# Questions tagged [axioms]

The axioms tag has no usage guidance.

The axioms tag has no usage guidance.

122
questions

-1
votes

0
answers

94
views

I have taken an interest in mathematical logic to understand to source of mathematics as we know it today. And in this regard I have the following two questions:
What would a logical structure/...

3
votes

1
answer

171
views

Let's say that a formula in the language of set theory is flexibly stratified iff there exists a function $f$ from variable symbols to $\omega$ such that if $x=y$ appears in the formula, then $f(x)=f(...

4
votes

1
answer

200
views

Given any linear space $L$ over an ordered field $F$, consider the equiproportion relation $${\sim}=\{((x,y,z),(a,b,c))\in L^3\times L^3: \exists t\in[0,1]\subseteq F\;(y{-}x=t(z{-}x)\wedge b{-}a=t(c{-...

10
votes

3
answers

924
views

Certain categories of mathematical structures have had synthetic axiom systems developed for them. One particularly well known such category is the category of sets and functions $\mathit{Set}$, which ...

5
votes

1
answer

129
views

Consider a set theory with the following axioms:
separation: $\exists y \forall x (x \in y \leftrightarrow \phi \land x \in a)$ where $y$ is not free in $\phi$
reflection: $\phi \to \exists u \phi^u$
...

7
votes

0
answers

178
views

It is well known that even if one works with say ZFC as a base theory, Grothendieck topoi do not in general satisfy even fairly weak axioms like countable choice or small violations of choice and one ...

1
vote

0
answers

96
views

Adopting the approach of Mono-sorted $\sf NBG$, define sets as elements of classes, then axiomatize:
Extensionality, Predicative Class comprehension, emptyset, in the usual manner along mono-sorted $\...

6
votes

1
answer

259
views

Work in a theory with (deep breath) a countable number of primitives denoted with capital letters from the end of the alphabet with numerical subscripts $\{X_n,Y_n,Z_n,\dots\}_{n<\omega}$ ...

4
votes

1
answer

131
views

In Mike Shulman's article Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, the fundamental axiom adopted for his real-cohesive homotopy type theory (axiom $\mathbb{R}\flat$), which ...

0
votes

0
answers

94
views

Background assumptions/definitions/questions (note that "S, A, B" stand for some sentence or other at issue per possible question; note also that these sentences can be imperatives, and in ...

1
vote

0
answers

101
views

This post is a follow up of this one posted to MathStackExchange.
Working in $\sf ZFC + \exists M \, (M \overset{trs} \models ZFC)$, lets define a theory $T_0$ as:
$$(\varphi \ \epsilon \ T_0) \iff \...

0
votes

1
answer

95
views

Peano Arithmetic consists of axioms $P_1, P_2, \ldots P_7$ plus first order classical logic. Let us call this theory $T$. This theory has its unprovable Gödel’s sentence $G$ such that
$$
G\...

1
vote

0
answers

177
views

The following is a reformulation of $\sf ZFC$, it is a version of a well known approach going back to Dana Scott (I suppose), it axiomatizes $\sf ZFC$ simply by "Specification, Reflection, and ...

3
votes

1
answer

243
views

In homotopy type theory, propositional excluded middle and the axiom of choice sets are both consistent with univalence, both of which yields type theoretic models for classical mathematics. However, ...

0
votes

1
answer

224
views

Fix a language $\mathcal{L}$ of first-order set theory. For this question, we can assume that $\mathcal{L}$ is the language described in Chapter 1 of “An introduction to set theory” [William A. R. ...

11
votes

1
answer

410
views

Recall the definition of cardinal definable, where every set being cardinal definable is proved consistent relative to ZF + V=HOD. To re-iterate it:
$Define: X \text { is cardinal definable} \iff \\\...

2
votes

0
answers

76
views

Working in mono-sorted first order logic with equality and membership:
Define: $\operatorname{set}(x) \equiv_\text{df} \exists y \, (x \in y)$
Axiomatize:
Extensionality: $(\forall x \, (x \in a \...

0
votes

1
answer

116
views

This posting comes as a possible salvage to this earlier presented reflective theory which was proved inconsistent. Attesting the particulars of the underlying language in reflection axioms.
Working ...

0
votes

1
answer

224
views

In Kanamori's Bernays and Set Theory pages 20-21, a first order reflection principle due to Bernays is mentioned, that of:
$$\sf \varphi \to \exists y \, (\text {Trans}(y) \land \varphi^y)$$ for ...

0
votes

0
answers

134
views

On 2012-06-14, the Kurt Gödel Research Center held a talk by Miguel Angel Mota titled A spectral approach to reflection principles. Abstract:
Gödel's response to the incompleteness of ZFC with ...

10
votes

1
answer

286
views

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 ...

0
votes

1
answer

369
views

[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 ...

1
vote

0
answers

152
views

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

236
views

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 ...

11
votes

0
answers

422
views

[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 ...

1
vote

0
answers

178
views

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

397
views

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.
...

2
votes

0
answers

191
views

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

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 ...

1
vote

0
answers

246
views

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 ...

2
votes

0
answers

297
views

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 ...

3
votes

2
answers

387
views

[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

3
answers

968
views

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, ...

26
votes

3
answers

3k
views

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 ...

11
votes

1
answer

1k
views

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: ...

1
vote

0
answers

422
views

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
votes

1
answer

692
views

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
votes

1
answer

343
views

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
votes

2
answers

422
views

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 ...

3
votes

2
answers

600
views

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
votes

0
answers

215
views

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 ...

9
votes

0
answers

356
views

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\...

-1
votes

1
answer

137
views

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
votes

1
answer

709
views

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 ...

3
votes

1
answer

364
views

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 ...

5
votes

0
answers

209
views

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
votes

0
answers

278
views

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 ...

6
votes

0
answers

322
views

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 ...

4
votes

0
answers

151
views

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, ...

8
votes

1
answer

503
views

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,...