All Questions
19 questions with no upvoted or accepted answers
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
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
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
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 ...
- 2,203
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 ...
- 6,962
5
votes
0
answers
278
views
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
4
votes
0
answers
424
views
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
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
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
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
1
vote
0
answers
82
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 ...
- 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
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 $\...
- 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
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 ...
- 11.3k
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 ...
- 11.3k
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: $\forall z (z \in x \leftrightarrow z \in y) \to x = y$
...
- 2,203
1
vote
0
answers
278
views
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
1
vote
0
answers
257
views
Is there a non-trivial consistency preserving transformation?
In set theory "equiconsistency" (and not "consistency") of the theories is the main part of researches. So we usually try to construct a new model using a given one. In the ...
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