Questions tagged [axioms]

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Mathematical Logic: A structure for Zermelo Frankel Axioms [migrated]

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/...
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3 votes
1 answer
171 views

Strengthening Quine's New Foundations with a more flexible stratification criterion?

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

Is the affine geometry a geometry of proportions?

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{-...
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10 votes
3 answers
924 views

Axioms for the category of groups

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

How strong is separation + reflection of unbounded quantifiers?

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: $\phi \to \exists u \phi^u$ ...
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7 votes
0 answers
178 views

Is any choice axiom other than WISC inherited by Grothendieck topoi?

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 ...
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1 vote
0 answers
96 views

Is $\sf \Gamma_0$ the proof theoretic ordinal of this kind of predicative class theory?

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

Consistency strength of an attempt at higher order set theory

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 $\{X_n,Y_n,Z_n,\dots\}_{n<\omega}$ ...
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4 votes
1 answer
131 views

Predicativity and axiom $\mathbb{R}\flat$ in real cohesive homotopy type theory

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

Deontic logic and axiomatic pluralism

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

What is the set of axioms for this theory extracted from what is provable in the minimal model of ZFC?

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

Reflection schema

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

Are there re-formulations of ZFC that more closely parallel large-cardinal extensions?

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

Consistency of Generalised Continuum Hypothesis and univalence in HoTT

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, ...
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0 votes
1 answer
224 views

How large is the supremum of minimal $V$-heights of all first-order set theories formulated in a particular language of FOST?

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

Is every set being cardinal definable consistent with ZF + negation of Choice?

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

Is restricting class parameters to be arguments of set functions in reflection consistent?

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

Is this reflection schema equivalent to second order Bernays reflection?

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

What is the proof of replacement from Bernays first order reflection?

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

A spectral approach to reflection principles

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 ...
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10 votes
1 answer
286 views

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 ...
0 votes
1 answer
369 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 ...
1 vote
0 answers
152 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
236 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 ...
11 votes
0 answers
422 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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1 vote
0 answers
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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: $\forall z (z \in x \leftrightarrow z \in y) \to x = y$ ...
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4 votes
2 answers
397 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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2 votes
0 answers
191 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 ...
1 vote
0 answers
246 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 ...
2 votes
0 answers
297 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 ...
3 votes
2 answers
387 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
3 answers
968 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, ...
user avatar
26 votes
3 answers
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 ...
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11 votes
1 answer
1k 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: ...
1 vote
0 answers
422 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 ...
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10 votes
1 answer
692 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 votes
1 answer
343 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 votes
2 answers
422 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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3 votes
2 answers
600 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 ...
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7 votes
0 answers
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$\Sigma^2_2$ 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 ...
9 votes
0 answers
356 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\...
-1 votes
1 answer
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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,.....
10 votes
1 answer
709 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 ...
3 votes
1 answer
364 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 ...
5 votes
0 answers
209 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 ...
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14 votes
0 answers
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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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6 votes
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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 ...
4 votes
0 answers
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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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8 votes
1 answer
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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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