For questions on axioms, mathematical statements that are accepted as being true, usually without controversy.

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6answers
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Axiomatic characterization of the rational numbers

We have the well-known Peano axioms for the natural numbers and the real numbers can be characterized by demanding them to be a Dedekind-complete, totally ordered field (or some variation of this). ...
8
votes
4answers
893 views

Why the axioms for a topological space are those axioms?

This question might have even been asked here before, I don't really know, so sorry if it's duplicate. I've started to study topological spaces and I've found the axioms for such spaces kind of hard ...
8
votes
6answers
3k views

Why accept the axiom of infinity?

According to my readings, Russell showed that a principle Frege used to reduce Peano arithmetic to logic lead to a contradiction. So, Russell tried to reduce mathematics to logic a different way but ...
19
votes
5answers
825 views

Why hasn't GCH become a standard axiom of ZFC?

I've never seen a text that includes GCH in the ZFC axioms. I presume this means that GCH has not achieved widespread acceptance. This seems surprising to me, given that: The cardinal numbers ...
15
votes
2answers
3k views

What happens if we remove the requirement that $\langle R, + \rangle$ is abelian from the definition of a ring?

Ever since I learned the definition of a ring, I've wondered why the additive group is required to be abelian. What happens if we allow $\langle R, + \rangle$ to be nonabelian as well as $\langle R, \...
10
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5answers
712 views

What are the postulates that can be used to derive geometry?

What are the various sets of postulates that can used to derive Euclidean geometry? It might be nice to have several different approaches together for comparison purposes and for ready reference. It ...
9
votes
3answers
1k views

Is the Bourbaki treatment of Set Theory outdated?

On page 5 of the following write up, the author asks why the Bourbaki did not notice that their system of Zermelo set theory with AC was inadequate for existing mathematics. Throughout the rest of the ...
6
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1answer
1k views

Explain Zermelo–Fraenkel set theory in layman terms

What does Zermelo–Fraenkel set theory mean? According to Wikipedia, Zermelo–Fraenkel set theory is a set theory that is proposed to overcome issues in naive set theory. I really appreciate if ...
16
votes
1answer
501 views

Is there an axiomatic approach to ordinal arithmetic?

I've always wondered, is there an axiomatic approach to the arithmetic of ordinal numbers? If so, I imagine it would be on par with set theory in terms of its proof-theoretic strength.
15
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4answers
1k views

Vector Spaces: Redundant Axiom?

Question Why are the axioms for vector space independent? More precisely $1x=x$ seems redundant... (I take the axioms from: Wikipedia) Explanation One has for zero vector: $$\lambda0+\lambda0=\...
11
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4answers
552 views

Kolmogorov's probability axioms

Why Kolmogorov's axioms are considered such a breakthrough in probability theory? They are just 3 simple statements everyone can agree with. When creating a system of axioms like this it's necessary ...
10
votes
3answers
883 views

Which statements are equivalent to the parallel postulate?

I would like to have a long-ish list of statements that are equivalent to the parallel postulate. If a line segment intersects two straight lines forming two interior angles on the same side that ...
6
votes
1answer
311 views

Topology: Opens vs Neighborhoods

Disclaimer: This thread is meant informative and therefore written in Q&A style. The problems are highlighted in bold face. The axiomatization of topology can be done in various ways all of ...
3
votes
2answers
70 views

Logical dependence between vector space axioms

My question is not very long, but I'd like to explain where it comes from. Consider the classical definition of vector spaces: $E$ is said to be a vector space over a field $F$ when: A) E is a ...
9
votes
1answer
145 views

What is needed to make Euclidean spaces isomorphic as groups?

Consider the abelian groups $G_n=(\mathbb R^n,+)$ for $n\geq1$. Claim: For any $n$ and $m$ the groups $G_n$ and $G_m$ are isomorphic. This claim is true if one assumes the axiom of choice, and I ...
8
votes
3answers
385 views

Is the “domain of discourse” in axiomatic set theory also a “set”?

The domain of discourse is defined by Wikipedia as the "set of entities over which certain variables of interest in some formal treatment may range." However, I believe we could not call the domain of ...
7
votes
1answer
2k views

What are the differences between Hilbert's axioms and Euclid's axioms?

