96
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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 ...
- 11.3k
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 $\...
- 20.5k
4
votes
1
answer
369
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 . \...
- 2,203
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 ...
- 63.1k
6
votes
0
answers
190
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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 ...
- 6,962
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
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 ...
- 2,203
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: $\phi \to \exists u \phi^u$
...
- 2,203
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 ...
- 11.3k
1
vote
0
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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 $\...
- 11.3k
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....
- 11.3k
1
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0
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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 ...
- 11.3k
1
vote
0
answers
94
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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 ...
- 11.3k
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 $\{X_n,Y_n,Z_n,\dots\}_{n<\omega}$ ...
- 10.1k
6
votes
1
answer
1k
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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 ...
- 10.1k
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 ...
- 20.5k
12
votes
0
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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 ...
- 2,203
1
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0
answers
192
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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: $\forall z (z \in x \leftrightarrow z \in y) \to x = y$
...
- 2,203
10
votes
1
answer
451
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Is material set theory conservative over structural set theory?
Suppose a statement $\phi$ that doesn't use the global $\in$-relation or the global $=$-relation in an essential way is provable in some material set theory, say bounded Zermelo with choice. (So that ...
- 101
6
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1
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309
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Set Theoretic Geology II: The structure of the directed partial order of grounds
In my previous question Set-theoretic geology: controlled erosion?
and the great answer by Jonas Reitz, I have learned a few things, starting from the awareness that I understand the fine-grain ...
- 7,853
5
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0
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278
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Class theory of ZF-minus-Powerset as classical predicative system?
I've been thinking about some mathematics in weaker foundational systems a little bit, largely from a structural viewpoint, and with particular attention to classes.
Some categories I've been keeping ...
- 35.5k
1
vote
1
answer
396
views
Complete and consistent first-order theories that contain interesting phenomena
Gödel has shown that a consistent recursively axiomatizable first-order theory that can interpret Robinson arithmetic is incomplete.
I think there is some sentimental value in working with a theory ...
user158636
74
votes
8
answers
14k
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Category theory and set theory: just a different language, or different foundation of mathematics?
This is a question to research mathematicians, as well as to those concerned with the history and philosophy of mathematics.
I am asking for a reference. In order to make the reference request as ...
- 6,937
19
votes
1
answer
942
views
Positive set theory and the "co-Russell" set
This is a more focused version of a question which was asked at MSE a couple years ago, but is still unanswered there. That question asks about a broad range of theories, whereas this version focuses ...
- 20.5k
3
votes
2
answers
720
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Shortest axiom of infinity for foundationless set theory
Let $T$ be the theory with a binary symbol $\in$, an unary symbol $S$, and the following axioms:
Axiom of extension:
\begin{equation}
\forall x \forall y (\forall z (z \in x \leftrightarrow z \in ...
- 2,203
6
votes
3
answers
2k
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How strong is this set theory?
In the spirit of this related question, consider a set theory with the following axioms:
Axiom of extension:
$$ \forall x \forall y (\forall z (z \in x \leftrightarrow z \in y) \rightarrow x = y) $$
...
- 2,203
38
votes
4
answers
6k
views
Could groups be used instead of sets as a foundation of mathematics?
Sets are the only fundamental objects in the theory $\sf ZFC$. But we can use $\sf ZFC$ as a foundation for all of mathematics by encoding the various other objects we care about in terms of sets. The ...
- 3,137
2
votes
1
answer
458
views
Set of definable real numbers?
Is there a set theory at least as strong as $KP\omega$ which has as a theorem that there is a set $\mathbb{D}$ of precisely the definable real numbers?
- 3,574
-1
votes
1
answer
291
views
Weak power set - what strength may it have? [closed]
In The Consistency of Classical Set Theory Relative to a Set Theory with Intuitionistic Logic in THE JOURNAL OF SYMBOLIC LOGIC Volume 38, Number 2, June 1973 page 316 Harvey Friedman's axiom 8* $Weak \...
- 3,574
1
vote
0
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278
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A countable set theory providing choice?
Instead of Zermelo set theory $Z$ take $Y$ = $Z$ minus the power set axiom plus
Enumerability: $\forall x(x\neq \emptyset \to\exists f[f:\mathbb{N}\overset{onto}{\frown}x ])$
$\imath$ is the ...
- 3,574
2
votes
0
answers
159
views
Why not replace reflection by bounded reflection in Muller's approach?
Bounded Reflection: If $\phi$ is a formula in the language of set theory [i.e.; $\small \sf FOL(=,\in)$], in which all and only symbols $``x,x_1,..,x_n"$ occur free, and $\phi^V$ is the formula ...
- 11.3k
6
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1
answer
994
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Which branches of mathematics can be done just in terms of morphisms and composition?
Consider the first-order language $L_{\omega\omega}$ of the signature $L:=\{\mathrm{dom}, \mathrm{cod}, \mathrm{comp}\}$, where $\mathrm{dom}$ and $\mathrm{cod}$ are unary function symbols and $\...
user103598
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
9
votes
1
answer
687
views
"Surjective cardinals" - using surjections rather than injections to define isomorphism classes of sets
Cantor used the notion of an "injection" to formalize the size of two sets: A is "smaller" than B if A injects into B.
Simply put, the question is - how does this situation change if we use ...
- 4,907
2
votes
0
answers
305
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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
3
votes
0
answers
283
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Formal foundations done properly [closed]
I would want to do mathematics properly, so that the proofs of results can be trusted on instead of them being just suggestions on which results could perhaps apply. This means formulating the math in ...
- 31
16
votes
3
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1k
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Consequences of foundation/regularity in ordinary mathematics (over ZF–AF)?
This is a followup question to Does foundation/regularity have any categorical/structural consequences, in ZF?
As shown in answers to that question, the axiom of foundation (AF, aka regularity) has ...
- 21.3k
10
votes
1
answer
1k
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Erroneous proof of recursion theorem examples
In his book Elements of Set Theory, Herbert Enderton defines (p. 70) a Peano system as a triple $(N, S, e)$ where $N$ is a set, $S$ is an $N$-valued function defined on $N$ and $e$ is a member of $N$ ...
- 209
15
votes
2
answers
1k
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Does foundation/regularity have any categorical/structural consequences, in ZF?
(Prompted by reflection on this old answer, and its suggestion of the “harmlessness” of the axiom of regularity.)
In ZFC, one may justify the axiom of foundation (AF, aka the axiom of regularity) as ...
- 21.3k
4
votes
0
answers
424
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What are the requirements of a foundational theory?
There are multiple languages to describe all of mathematics, and there are some equivalences between them, some more successful then others.
My question is can we describe some requirements (in some ...
- 773