# Questions tagged [alternative-set-theories]

For questions about various alternative set theories substantially different from ZFC. For example, NF and NFU, IST, ETCS, SP, AST.

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### Proof that universal class is a proper class?

I've been trying to work out this proof, but I think there's something I'm missing out because with the deductions I made, it much looks like the universal class is a set... (help). So in my class we'...
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### What set theory do we get if we take the axioms of ZF and add the compactness theorem for propositional logic?

I was re-reading Noah Schweber's answer to a question I wrote a bit over a year and a half ago. One part of this answer uses Kőnig's lemma to prove (countable?) compactness of classical propositional ...
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### Does separation + reflection prove ZF - extensionality - foundation?

Assume a first-order context. Let $\mathrm{tran}(x)$ denote that $x$ is transitive: $$\mathrm{tran}(x) \leftrightarrow (\forall y \in x) (\forall z \in y) (z \in x)$$ Let $\mathrm{suptran}(x)$ denote ...
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### Is this Class\set theory equivalent to MK or NBG?

Working in bi-sorted \ FOL, where lower cases stand for sets and upper cases for classes; add all axioms of ZFC-Ext.-Reg. written completely in lower case, and add axioms of Ext. and Reg. over all ...
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### Costs and benefits of using non-well-founded set theories instead of ZFC, or ZFC instead of non-well-founded set theories?

What are some advantages and disadvantages of using ZFC as opposed to non-well-founded set theories, and what are some advantages and disadvantages of using non-well-founded set theories as opposed to ...
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### Can a free complete lattice on three generators exist in $\mathsf{NFU}$?

Also asked at MO. It's a fun exercise to show in $\mathsf{ZF}$ that "the free complete lattice on $3$ generators" doesn't actually exist. The punchline, unsurprisingly, is size: a putative ...
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### How many axioms can you remove from ZF set theory and still have an "interesting" version of mathematics?

Obviously you can remove the axiom of choice from ZFC set theory to get ZF set theory. Using ZF only you can still construct most of mathematics and proofs. This led me to wonder how many more axioms ...
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### Is this fragment of Anti-Cantorian bi-hierarchy theory consistent?

Below I'll re-iterate the exposition [with modification to suit this question] of a theory of mine given on Mathoverflow lately, which is unsolved yet. Howevere, here, I want to understand the matters ...
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### Can we have a downward power stage sequence in stratified Z?

See this question on Mathoverflow. Can we have this with stratified Zermelo. Where the latter is Zermelo set theory but with separation restricted to stratified formulas only. To present the question ...
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### Is set theory with "homogeneous comprehension" (no set can contain both sets and atoms) consistent?

Is this single-sorted set theory with sets and atoms consistent? An atom is an element of the domain that is not a set. The idea is that the elementhood predicate $\in$ is not constrained very much by ...
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### Does the degree of impredicativity always matter in type theory?

My question here is actually about whether different degress of impredicativity matter? To show that, lets confine ourselves with the following predicative formalism. Language: multi-sorted first ...
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### Why doesn't the Kuratowski pair work in $\mathrm{NF3}$?

According to the Wikipedia article on the set theory New Foundations, the variant of New Foundations but with the comprehension axiom limited to theories that can be stratified using only three types (...
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### finite set is countable (in ETCS)

When I originally asked this question (see below), I was looking for any solution, but I was not completely satisfied with Saving's answer as some of the details were unclear to me. So I came up with ...
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### Does permitting comprehension for all (and only) contingent formulas result in paradoxes?

Does permitting comprehension over all well-formed formulas that are neither contradictions nor tautologies result in paradoxes? I have a hunch that a simple extensional set theory with the "...
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### Is the class of urelements in NFU a set or contained in the universal set?

I'm wondering about the urelements in $\mathrm{NFU}$. How many of them are there? Can the urelements be collected into a set? Are the urelements, whether they form a set on their own or not, ...
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### What is the proof of Replacement in iterating functions over the empty set?

Define (ordinal): \begin {align} ordinal(x) \iff & trs(x) \land \forall y \in x (trs(y)) \,\land \\& x \, \text {is} \in \text{-well-founded} \end{align} Where $trs(x)$ means $x$ is ...
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### ZF-{regularity, comprehension} with "reflexive" comprehension

ZF without regularity with "reflexive" comprehension. Can we successfully defuse known paradoxes (and produce a consistent theory) by using a comprehension schema that limits comprehension ...
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### Reference request: is axiom of choice motivated along type-set lines?

Language: bi-sorted first order logic with equality and its axiom and additionally the extra-logical primitives: $\tau, < , \in"$, the first is a total unary function on sets denoting is the ...
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### Exercises for NF(U)

Are there any good sources for exercises (or exercises themselves) for New Foundations, with or without urelements? I don't know much about New Foundations besides what's listed on its Wikipedia ...
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### Can ZFC be interpreted in the following type-set theory?

Language: first order logic with equality and its axiom and additionally the extra-logical primitives: $\tau, < , \in"$, the first is a total unary function denoting is the type of, the rest are ...
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### Is ZFC a fragment of this theory?

Let $\mathcal T x. \phi"$stands for the totality of all objects $x$ satisfying $\phi$. So it is a term of the language as long as $\phi$ has $x$ free and never occur as bound. It cannot be ...
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### What's the exact consistency strength of this fragment of ZFC?

What's the consistency strength of restricting parameters in axioms of ZFC (other than Extensionality and Foundation schema) to parameter free definable sets? So for example, the axiom of pairing ...
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### Is BZC inconsistent with Reinhardt cardinals

A Reinhardt cardinal is defined as the critical point of a nontrivial elementary embedding $j: V\rightarrow V$ from the universe $V$ to itself, and is known to be inconsistent with the axiom of choice ...
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### Which large cardinal this theory stops at proving its existence?

The language of this theory is first order predicate calculus with extra-logical primitive symbols of $=; \in, W"$, where $W"$ is a constant symbol. Axioms: those for identity theory + ...
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### Why do we *restrict* to universes instead of *surrounding* us with them?

In set theory and category theory one easily runs into the problem of size. For example Russell's paradox tells us that it is impossible to have consistent set theory allowing a set of all sets. ...
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### Set Theory without Powerset

I'm looking for references for a set theory which does not include the axiom of powerset but includes axioms allowing taking cartesian product. Please refer me to such if you know about any.
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### Does $X^A = Y^B \Longrightarrow X=Y \wedge A=B$? (ETCS)
I'm trying to prove that $f^\flat$ is well-defined, where $f^\flat:Z\times A \to X$ is defined to be the composition $e_X \circ 1\times f$ where $1\times f: A\times Z \to A\times X^A$ and \$e_X: A\...