Euclid had his axioms. Why would we need Hilbert's modern axiomatization of Euclidean geometry? What are key differences between the two sets of axioms?
6
votes
1answer
698 views

A model of geometry with the negation of Pasch’s axiom? [duplicate]

How would a geometry with all the usual axioms of Euclidean geometry, except that instead of Pasch’s axiom, we take it negation, look like?
4
votes
1answer
168 views

Two congruent segments does have the same length?

The answer to the question in the title seems an obvious ''Yes by definition !''. And this really is the definition from Wikipedia: Two line segments are congruent if they have the same length. ...
3
votes
3answers
153 views

Why do we want the Axiom of the Power Set?

I'm just learning a bit about axiomatic set theory, and I'm kind of confused as to why we need/want this axiom? Does not accepting it imply that there exists some set which doesn't have a power set? ...
2
votes
2answers
133 views

Different axiomatizations of set equality

I've seen two definitions (or axioms?) of set equality: $a=b \Leftrightarrow (\forall x : x \in a \Leftrightarrow x \in b)$ $a=b \Leftrightarrow (\forall x : a \in x \Leftrightarrow b \in x)$ That ...
2
votes
1answer
285 views

What makes Tarski Grothendieck set theory non-empty?

I'm fighting with Grothendieck set theory for some time now. This is the framework for the automated proof checking system of Mizar and hence there is a formalized version of the axioms here too, and ...
14
votes
2answers
2k views

Axiom of Regularity

I am having difficulty understanding Axiom of Regularity : Every non-empty set $\rm A$ contains an element $\rm B$ which is disjoint from $\rm A.$ So from my understanding if I have a set like: ...
6
votes
1answer
204 views

Can the generalized continuum hypothesis be disguised as a principle of logic?

A cool way to formulate the axiom of choice (AC) is: AC. For all sets $X$ and $Y$ and all predicates $P : X \times Y \rightarrow \rm\{True,False\}$, we have: $$(\forall x:X)(\exists y:Y)P(x,y) \...
6
votes
3answers
126 views

Is my proof of the uniqueness of $0$ non-circular?

Please try to avoid jumping directly the proof, the text before it is crucial to my question as well. I had a proof of this here, but I have come to realize that the proof is circular since I implied ...
6
votes
2answers
299 views

“Completeness modulo Godel sentences”?

So this has been bugging me for roughly four years. When I was an undergraduate, I attended a colloquium in which the speaker was a 'cheerleader' for AD (the axiom of determinacy- an alternative to ...
6
votes
3answers
328 views

Why is the postulate $1$ not equal to $0$ not superfluous? [duplicate]

Possible Duplicate: Explanation for why $1\neq 0$ is explicitly mentioned in Chapter 1 of Spivak's Calculus for properties of numbers. I am self-studying the wonderful book, Elementary ...
5
votes
4answers
8k views

What is the difference between an axiom and a postulate?

I hear about axioms in set theory and postulates in geometry, but they seem like the same thing. Do they mean the same thing but then are used in different instances or what? Is one word more ...
5
votes
2answers
5k views

Proof for SSS Congruence?

I'm hoping that someone can provide a method for deducing the commonly known SSS congruence postulate? The postulate states If the three sides of one triangle are pair-wise congruent to the three ...
5
votes
1answer
813 views

Is Euclid's Fourth Postulate Redundant?

Euclid's Elements start with five Postulates, including the fifth one, the famous Parallel Postulate. Less well, known however is the Postulate that forms the basis for motivation behind the fifth: ...
4
votes
1answer
162 views

ZFC + not-CH, can a set with cardinality between $\aleph_0$ and $2^{\aleph_0}$ be defined?

Continuum hypothesis states, there is no set with cardinality between the integers and the reals. There is a milestone result, that CH is independent from ZFC. That means, both of ZFC + CH, and ZFC + ...
4
votes
2answers
302 views

Why does any transitive model satisfy extensionality?

I see it stated as something very clear, but i can't figure it out. i found a proof in Jech(old version) which goes through the concept of restricted formulas, which i don't quite understand. (i'm not ...
4
votes
3answers
1k views

What properties are allowed in comprehension axiom of ZFC?

I am trying to understand the axioms of ZFC. The axiom schema of specification or the comprehension axioms says: If A is a set and $\phi(x)$ formalizes a property of sets then there exists a set C ...
3
votes
3answers
332 views

Defining the integers and rationals

What axiomatic system is most commonly used by modern mathematicians to describe the properties of the integers and the rationals? Properties like $a+0=a$, $a*1=a$, $a+b=b+a$, Also given these ...
3
votes
2answers
842 views

What are the primitive notions of real analysis?

