Questions tagged [axioms]

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51
votes
10answers
9k 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 ...
47
votes
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 ...
36
votes
5answers
3k 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 ...
33
votes
6answers
5k views

Does finite mathematics need the axiom of infinity?

A statement referring to an infinite set can sometimes be logically rephrased using only finite sets/objects. For example, "The set of primes is infinite" <-> "There is no largest prime". ...
31
votes
3answers
4k views

Counterintuitive consequences of the Axiom of Determinacy?

I just read Dr Strangechoice's explanation that if all subsets of the real numbers are Lebesgue measurable, then you can partition $2^\omega$ into more than $2^\omega$ many pairwise disjoint nonempty ...
28
votes
11answers
6k views

Does the Axiom of Choice (or any other "optional" set theory axiom) have real-world consequences? [closed]

Or another way to put it: Could the axiom of choice, or any other set-theoretic axiom/formulation which we normally think of as undecidable, be somehow empirically testable? If you have a particular ...
27
votes
3answers
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 ...
19
votes
1answer
2k views

What's the difference between ZFC+Grothendieck, ZFC+inaccessible cardinals and Tarski-Grothendieck set theory?

Say that "U" is the axiom that "For each set x, there exists a Grothendieck universe U such that x $\in$ U", where Grothendieck universes are defined in the usual way (or, if that'...
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 ...
17
votes
6answers
4k views

Axiomatic definition of integers

The real numbers can be axiomatically defined (up to isomorphism) as a Dedekind-complete ordered field. What is a similar standard axiomatic definition of the integer numbers? A commutative ordered ...
17
votes
3answers
4k views

How would calculus be possible in a finitist axiom system?

I am interested in learning a little more about finitism, currently about which I only know a few encyclopedic paragraphs. I know that during some time, some mathematicians like Kronecker thought ...
15
votes
3answers
1k views

Any paradoxical theorems arising from large cardinal axioms?

If we accept the axiom of choice, we take the responsibility of living in a world in which, e.g., a ball in Euclidean 3-space is equiscindable to two isometric copies of itself (Banach-Tarski). So we ...
14
votes
5answers
5k views

getting rid of existential quantifiers

It seems to me that for most of the twentieth century, axiomatic foundations for mathematical theories were constructed with the (mostly allied) goals of minimizing the number of primitive notions and ...
14
votes
0answers
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 ...
13
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0answers
757 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, ...
12
votes
1answer
711 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, ...
11
votes
5answers
3k views

Minimal subset of axioms for ZFC

Hello all, one may look for "minimal system of axioms" for ZFC (or any other theory) in the following (unusual) sense : say that a subset S of ZFC is "sufficient" if there is an explicit procedure ...
11
votes
2answers
6k views

Is it possible to show that an infinite set has a countable (infinite) subset, without using the Axiom of Choice?

Let X be an infinite set. Is it possible to show the existence of a countably infinite subset of X without using the Axiom of Choice?
11
votes
3answers
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, ...
10
votes
1answer
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-...
10
votes
2answers
397 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 ...
10
votes
1answer
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 ...
10
votes
1answer
945 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: ...
10
votes
1answer
692 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 ...
10
votes
0answers
342 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 ...
9
votes
2answers
894 views

Are there any complete, first-order and unstable theories which have non-categorical second-order formulations?

Since it's not stable, $PA$ fails at being categorical in a power in the worst possible way, having $2^{\lambda}$ models in any uncountable $\lambda$. But $PA$ regains its categoricity in the move to ...
9
votes
1answer
799 views

Axiom of class collection

One version of the Axiom of Collection says that any surjection $A\to B$ from a class $A$ to a set $B$ is factored through by some surjection $C\to B$ where $C$ is a set. Note that assuming $B$ is a ...
9
votes
0answers
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\...
8
votes
1answer
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 ...
8
votes
1answer
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,...
8
votes
0answers
215 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 ...
7
votes
8answers
923 views

Result that follows from ZFC and not ZF but are strictly weaker than choice

A number of results that people use that require the axiom of choice (i.e. do not follow from ZF alone) are known to actually imply the axiom of choice. Therefore, one might naturally wonder whether ...
7
votes
2answers
1k views

Are there uncountably many essentially inequivalent versions of Mathematics?

Hi everyone, Disclaimer 1: logic and set theory are a long way from my field, so apologies in advance if I demonstrate extreme ignorance or stupidity, and please correct me if (when?) I write stupid ...
7
votes
2answers
286 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 ...
7
votes
1answer
143 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 ...
7
votes
0answers
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 ...
6
votes
1answer
447 views

What is known about size-restricted power set axioms?

What is known about ZF without powerset but with an axiom "every set has a set of all its countable subsets"? This seems stronger than positing that the set of natural numbers has a powerset, though ...
6
votes
1answer
842 views

theorems equivalent to the parallel postulate

Is there a good survey article listing all the theorems of Euclidean geometry that are equivalent to the parallel postulate?
6
votes
0answers
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 ...
6
votes
0answers
196 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 ...
5
votes
2answers
2k views

Axiom of Computable Choice versus Axiom of Choice

What would be the consequence of requiring that any choice function be computable; i.e. using as the foundational basis ZF + ACC? Does it make a difference if we admit definable functions? I guess I ...
5
votes
2answers
623 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 ...
5
votes
0answers
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 ...
5
votes
0answers
401 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 $\{\...
5
votes
0answers
682 views

two versions of the nested interval property

There appear to be two different nested interval properties for the reals with the punchline "... then the intersection of the intervals is non-empty", and I'd like to know their respective histories (...
4
votes
2answers
312 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. ...
4
votes
2answers
405 views

On wild behavior of $\omega_{1}$ in the absence of some essential axioms of $ZFC$

The regularity of $\omega_{1}$ is one of the most well known facts of set theory. But it seems that in order to prove this simple fact we need the "full power" of mathematics! For example by an ...
4
votes
4answers
754 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
430 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)) \...
4
votes
3answers
1k views

Survey of finite axiomatizability for relational theories?

An $L$-theory $T$ is finitely axiomatizable if there is a finite set $A$ of $L$-sentences with the same consequences as $T$, i.e. such that $M \models T$ iff $M \models A$ for every $L$-structure $M$. ...