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.

For questions about various alternative set theories substantially different from ZFC. Examples might include:

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Positive set theory, antifoundation, and the "co-Russell set"

A more focused version of this question has now been asked at MO. Tl;dr version: are there "reasonable" theories which prove/disprove "the set of all sets containing themselves, contains itself"? Inspired by this question, I'd like to ask a…
Noah Schweber
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Why use ZF over NFU?

Forgive me if this question is quite naïve; I've studied axiomatic set theory in the context of ZF, but my knowledge of NF(U) goes little beyond its axioms, what it means for a formula to be stratified, and stuff that I've read on websites here and…
Clive Newstead
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Alternative set theories

This is a (soft!) question for students of set theory and their teachers. OK: ZFC is the canonical set theory we all know and love. But what other, alternative set theories, should a serious student encounter (at least to the extent of knowing that…
Peter Smith
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Is there a set theory in which the reals are not a set but the natural numbers are?

Is there any known axiomatization of set theory in which the real numbers are not a set, but the natural numbers and other infinite sets do exist? Such a set theory would have an Axiom of Infinity, but not an Axiom of Power Set. I know that…
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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 free complete lattice on $3$ generators would surject onto…
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Is there a non-standard set theory that makes use of a null element?

Has there ever been something like an "empty element" or a "zero entity" been proposed, or at least is there any more or less standardized symbol to denote such a (no)thing? E.g., if $\epsilon$ denoted what I am interested in, it would be that…
lemontree
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Set theory with antielements?

This is an idea which I've had sitting on my desk for a while, but hadn't gotten to until now. Suppose that there two types of set-theoretic objects, which we'll call "elements" and "antielements." For any element $a$, there is a unique antielement…
R. Burton
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Is there a contradiction hiding in this alternative set theory with 3 axioms?

Let us take as a vocabulary the $\in$ relation (is an element of), and a single unary predicate $C$, where $Cx$ is read "$x$ is constructible" or "$x$ is a constructible set" (I'm making this up, but the term seems appropriate). We may then write…
6005
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Can one-element set be considered equal to its element?

Are there "interesting" (that is non-trivial, for example not containing only one set) set theories with one element set being equal to their element ($\{x\}=x$ for every $x$)? This question arose from the practical problem: Is it possible without…
porton
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ZF Set Theory and Law of the Excluded Middle

I know that the law of the excluded middle is implied in ZFC set theory, since it is implied by the axiom of choice. Taking away the axiom of choice, does ZF set theory (with axioms as stated in the corresponding Wikipedia article), imply the law of…
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How much maths can we do in NF(U)?

I have recently become interested in non-standard set theories, particularly in NF and NFU and have been reading some things here and there. Of course, I don't know much about it and I'm still trying to get my head around the very basics. There is…
Cassidy
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About ordinals and cardinals in structural set theory

Among the most important concepts of set theory for mathematical real life applications are ordinal numbers and cardinal numbers. In material set theory, ordinal numbers are defined as transitive sets, well-ordered by the membership relation. It can…
user158047
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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 $\{f(a), f(b), f(c) \}$ is called the direct image…
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Infinite, totally ordered and well ordered sets

It's quite easy to show that a finite set is well ordered iff it is totally ordered. Is the converse also true? That is: is it true that a set is infinite iff it admits a total order which is not a well order? (For the sake of brevity, I shall write…
Caffeine
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How badly does foundation fail in NF(etc.)?

The strongest antifoundation axiom I know is due to Boffa. Roughly, it asserts that every graph which could represent a set, does. For example, considering a graph consisting of (a root connected to) arbitrarily many "one-vertex loops" leads to the…
Noah Schweber
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