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'...
tesseract0's user avatar
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Is there a universal group in NFU?

This is a follow-up to is there a universal group. I'm curious about whether this holds in NFU (+ pairing, infinity, and choice). I haven't seen a reference for pairing before, but I'm going to make $(...
Greg Nisbet's user avatar
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Infinitary logic positive set theory

I've been somewhat interested in positive set theory recently, which solves Russell's paradox by changing the comprehension axiom schema to only allow positive formulas. I had the thought that an ...
Mike Battaglia's user avatar
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functional analysis with the axiom of determinacy

I was wondering if any mathematician has tried to develop functional analysis with the axiom of determinacy.
Jello's user avatar
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Can this rule of inference do replace axiom of extensionality?

If I replace the below axiom in ZFC (or NBG) by the below inference rule, there are any consequence in what can be demonstrated? Axiom: If two sets (or classes) have the same elements then their are ...
I.F.F. dos Santos's user avatar
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Set theories combining ZFC and NFU

I know very little set theory, but I have heard of NFU before and am wondering whether it can be used as a "platform" to "host" an arbitrary first-order theory in its urelements, ...
Greg Nisbet's user avatar
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Is this proof wrong?

I am reading Decio Krause's quasi set theory (2010). It allows ZF axiomatic set theory to exist within it but with a couple new fundamentals. One of the theorems had a proof which I can't make sense ...
Isaac Sechslingloff's user avatar
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Classifying all vacuously true FOL statements in set theory?

Although ZFC stands tall as the standard set theory, there is no shortage of alternative set theories. Many of which differ on fundamental theories. This has me wondering, assuming first order logic (...
Graviton's user avatar
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Constructing Algebraic Closures in ETCS

It seems to be folklore that "ETCS suffices to develop most of algebraic geometry", formally backed by the fact that "ETCS is equivalent to Bounded ZFC". But I have some doubts ...
Jonas Linssen's user avatar
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Can this theory be defined in FOL(=,$\in$)?

The question I'm asking here is purely technical. For the below mentioned theory, can it be defined in first order logic with equality and membership, or it necessitates adding '$\operatorname {stage}$...
Zuhair's user avatar
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Extending empty set + adjunction to interpret PA

Let N = empty set + adjunction. N interprets Q.1 Q + induction yields PA. Does N + epsilon-induction interpret PA? If so: Are they mutually interpretable, sententially equivalent, and/or bi-...
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Can we restrict the interpretation of ' belongs to' relation in set theory?

In axiomatic set theory, the relation ' belongs to' is undefined, subject to the restriction imposed by other axioms, for example, the Axiom of Regularity. In common parlance, the notion of 'belonging ...
Sudhir's user avatar
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Set theory with infinite subsets of N but without an uncountable power set

If we allow infinite subsets of $N$, do we necessarily have an uncountable power set$?$ Do we have any axiomatisation which allows for infinite subsets of $N$ $($ or any countably infinite set$)$, but ...
Sudhir's user avatar
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Where do subsets of a stage in L stop to newely appear?

At the constructible hierarchy $L$, for each stage $L_\alpha$ we keep having subsets of it newly arising high up the hierarchy: is it a theorem of $\sf ZFC + V=L$ that this stops by $L_{|\alpha|^+}$? ...
Zuhair's user avatar
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"Denseness Axiom"

I am reading some set theory right now and started to read about the Axiom of Choice and that ZF is consistent iff ZFC is consistent. So, since the axiom of choice causes the cardinalities of sets to ...
therealsarecountable's user avatar
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Reference Request: Has this line on rudimentary relations been worked out before in relation to defining sets and their membership?

Language: Bi-sorted FOL, upper cases for classes, lower for elements. Primitives: equality '='; rudimentary membership '$\in' $'; Ordered pairing '$(,)$'. The first between any variables, the second ...
Zuhair's user avatar
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Has this logic of relations been done before?

First Order Logic of Relations "FOLR": Language: first order logic with Equality "$=$"(and its axioms), and Membership "$ \in $", nLinks "$\overrightarrow {x_1,..,...
Zuhair's user avatar
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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 ...
Greg Nisbet's user avatar
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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 ...
user76284's user avatar
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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 ...
Zuhair's user avatar
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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 ...
Rui Han's user avatar
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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 ...
Noah Schweber's user avatar
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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 ...
zooby's user avatar
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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 ...
Zuhair's user avatar
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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 ...
Zuhair's user avatar
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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 ...
Greg Nisbet's user avatar
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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 ...
Zuhair's user avatar
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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 (...
Greg Nisbet's user avatar
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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 \...
Zuhair's user avatar
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6 votes
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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 $\{...
Yes's user avatar
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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 ...
Squirtle's user avatar
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3 votes
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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 "...
Greg Nisbet's user avatar
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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, ...
Greg Nisbet's user avatar
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3 answers
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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 ...
Zuhair's user avatar
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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 ...
Greg Nisbet's user avatar
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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 ...
Zuhair's user avatar
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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 ...
Greg Nisbet's user avatar
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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 ...
Zuhair's user avatar
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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 ...
Zuhair's user avatar
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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 ...
Zuhair's user avatar
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1 vote
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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 $...
Zuhair's user avatar
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2 votes
2 answers
170 views

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?
user123's user avatar
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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: ...
Zuhair's user avatar
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3 votes
2 answers
269 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 (...
Zuhair's user avatar
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2 votes
1 answer
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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 + ...
Zuhair's user avatar
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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. ...
Jonas Linssen's user avatar
1 vote
1 answer
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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.
Troy McClure's user avatar
1 vote
1 answer
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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 ...
Ioachim Drugus's user avatar
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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\...
Squirtle's user avatar
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