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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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Would adding abstractions over sets be inconsistent or otherwise increase the consistency strength?

Is adding abstractions (in the below mentioned manner) over $ZFC$ inconsistent? if not then does it result in increment of consistency strength? The idea is to add a new primitive binary relation "is ...
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Relative consistency of $\mathsf{ZFC}-$Ext$+\neg$Ext+“every set has a unique powerset”

Let $T$ be the $\mathcal L_\in$-theory whose axioms are the axioms of $\mathsf{ZFC}$, with extensionality replaced by its negation, and an additional axiom specifying that every set has a unique ...
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Is there a clear inconsistency with this Lewis like Mereological foundation of set theory?

Lewis's approach to Mereological foundation of set theory, is very interesting by itself. The following shows that it can indeed provide an interpretation for Ackermann's set theory! In brief let our ...
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Whats the consistency strength of this theory?

Language: First order logic with identity and extra-logical primitives of membership $``\in"$, and $``W"$, the last is a constant symbol. Axioms: ID axioms + 1.Extensionality: $\forall z (z \in x \...
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Reference Request: had relative part-hood been investigated before?

I want here to introduce some rather strange relation, it's a little bit made up, possibly ad-hoc. However, the point behind it is that it can straight forwardly interpret the membership relation of ...
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Is Ackermann set theory [minus class comprehension] equi-interpretable with Mereological Logicism?

Terminology: I'll denote this theory by "Mereological Logicism", because its primitive are Part-hood and, Predication, which are the primitives of Mereology and Logic. However those are here extending ...
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Is definable power sufficient to interpret ZFC?

This post is actually related to posting titled What is the strength of definable ZFC?. However the presentation there was 'semiformal', and here I'll present a complete formal exposition of that ...
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Why the restriction to "elements of $V$' in the output of formulas used in Reflection axiom schema of Ackermann class theory?

Can we upgrade the Reflection axiom schema in Ackermann to the following: Modified Reflection axiom schema: if $\psi(y)$ is a formula that doesn't use the symbol $V$, in which only symbols $y,x_1,..,...
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Is class comprehension schema needed for the equivalence of Ackermann set theory with ZF?

William N. Reinhardt had presented a proof of equivalence of Ackermann's set theory with ZF. Does this proof make essential use of the class comprehension axiom schema? In other words, is it still ...
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Can this extension of a fragment of Ackermann set theory survive inconsistency?

Lets work in the language of Ackermann set theory., which is first order logic with equality $``="$, class membership $``\in"$, and sub-world $``V"$, where the last is a constant symbol informally ...
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114 views

Can one combine proper classes into a set?

Let's assume you have two non-set classes $C_1$ and $C_2$ of objects. (E.g., $C_i$ is the class of all algebras of some distinct signature $S_i$ for $i=1,2$.) Now, is there a reasonable variant of ...
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Reference request: Would this axiom motivate a Mereological foundation of set theory?

If $\psi(s,x)$ is a formula in which symbols $``s,x"$ occur free, in which the symbol $``u"$ doesn't occur free, then all closures of: $$\forall S \ [ \forall s (s \subseteq S \exists! x (\psi(s,x))) ...
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Can completeness due to adding $\omega$-rule be extended to all recursive ordinals?

Is completeness due to $\omega$-rule effected by restricting matters to what the meta-theoretic indices confer? I mean in the case of the ordinary $\omega$-rule, we can do it over the world of finite ...
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Are finite sets of standard finite sets, standard finite sets?

If we add the unary primitive $std$, denoting standard to the primitives of $ZFC$, now we add the following $\omega$-rule: From, for $n=0,1,2,3,...; \forall x (x=\{y_1,..,y_n\} \to \psi(x)) $ We ...
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1answer
55 views

Is “ZFC + Omega rule for finite sets” a complete theory?

Omega rule for finite sets $\omega^{fin}$: if $\phi(y), \psi(y)$ are formulas in one free variable symbol $y$, then: From: for $ n=0,1,2,3,...; \text { we have: } \forall y \ [y=\{x_1,...,x_n\} \...
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Is ZF+Omega rule for sets a complete theory?

Omega rule for sets$``\omega^{ Set}"$: if $\{\varphi_0(y),\varphi_1(y), \varphi_2(y),...\}$ is the set of all formulas in the language of $ZF$ in one free variable symbol $``y"$, and if $\psi(x)$ is a ...
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Can Ackermann - Heredity + Limitation of size prove set union over $V$?

If we remove the first completeness axiom for $V$ (i.e. the axiom of Heredity), in Ackermann set theory, and add an axiom of Limitation of size on $V$, formally this is: $\forall x (x \subseteq V \...
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Why not weaken reducibility in $K_2(W)$ instead of sub-world separation?

In this theory $K_2(W)$ (page 7) Harvey Friedman argue for weakening Sub-world Separation into SS-. He did that in order to evade making $W$ transitive. But he could have done that by simply reverting ...
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Does the primary free variable in Ackermann set comprehension need to range over all classes?

According to the Stanford Encyclopedia of Philosophy page on alternative axiomatic set theories, Ackermann set theory includes (among others) the following two axioms: Axiom of class comprehension: ...
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Is this basic number-set theory equivalent to PA?

Informal account: the following theory copies one of the most basic understanding of "natural number", that of it being an indicator of how many members are there in a finite set, all sets are fully ...
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Is Reflection consistent with Resemblance?

