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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77 views

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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116 views

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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86 views

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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134 views

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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93 views

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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Can there exist a world where every consistent effective FOL theory can be written and satisfied in some part of it?

is it consistent to postulate the existence of a set $W$ and a membership relation $\in$ and the following axioms about it: Transitivity: $\forall x \in W: x \subset W$ Extensionality: $\forall x,y \...
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Is PA interpretable in this simple number-set theory?

Working in mono-sorted first order logic, add primitives of equality and its axioms, set membership $\in$, and a total unary function symbol $``||"$ denoting the cardinality of Axioms: Extensionality: ...
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What do you call the generalisation of the direct image?

Informal Description Let me start with an example. Let $X$ be the set $\{a, b, c, d, e\}$ and $E$ be the set $\{a, b, c\}$. Let $f$ be a function with domain $X$. Then the mapping that sends $E$ to $\{...
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317 views

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 those definitions capture the cumulative hierarchy?

Working in first order logic with identity and membership: Definitions: $ \begin {align} hierarchy(x) \iff \forall y \in x : y=\{z \subseteq k \mid k \in x \, ( k \subsetneq y)\} \end {align}$ $level(...
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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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105 views

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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115 views

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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Can sets be of equal cardinality with their stratified powers?

I Have transferred this posting from Mathoverflow, because I think it is not research level, though it is not too elementary I suppose. Define $\mathcal P^\equiv (A)$, the stratified power set of set $...
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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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32 views

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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60 views

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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234 views

Are those two theories about universes equivalent?

If we define a universe as a well founded extensional transitive set that is closed under power, union, and non-bigger than, formally this is: $\mathbf U (X) \iff \forall a \in X: \\ \forall m \in a (...
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129 views

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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Is there any successful approach to algebraization of set theory other than the category-theoretical approach?

The approach towards algebraization of set theory which started to be developed in 1988 by André Joyal and Ieke Moerdijk was presented in their book titled "Algebraic Set Theory" which ...
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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\...
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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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306 views

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) ...
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Is bounded finite set theory equivalent to bounded arithmetic?

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 ...
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Would mathematics based on lists obviate the need for the axiom of choice?

I'm trying to wrap my head around the axiom of choice and its equivalent well-ordering theorem. Imagine a mathematics founded on ordered lists rather than sets. So by construction, wouldn't every ...