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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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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Reference request: Boffa's proof of NF being interpretable in NFU with Ur elements being at most as many as Sets?

It is known that if we add to axioms of $\sf NFU$ the following axiom: $\exists f: Ur \to Sets , f \text{ is an injection}$ In English: there exists an injection from the set of all Ur-elements to the ...
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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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Can a set have a complement in intuitionistic ZF?

Does IZF (ZF formulated in intuitionistic logic) prove that for any set $a$, $\{x: x \notin a \}$ does not exist?
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Is the following a reformulation of MK?

Would the following system succeed in constituting a re-formulation of MK? Language: mono-sorted first order logic with equality $="$ and membership $\in"$, with the following axioms added: ...
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The reflection axiom and large cardinals?

Extensionality: as in Zermelo set theory Separation: as in Zermelo set theory Reflection: if $\phi$ is a formula having all of its free variables among $\vec{x}$ that doesn't use the symbol $v$ ; and ...
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Is strengthening foundation over Ackermann and withdrawing class comprehension still equivalent to ZFC?

Can ZFC prove the consistency of the following system Language: mono-sorted first order logic with primitives of equality, membership, and one place predicate symbol $set$. Axioms: ID axioms + ...
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Elementary Embeddings of Transitive Models of ZFA

My question is related to https://mathoverflow.net/questions/289643/does-every-elementary-embedding-jv-to-v-in-zfa-arise-from-a-self-injection-o Let $M$ be a transitive model of ZFCA (ZFC with atoms) ...
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Substitution In Logic

I am trying to understand how substitution of variables with terms works in Toposes and Local Set Theories by Bell. However I don't think the details of that book are important for my question, and I ...
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Unique Existence In Local Set Theories

I am working through Toposes and Local Set Theories by Bell. I would like to know the proof of this expression from page 82: $$(\exists ! w)\alpha, \alpha(w/true),\alpha \vdash w = true$$ I would ...
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Is there an obvious inconsistency with this set theory?

Is there an obvious inconsistency with the following first order set theory? Axioms: Extensionality as in ZF. Foundation as in ZF. Hereditary comprehension: if $\leq$ is a binary relation symbol, ...
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Is $x=\{x,\{x\},\{x,\{x\}\},...\}$ a thing? $\lor$ Do fractal sets exist?

I am currently learning about hypersets. So far I have learned that: Non-well-founded set theories are variants of axiomatic set theory that allow sets to contain themselves and otherwise violate the ...
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How does Reinhardt's extension of the set-theoretic universe beyond $V_\Omega$ work?

In this answer it is stated that in William Reinhardt's paper 'Remarks on reflection principles, large cardinals, and elementary embeddings' (1974) Reinhardt suggests extending the set-theoretic ...
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Math With Base 3 Truth Values

I was wondering if there is any work towards a system of mathematics were truth values could take on 3 values. Traditionally, we view every statement as either true or false. Computers are built in ...
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Can Cantor's theorem survive this kind of parametric type predicative restriction on set formation?

Language: mono-sorted first order logic with equality Extra-logical primitives: $T, \in, <$; the first signifies "the type of" and its a total one place function symbol, the second is set ...
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If $S=\{x:x\in x\}$, is $S\in S$ knowable? (Naive set theory)

Assuming the set theory we're working with allows self-containment, as well as arbitrary set building of the form $\{x:\Phi(x)\}$, if we define $S=\{x:x\in x\}$, is $S\in S$ knowable? As we see from ...
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Can we go further than (surreal) numbers, games & gaps?

It is my understanding that numbers & games have as there left & right options sets. I think this means that numbers & games are themselves sets. Gaps on the other hand can have (proper) ...
Bounded finite set theory is: Extensionality + $\Delta_0$-Separation + Adjunction? Where Adjunction is: $\forall x \forall y \ (x \cup \{y\} \text { exists})$ Would this theory be equivalent to ...