The following theory is a class theory that combines two principles that of reflection and resemblance, informally it says that the class $V$ of all sets resembles a set $W$ that stands as a sub-world ...
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58 views

Can we extend a version of MK with that coding function?

Lets weaken the limitation of size axiom of MK to that of sets, i.e. every class that is subnumerous to a set is a set, now to that version of $MK$ lets add to the language of it a primitive unary ...
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Limit of a sequence related to non-well founded set theory [closed]

Consider the following: $$\alpha_0 = \{\} $$ $$\alpha_1 =\{\{\}\}$$ $$\alpha_2 =\{\{\{\}\}\}$$ $$...$$ Clearly $\alpha_i$ is considered a set for all $i \in \omega$. Then consider $ \alpha=\lim_{i \...
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Relative consistency of ZF with respect to IZF

Is there a forcing argument of this fact? Can anybody point me to the place? The reason I'm asking is because I was reading Heyting-Valued Models for Intuitionistic Set Theory by R.J. Grayson, yet ...
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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) ...
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Principia Mathematica, chapter *117: a false proposition?

I was reading Principia Mathematica of Whitehead and Russell and I have found what I think is a false proposition. The proposition in question is *117.632 click on the link to see the formula and the "...
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Soft question - recommendations concerning basic topics inside rough set theory

(I hope this will be phrased alright, apologies if it isn't, it's my first soft question!) I'm looking into rough set theory because it seems like a very interesting concept. I'm still a beginner in ...
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Sets with a time dynamic?

I am looking for advice regarding if there is any literature wrt set theory that also has a inter temporal aspect to the set theory notation. E.g. an element can exist in one set at t=1, but move to ...
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Ackermann set theory appears to proof inaccessible cardinals exist?

I know this proof must be wrong, as it would mean that ZF proves inaccessible cardinals exist, which it doesn't. Let A(x) = ∀I, I is an inaccessible class ⇒ x∈I Now the class of all sets V must be ...
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Subtyping of Prop in Coq. Implementation of Ackermann class theory. First-order theories.

I am trying to implement Ackermann set theory. The first approach is the code below. But there is an incorrect axiom "rs_ax". That's because for every x formula (F x) shouldn't contain predicate M. $\...
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1answer
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Asking for refs: formalisms that admit {x}={{x}}

I have some interest in alternative formulations of set theory which blatantly admit {x}={{x}}. What gave rise to such interest is the following. Take the set equation A={A}. The RHS dictate that {A} ...
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Is the second completeness axiom for V really needed for Ackermann set theory to interpret ZF?

If from the axioms of Ackermann set theory we remove the second completeness axiom for $V$, and replace it with an axiom of power sets in $V$, that is: For every element $x$ of $V$: the class of all ...
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Question regarding Paraconistent valued models

Hey could somebody please check the following "proof". I am using generalized valued models for set theory in order to build a model for paraconsistent set theory. The approach i am following is very ...
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Are there set theories making the class-as-many and class-as-one distinction?

I always imagined a set as an "abstract container" which contains things said to be its elements. I treat the "abstract container" as part of the set and not as element of the set, and thus, one ...
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1answer
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Cardinality of the set of truth values in a Heyting-valued model of IZF

I've very recently become interested in intuitionistic set theory. I'm now trying to get acquainted with the very basics, and I have a number of small silly questions, that might not make sense due to ...
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Positive set theory, antifoundation, and the “co-Russell set”

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 question which has been ...
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2answers
175 views

Considering a non-standard set theory

I'm thinking of creating a non-standard set theory with urelements that has a certain restriction on a specific kind of sets which ZF set theory does not have. Informally, I would like for all set ...
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1answer
330 views

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 $\epsilon$ ...
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1answer
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Good books on non-standard logics/set theories.

So, I am looking for good books on intuicionistic logic & set theory theory and on constructive set theories. Does anyone has suggestions? Background notes: I am familiar with work within ZFC, ...
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Why is V a proper class in positive set theory?

This is my question, I do not have found any paper or book explaining this but this is repeated a lot. So why is V a proper class in this set theory?
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1answer
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Is there a conservative extension of IZF that extends IZF by a weak form of the axiom of choice?

The full axiom of choice implies the LEM, and so is incompatible with constructive mathematics, although there are some weaker forms of the axiom of choice, such as the axiom of dependent choice, or ...
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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,...
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Cardinality of universal set?

I read that there are some non-standard versions of set theory that allow for the existence of a universal set. My first question is: what (if anything) can be said about the cardinality of the ...
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Surreal numbers in set theories other than ZFC

This isn't really a question rather than my thoughts on these things; I initially had questions but believe I managed to answer them. Regardless, here goes. Feel free to correct any mistakes, as I'm a ...
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1answer
580 views

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 ...
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1answer
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Is $\mathcal{P}(\omega)$ bigger than $\omega$ in $NFU$ + infinity?

$NFU$ doesn't prove Cantor's theorem (that $\mathcal{P}(S)$ is cardinally greater than $S$) by a stratification dodge: the proof's critical step makes use of the unstratified formula $x \not\in f(x)$, ...
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1answer
179 views

Examples for intensional set theories.

The normal set theory of todays mathematics (ZFC) is extensional, i.e. it has the axiom of extensionality $$\forall A \, \forall B \, ( \forall X \, (X \in A \iff X \in B) \Rightarrow A = B)$$ Are ...
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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 ...
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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 ...