# Questions tagged [axioms]

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103
questions

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### Is choice over definable sets equivalent to AC over axioms of ZF-Reg.?

If we add the following axiom schema to ZF-Reg., would the resulting theory prove $\sf AC$?
Definable sets Choice: if $\phi$ is a formula in which only the symbol $``y"$ occurs free, then:
$$\forall X ...

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**1**answer

309 views

### Is this theory equivalent to MK?

[EDIT] The older exposition of this theory was proved inconsistent by EmilJeřábek (see comments). Here, this is a possible salvage. (the new information over the older post shall be put in square ...

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146 views

### Can we have a bijection between a set and its powerset with the following properties?

This question is related to a question Is this internalization of a bijection between a set and its powerset possible? lately posted to $\cal MO$. Similarly, we add one primitive unary partial ...

**2**

votes

**1**answer

201 views

### Can we internalize a bijection between a set and its powerset in this way?

This question is related to a question lately posted to $\cal MO$. Here, we add two partial unary functions $``j,f"$ to the language of $\sf ZF$.
The question is about if we can add the following on ...

**10**

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344 views

### Harvey Friedman's minimalist axioms for set theory

[This is a question on the FOM mailing list.]
In 1997, Harvey Friedman introduced the following theory: Let $\in$ be a binary predicate and $U$ be a constant. Add the following axioms:
Subworld ...

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147 views

### Does foundationless Ackermann set theory prove replacement?

From Ackermann's set theory equals ZF (1970) by William N. Reinhardt:
Let A be the theory determined by the following axioms:
Extensionality: $\forall z (z \in x \leftrightarrow z \in y) \to x = y$
...

**4**

votes

**2**answers

313 views

### Applications of ZFA-Set Theory

The set theory with atoms (ZFA), is a modified version of set theory, and is characterized by the fact that it admits objects other than sets, atoms. Atoms are objects which do not have any elements.
...

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183 views

### What is the strength of the single Replacement sentence?

What's the consistency strength of adding the following single sentence replacement like statement to $\sf Z + \forall x \exists \alpha: x \in V_\alpha$ ?
$$\forall \varphi \forall A \ [\forall x \in ...

**8**

votes

**1**answer

1k views

### 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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243 views

### 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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92 views

### 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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266 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 ...

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**2**answers

361 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 ...

**11**

votes

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774 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, ...

**27**

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3k views

### 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 ...

**10**

votes

**1**answer

946 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: ...

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365 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 ...

**10**

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640 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-...

**3**

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269 views

### Is axiom of constructibility $V = L$ consistent with Tarski–Grothendieck set theory?

I wonder what is the relationship between ZF + $V = L$ and Tarski–Grothendieck set theory, because I haven't found any bibliographic references.
If they are compatible, it is possible to introduce $V =...

**10**

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398 views

### Does bounded Zermelo construct any cumulative hierarchy?

ZF is sufficient to construct the von Neumann hierarchy, and prove that every set appears at some stage $V_\alpha$. This is the basis for Scott's trick, for instance. But how much of ZF is needed? Is ...

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562 views

### Shortest axiom of infinity for foundationless set theory

Let $T$ be the theory with a binary symbol $\in$, an unary symbol $S$, and the following axioms:
Axiom of extension:
\begin{equation}
\forall x \forall y (\forall z (z \in x \leftrightarrow z \in ...

**7**

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**0**answers

194 views

### $\Sigma^2_2$ absoluteness and $\diamondsuit$

This question touches on recent questions concerning the role of $\diamondsuit$ and the Continuum Hypothesis. In particular, I would like to know more information about a conjecture of Woodin on the ...

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343 views

### On the role of $\diamondsuit$

The well-known axiom $\diamondsuit$ states that there is a sequence $\langle A_\alpha:\alpha<\omega_1\rangle$ (a $\diamondsuit$-sequence) of countable sets with the property that for any $A\...

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135 views

### 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,.....

**10**

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**1**answer

533 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 ...

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**1**answer

321 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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202 views

### 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 ...

**14**

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272 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 ...

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299 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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147 views

### 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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458 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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402 views

### The surreal numbers under a change of universe

Suppose we start with a model $\mathcal{M}$ of $ZFC$ (or $GBC$ or $MK$ if you prefer), and let $N_0^\mathcal{M}$ denote the surreals in $\mathcal{M}$. If we add some large cardinal assumptions $\{\...

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163 views

### What is the strength of this strict constructible iterative hierarchy?

Begin with the empty set then construct the set of the empty set, then construct the set of all subsets of the latter set, then at each level of construction construct the next level as the set all ...

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290 views

### Is Replacement motivated by ranked iterative conception of sets?

When one reads the Wikipedia article on the Von Neumann Universe, one gets the impression that the idea of "the cumulative hierarchy" serves as a motivation for $ZFC$. I don't see really how ...

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176 views

### Are Regularity schema and $\in$-induction schema equivalent in intuitionistic logic?

In posting "Does Regularity schema imply $\in$-induction when added to first order Zermelo set theor?"
the answer was that they are equivalent in classical first order logic with membership "$\in$".
...

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247 views

### Does Regularity schema imply $\in$-induction when added to first order Zermelo set theory?

That $\in$-induction fails to be a theorem schema of first order Zermelo + Foundation (see here), then it appears that it is more eligible to replace axiom of Regularity (Foundation) by a Regularity ...

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346 views

### Is there a known shorter axiomatization of NF than this?

Is there an already known axiomtization of $NF$ that is shorter than the following axiomatic system in first order logic with equality $``="$ and membership $``\in"$? And what is exactly meant by ...

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216 views

### $MK+CC$ as a foundation for category theory

Has any work been done on what $MK+CC$ looks like as a foundation for category theory? Is it 'the same' as restricting to inaccessibles in some precise manner?
According to wikipedia, any category ...

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143 views

### theories where angles exist without a metric

The underlying basic question, which I'm sure I'm not the first to ask, is what are the possible exotic/nonintuitive models of Euclid's axioms/postulates, outside the one where "lines" are interpreted ...

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123 views

### Is this schema equivalent to Replacement under removal of Extensionality?

If $\phi(x)$ is a formula in which only symbol $``x"$ occurs free, and it only occurs free, and in which symbol $``y"$ never occurs; and if $\phi(y)$ is the formula obtained from $\phi(x)$ by merely ...

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159 views

### Can we have a nearily unrestricted class comprehension over predicates that do not mention the class membership symbol

Suppose that $T$ is a consistent first order theory. Now let the language of $T$ be $L_T$.
Question: is it always consistent to add a new primitive constant $D$, and a new primitive binary relation $...

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297 views

### 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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**1**answer

241 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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790 views

### 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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381 views

### Is full Replacement provable in Z + Ordinal Replacement?

$\text{Ordinal Replacement:}$ 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 \...

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**1**answer

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 ...

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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 $...

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277 views

### 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 ...

**0**

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206 views

### Is the following injectivity schema provable in ZF-foundation?

Is the injectivity scheme present in the below axiomatic exposition of a first-order set theory provable in $\text{ZF}$?
Extensionality: $\forall A,B \ [\forall x \ (x \in A \leftrightarrow x \in B) \...

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328 views

### Does the axiom schema of Replacement follow from the abstract notion of the iterative conception of sets?

Let's define an iterative function $V^F$ to indicate a iterative hierarchy building function that iterates a function $F$ starting from $\emptyset$ after a well ordering relation set $R$ whose domain ...