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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.

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

132 questions
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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 elements to be "equally nested" e.g. $\{a, b\}$ and…
Fomalhaut
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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 universal set in such theories? Is it greater than…
user171348
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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 could you remove and still form some kind of…
zooby
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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 appeared in 1995. This approach is sometimes called…
Ioachim Drugus
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Ackermann set theory appears to prove 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 an inaccessible class, if it was not then it would…
DanOc004
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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 cannot say that my treatment is incompatible with…
Ioachim Drugus
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Is there a set theory based on ordered lists?

It seems like a lot of the difficulties in set theory come from trying to make sense of ordered lists just starting with unordered sets. Wouldn't it be easier to have ordered lists as the fundamental objects? Or even just an ordered pair? Then the…
zooby
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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?
Rodrigo Tavares
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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} contains one element. So since A={A}, A is also…
Troy McClure
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