My dad introduced me to primitive notions in geometry in high school. It's come back to haunt me as I study real analysis; I find myself wondering, Have we given this a formal definition? ...
3
votes
3answers
264 views

An elementary question regarding the uniqueness of a set, viewed with different cardinality

Does the cardinality of sets, like for example the real numbers, depend on the fundamental axioms one is working with? If so, what does it mean to speak of such a set if it is not really one single ...
2
votes
5answers
2k views

Are (some) axioms “unprovable truths” of Godel's Incompleteness Theorem?

Like any math newbie, Godel's Incompleteness Theorems are easy to understand in general layman's terms, but difficult to understand beyond the typical "liar's paradox" and "barber's paradox" type ...
2
votes
2answers
169 views

Axioms for sets of numbers

What is the most common axiomatic system used by modern mathematicans for the properties of the integers, rationals, reals, and complex numbers? Or does one commonly use a single axiomatic system ...
1
vote
1answer
402 views

Axiom of extension

I am learning Set Theory from the book Naive Set Theory by Halmos as part of my course. The first chapter is on the Axiom of Extension. I understand what it is but what I don't understand is why it ...
1
vote
1answer
93 views

Hilbert projection theorem without countable choice

All the proofs of the Hilbert projection theorem, existence part, that I have seen so far use countable choice (usually implicitly). Is this necessary? It seems like you might be able to leverage the ...
1
vote
1answer
274 views

Do the proofes in set theory rely on the semantics of the formulas used in the axioms?

Motivation: The Axiom of separation $$\forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow [ x \in A \wedge \phi(x, w_1, \ldots, w_n, A) ] )$$ is used to ...
7
votes
1answer
70 views

A familiar quasigroup - about independent axioms

A quasigroup is a pair $(Q,/)$, where $/$ is a binary operation on $Q$, such that (1) for each $a,b\in Q$ there exists unique solutions to the equations $a/x=b$ and $y/a=b$. Now I want to extract a ...
7
votes
2answers
1k views

How can I write the Axiom of Specification as a sentence?

I began reading Paul Halmos' "Naive Set Theory", and encountered the "Axiom of Specification". To every set $A$ and to every condition $S(x)$ there corresponds a set $B$ whose elements are exactly ...
6
votes
1answer
319 views

Is it possible to prove the axiom of infinity from the real number axioms?

What I mean by the title is whether, given that there is a class $\Bbb R$ and operations $+,\cdot,<$ that satisfy the ordered field axioms and the least upper bound axiom, you can prove the ...
4
votes
3answers
257 views

If we abandon the axiom of regularity, can the cumulative hierarchy just become a definition?

In ZFC, the axiom of regularity is used to prove that every set is an element of some stage of the cumulative hierarchy. The index of the least such stage is, by definition, the rank of that set. Now ...
4
votes
2answers
472 views

Problem understanding the Axiom of Foundation

I am just beginning to learn the ZF axioms of set theory, and I am having trouble understanding the Axiom of Foundation. What exactly does it mean that "every non-empty set $x$ contains a member $y$ ...
3
votes
2answers
825 views

Why is matrix multiplication defined a certain way? [duplicate]

Why is it that when multiplying a (1x3) by (3x1) matrix, you get a (1x1) matrix, but when multiplying a (3x1) matrix by a (1x3) matrix, you get a (3x3) matrix? Why is matrix multiplication defined ...
3
votes
2answers
648 views

axioms of equality

Every text I've come across uses the Axiom of Extensionality to describe the fact that sets are equal iff they contain each other's elements. $$(\forall x)(\forall y)\bigl((\forall a)(a\in x \iff a\in ...
3
votes
2answers
98 views

Module over a ring which satisfies Whitehead's axioms of projective geometry

I asked this question(Characterization of a vector space over an associative division ring). It was pointed out that the answer was negative. So I reconsidered the problem and have come up with the ...
2
votes
2answers
90 views

How do we introduce subtraction from these field axioms?

I am familiar with two different sets of field axioms. The first one is from "Mathematical Analysis" by Apostol. It has the first $3$ usual axioms, but the $4^{th}$ one is different: Axiom 1: ...