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Questions tagged [axioms]

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7
votes
1answer
298 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,...
4
votes
0answers
335 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
vote
1answer
146 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
vote
1answer
185 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
votes
0answers
60 views

Can MK be interpreted in a class theory about an abstract hierarchy principle + an accessibility principle?

The following is a first order MONOSORTED class theory, that is primarily motivated by an abstract hierarchy principle. It extends first order logic with equality, its language has only two extra-...
0
votes
2answers
138 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 "$\in$". ...
1
vote
2answers
162 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 ...
0
votes
0answers
229 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
votes
0answers
194 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
votes
0answers
115 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 ...
0
votes
0answers
107 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
vote
1answer
134 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 $...
-3
votes
1answer
263 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
votes
1answer
189 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 ...
-1
votes
1answer
760 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 ...
3
votes
1answer
283 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 \...
3
votes
1answer
304 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 ...
-1
votes
3answers
573 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 $...
0
votes
1answer
243 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
votes
1answer
174 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
votes
1answer
258 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
votes
4answers
599 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
votes
1answer
344 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
votes
2answers
280 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 ...
1
vote
0answers
67 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
votes
0answers
174 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
votes
1answer
232 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
votes
1answer
174 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
votes
2answers
375 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 ...
50
votes
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
votes
1answer
133 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
votes
1answer
287 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, ...
1
vote
0answers
289 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
votes
0answers
652 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, ...
33
votes
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 ...
42
votes
4answers
4k 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
votes
1answer
445 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
votes
0answers
94 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
votes
1answer
150 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 \...
7
votes
2answers
246 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 ...
6
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0answers
177 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 ...
-3
votes
1answer
238 views

An axiomatic system with a set of constants that form a complete ordered field [closed]

I am developing a ZFC axiomatic system where together with the empty set, there is a singular (and huge) set of constants that are themselves sets and form a complete ordered field (cof) these ...
18
votes
3answers
1k views

Characterize the category of rings

(Sub)categories of many well-studied mathematical objects have been characterized purely in terms of their morphisms. Some (famous) examples: Sets and functions, due to Lawvere. Modules over some ...
12
votes
1answer
629 views

Transfinitely extending $\sf PA$ — can we get stronger than $\sf ZFC$?

Let $\sf PA$ denote the theory of natural numbers with constants $(0, 1)$ and binary operators $(+,\times)$ based on the first-order predicate calculus with equality, having the following axioms, ...
0
votes
1answer
206 views

Possible no standard use of replacement axiom

The idea is to build in ZFC using replacement, a set REPLACEMENT(x ∈ A: TERM(x)) from a set and a term in the same way the set {x ∈ A: FORMULA(x)} is built using specification from a set and a formula....
2
votes
1answer
394 views

Consistency of: “The continuum function is injective, and for all infinite cardinals $\kappa$ we have that $2^\kappa$ is weakly inaccessible.”

I asked here about "large powerset axioms" and to my delight, learned that such axioms are being taken seriously. I've been toying with them ever since. My favourite is: "The continuum function is ...
4
votes
1answer
159 views

Class theory with support for self-application of class functions?

To every natural number $n$, we can assign its Church numeral $\underline{n}.$ A formal definition would be: $\underline{0}(f)=\mathrm{id}_{\mathrm{dom}(f)}$ $\underline{n+1}(f) = \underline{n}...
2
votes
2answers
500 views

Have axioms / axiom schemata of this flavour been proposed or otherwise considered?

With the exception of a few miscellaneous cases, the axioms (and/or schemeta) of ZFC can roughly be divided into two kinds: Those that guarantee the existence of more complicated sets, given that ...
3
votes
1answer
640 views

Please recommend a nice and concise math book on probability theory [closed]

My intention is neither to learn basic probability concepts, nor to learn applications of the theory. My background is at the graduate level of having completed all engineering courses in probability/...
5
votes
1answer
224 views

Order Types and Replacement Schema

Using Replacement Schema we can prove any well-ordering is isomorphic to an ordinal number. Q: Is the following consistent? $ZFC-Rep+\neg Rep+\text{Any well-ordering is isomorphic to an ordinal ...