# Questions tagged [axioms]

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### Was there a time in mathematics when a counterexample was wrong? [closed]

I am doing an essay on the knowledge of Mathematics and how we know what we know to be true. I was just wondering if there was an example in mathematics of some theorem that was disproven by a ...
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### Is this theory using defined notions of classes, sets, and membership interpretable in ZFC?

The main difference with this formal theory is that it depends in an essential manner on a defined notion of class, set, and set membership $\in$, rather than the usual appraoch of leaving them ...
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### What is the exact consistency strength of this type-set theory?

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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### Can we have $\sf V=HTD$? How it relates to $\sf V=HOD$?

In my search for some type-set motivated line of thought that might prove axiom of choice, I was thinking of a concept that looks like $\sf V=HOD$, but in terms of types instead of ordinals, that is ...
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### Can Z + Ranks + Successor cardinals + Ordinal inaccessibility be equal to ZF?

[EDIT: The axiom of successor cardinals was found by an answer by Greg Kirmayer, not to be capturing the intended meaning of it, which is simply reflected by its name, i.e. the existence of a ...
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### Elementary theory of the category of groupoids?

One axiomatisation of set theory, the Elementary Theory of the Category of Sets, or ETCS for short, comes from category theory and states that sets and functions form a locally cartesian-closed, ...
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### What sorts of extra axioms might we add to ZFC to compute higher Busy Beaver numbers?

First, some context. Ever since I was a high schooler, I have been fascinated with large numbers. As I have grown in mathematical maturity, I have become both disappointed and fascinated to see that ...
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### Is this set theory equivalent to ZFC?

Consider a variant of set theory with these axioms: Extensionality, Regularity (foundation), Separation, Powerset, Axiom of Choice, and Transitive closure of a set-like relation is set-like. Update: ...
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### Is this a good way of conceptualising the current status of Foundation of Maths projects?

I have been reading 'A Very Short Introduction to Mathematics' by Timothy Gowers and at one point he mentions that most of the mathematical proofs can be finally resolved to a set of logical ...
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### When does a topos satisfy the axiom of regularity?

In categorical set theory, we observe that certain topoi satisfy (suitable versions of) certain axioms from set theory. For example, Lawvere's $\mathsf{ETCS}$ asserts that $\mathbf{Set}$ is a well-...
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### Is cyclic replacement inconsistent with ZFC-Foundation?

Replacement: if $\phi(x,z)$ is a formula in which all and only symbols $x,z,x_1,..,x_n"$ occur free, and non of them occur as bound, and in which the symbol $B"$ never occur; then: \forall x_1,.....
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### Logical completeness of Hilbert system of axioms

This is really a question about references. The entry in Russian Wikipedia about Hilbert's axioms states, in particular, that completeness of Hilbert's system was proven by Tarski in 1951. The ...
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### (ZC + $\Sigma_2$ replacement + inaccessible cardinal) equiconsistent with (ZFC + inaccessible cardinal)?

Randall Holmes has made a quite convincing argument against the fact that the full axiom schema of replacement should be considered as “intuitively obvious”—even though he does believe ZFC to be ...
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### Are any formal systems based upon the idea of "iterated characterization pushing" currently in existence? If not, is anyone working on them?

I had an idea in regards to the design of formal systems with foundational aspirations. To convey the idea, let's talk a bit about the second-order Peano axioms. The way these axioms work, we have a ...
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### Complement-like operator and the axiom of choice

We say that an operator $^*$ on ${\cal P}(A)$ is $\star$-complement if $^*$ is not the complement operator and for all $X⊆A$ we have: $X^*∪X=A$ $X^{**}=X$ We say that $^*$ is $\star$-strong ...
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### What is the status of the Born Rule in axiomatic QM?

While physicists have tried multiple times and failed to derive the Born Rule (for example: https://arxiv.org/pdf/quant-ph/0409144.pdf). I was wondering what axiomatic Quantum Mechanics had to say ...
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### Apart from Tarski's study, is there any other source that has been looking at the parallelism of concepts and theorems?

Alfred Tarski in his next study (Some Methodological Investigations on the definability of concepts, TARSKI, Logic, Semantics, Metamathematics. Papers from 1923 to 1938. Clarendon Press, Oxford, 1956, ...
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### Do any finite predictions of Quantum Mechanics depend on the set theoretic axioms used?

I was wondering if any of the finite predictions of Quantum Mechanics depend on what set theoretic axioms are used. We will say that Quantum Mechanics makes a finite prediction about an experiment if,...
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### What is the intuitive notion that ZF-Extensionality-Foundation+Collection can be said to capture? [closed]

This question has been moved to philosophy.stackexchange.com I'll try to abbreviate it here: the question asks about the "informal notion" that the fragment of $\text{ZFC}$ that is axiomatized by ...
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### Concrete mathematical statements in relation to Choice versus Reinhardt cardinals?

Harvey Friedman is well known for investigating concrete mathematical statements that requires strong assumptions, i.e. those that can only be interpreted in a strong extension of $\text{ZF(C)}$. My ...
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### Is it possible to derive the rules of set theory as transfers from the pure finite set world, and can we extend this further?

Informally the idea of this question is about whether the rules of set theory can be derived as a transfer of some rules from the hereditarily finite set realm, and whether this transfer principle ...
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$\text{Ordinal Replacement:}$ if $\phi(x,y)$ is a formula in two free variables $x,y$, then: $\forall x \ [ordinal(x) \to \exists! y \ (ordinal (y) \wedge \phi(x,y)) ] \to \forall A \ (\forall x \... 1answer 442 views ### Can rules of set theory be founded by paralleling parts of atomic Mereology? If we work in General Extensional Atomic Mereology [without bottom], so the primitives of the language are$P$standing for "is a part of", and equality, now we add to it membership$\in$relation ... 3answers 618 views ### What is the consistency strength of Z+ Accessibility? Informally the axiom schema of accessibility states that for each unary function$F$that is definable over the whole universe of discourse "in the language of set theory", like the powerset function$...
Let's assume all axioms of $\text{Z}- \text{Infinity}$. Now let $F$ be any function that is definable over the whole universe of discourse by a formula in the language of $\text{Z}$. Now we define ...