Questions tagged [axioms]

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3
votes
2answers
498 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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0answers
151 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 ...
9
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0answers
311 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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1answer
126 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,.....
9
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1answer
414 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 ...
2
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1answer
197 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
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0answers
186 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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0answers
241 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 ...
6
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0answers
237 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 ...
4
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0answers
143 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, ...
8
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1answer
385 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,...
5
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0answers
369 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 $\{\...
1
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1answer
158 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 ...
1
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1answer
237 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 this is ...
0
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2answers
153 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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2answers
207 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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0answers
334 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 ...
4
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0answers
205 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 ...
2
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0answers
131 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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0answers
120 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 ...
1
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1answer
152 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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1answer
279 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 ...
2
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1answer
210 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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1answer
785 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 ...
4
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1answer
344 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 \...
2
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1answer
389 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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3answers
610 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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1answer
259 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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1answer
183 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) \...
2
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1answer
286 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 ...
5
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4answers
688 views

Can we add set complements on top of ZF?

Can we introduce complements on top of the standard set theory $\text{ZF}$ and have some comprehension axioms about them, like in defining a "small set" as an element of a stage of the Cumulative ...
4
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1answer
387 views

What would be the effect of replacing Separation by Injective Replacement?

Let "Injective Replacement" be the following schema: If $\phi(x,y)$ is a formula in which only x,y occur free, and only free, then: $\small \forall A \ [\forall x \in A \exists y (\phi(x,y)) \...
5
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2answers
439 views

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 ...
3
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1answer
134 views

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 ...
8
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0answers
194 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 ...
2
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1answer
261 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 ...
0
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1answer
224 views

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 ...
3
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2answers
399 views

Relation between AC and the axiom of foundation

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 ...
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10answers
8k views

How should a “working mathematician” think about sets? (ZFC, category theory, urelements)

Note that "a working mathematician" is probably not the best choice of words, it's supposed to mean "someone who needs the theory for applications rather than for its own sake". Think about it as a ...
3
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1answer
137 views

Relationship between computational undecidability and axiomatic undecidability

On surface, these seem two completely different class of problems. One class represent statements which can't be proved or disproved in an axiomatic theory. For example One can write down a ...
0
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1answer
324 views

Pasch axiom and Pythagorean field condition?

I am looking for a reference for the claim that the Pasch axiom is equivalent to the Pythagorean field condition, and with respect to what base theory this should be true. Since posting the question, ...
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0answers
293 views

Can we strengthen the axiom of choice to settle the generalized continuum problem?

By the generalized continuum problem, I mean the following: given an infinite cardinal $\kappa$, find the order type of the set of all cardinals strictly between $\kappa$ and $2^\kappa$. Now whenever ...
13
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0answers
682 views

Arguments against Freiling's argument against Continuum Hypothesis

Freiling's axiom of symmetry ($\sf AS$) is known as a justification for falsity of Continuum Hypothesis. Freiling in his 1986 paper, Axioms of symmetry: throwing darts at the real number line, ...
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5answers
2k views

What are the advantages of the more abstract approaches to nonstandard analysis?

This question does not concern the comparative merits of standard (SA) and nonstandard (NSA) analysis but rather a comparison of different approaches to NSA. What are the concrete advantages of the ...
45
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4answers
5k views

How undecidable is the spectral gap?

Nature just published a paper by Cubitt, Perez-Garcia and Wolf titled Undecidability of the Spectral Gap, there is an extended version on arxiv which is 146 pages long. Here is from the abstract:"Many ...
8
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1answer
599 views

Is the axiom schema of replacement used in algebraic number theory (or more generally outside logic)

Here's a precise question. Does Wiles' proof of FLT run just fine in the set theory that logicians would perhaps call "Zermelo + choice" -- i.e. drop the axiom schema of replacement but assume the ...
2
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0answers
98 views

Quasigroups extracted from the rational numbers and division

Consider a quasigroup $(Q,/)$, that is, Q is a set and for $\forall a,b\in Q$ there are unique solutions to the equations $x/a=b$ and $a/y=b$. How to find a maximal set of independent representants of ...
4
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1answer
151 views

Does this axiom (a weak form of class valued choice) has a name?

At some point in my work (which has nothing to do with set theoretics foundation) I need to consider the following axiom: For any set $X$, any class $V$ with a surjective map $f : V \...
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2answers
282 views

Axiomatic approach to means

Recently I have been contemplating on a talk for high school children. One of my favorite topics in high school was the inequality of means. I had a great high school teacher who wrote some very nice ...
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0answers
188 views

A generalization of SOCA

Roughly speaking, SOCA (Semi Open Coloring Axiom) says that for an open coloring of the unordered pairs over an uncountable separable metric space you can always find an uncountable homogeneous subset ...