# Questions tagged [axioms]

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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 ...
205 views

### 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 ...
308 views

### 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 ...
663 views

### 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 ...
900 views

### 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: ...
344 views

### 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 ...
569 views

### 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,.....
480 views

### 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 ...
301 views

### (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 ...
254 views

### 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 ...
278 views

### 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, ...
437 views

### 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 ...
231 views

### 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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### What is the consistency strength of F accessibility?

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 ...
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### An axiom for collecting proper classes

I'm currently working on some universal algebra using proper classes (in MK class theory), and I repeatedly run into situations where I want to collect together some proper classes as the members of a ...
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### Is there a complete countable axiomatization of conditional independence? (Graphoids)

Note: A pointer to a reference, or a yes/no answer with a 1-2 sentence incomplete/non-rigorous justification would suffice for answers. I am just curious about whether the result is true; it is fairly ...
210 views

### Second-order separation schema in Zermelo and Zermelo--Fraenkel

It is a nice theorem of Zermelo that if we replace the Replacement schema with its second-order counterpart, "The image of a set under any function is again a set", then we necessarily get a model ...
279 views

### Can you formulate a theory stating that a truth predicate does not exist for first order set theory?

A truth predicate for first order set theory would allow you to determine the truth of statements in first order set theory. A definition is given here. My question is, can you formulate a statement ...
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### Understanding Polish notation in Lukasiewicz's axioms [closed]

In a paper about Presburger Arithmetic, Lukasiewicz's axioms of propositional calculus are written as follows: CCpqCCqrCpr CCNppp CpCNpq I am having a hard time understanding what these axioms ...
The fact that the axiom of foundation doesn't imply the axiom of choice is pretty standard (the model Cohen created to prove the consistency of $\neg AC$ models the axiom of foundation as well), and ...