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

learn more… | top users | synonyms

10
votes
3answers
684 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 ...
8
votes
6answers
2k 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 ...
6
votes
1answer
285 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 ...
6
votes
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 ...
3
votes
2answers
64 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 ...
14
votes
4answers
896 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: ...
7
votes
1answer
129 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 ...
7
votes
1answer
1k 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
3answers
122 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
283 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 ...
5
votes
2answers
121 views

How can I define $\mathbb{N}$ if I postulate existence of a Dedekind-infinite set rather than existence of an inductive set?

Suppose in the axioms of $\sf ZF$ we replaced the Axiom of infinity There exists an inductive set. with the Axiom of Dedekind-infinite set There exists a set equipollent with its proper ...
5
votes
2answers
6k 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 ...
4
votes
1answer
669 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
325 views

System with infinite number of axioms

Assume we have a set of axioms $A_0$. There exists a statement that can be formulated with these axioms that cannot be proven to be true with this system. Assume we give such a statement axiomatic ...
3
votes
3answers
108 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? ...
3
votes
3answers
320 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
3answers
261 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
1answer
247 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 ...
2
votes
2answers
166 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
91 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
271 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
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
302 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 ...
6
votes
4answers
1k views

which axiom(s) are behind the Pythagorean Theorem

There are many elementary proofs for the Pythagorean Theorem, but no matter they use areas, similarities, even algebraic proofs, it is not straightforward to tell why it is true tracing back to the ...
6
votes
1answer
204 views

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

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
3answers
228 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 ...
3
votes
2answers
781 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? ...
2
votes
2answers
84 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: ...
1
vote
3answers
107 views

$\exists x ~ \forall y ~ f(x, y) \iff \forall y() ~ \exists x~ f(x, y(x))$ Name? Proof?

I was looking at potential theorems, and this one came up: $$\bigg(\exists x ~ \forall y ~:~ f(x, y) \bigg) \iff \bigg(\forall y() ~ \exists x ~:~ f(x, y(x))\bigg)$$ (where the second $y$ is ...
1
vote
0answers
72 views

Using definitions instead of axioms.

Lets take (classical) first-order logic for granted, including an equality symbol and its associated axioms. Given all this, a rigorous work of mathematics will typically begin with a signature - ...
1
vote
1answer
164 views

What should I be able to do with this chapter on Axiomatic Set Theory in order to check if I've learned it decently? [closed]

I've just read a chapter on axiomatic set theory, from Comprehensive Mathematics for Computer Scientists 1. It comes with basic notation on sets and some axioms: Axiom 1 (Axiom of Empty Set) Axiom 2 ...
7
votes
1answer
67 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 ...
6
votes
3answers
313 views

Question Regarding the Axiom of Extensionality

Jech's text on Set Theory states the following: If X and Y have the same elements, then X = Y : ∀u(u ∈ X ↔ u ∈ Y ) → X = Y. The converse, namely, if X = Y then u ∈ X ↔ u ∈ Y, is an axiom ...
5
votes
3answers
325 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 ...
4
votes
1answer
150 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 + ...
3
votes
2answers
97 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 ...
3
votes
2answers
548 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
729 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 ...
2
votes
0answers
71 views

Prove if $x<y$ then $x+z<y+z$ [duplicate]

Note I am looking a proof using Peano's axiom as I am working on the Natural Numbers, not the Real Numbers. I need help proving "For each x,y,z in N, if $x<y$ then $x+z<y+z$ This must be done ...
1
vote
4answers
61 views

Why proof by induction is working. [duplicate]

I know how to make proofs by induction, but I don't understand why it prove that the propriety is true. It's in fact logical, but how to prove that proof by induction really prove the assertion.
1
vote
1answer
273 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
326 views

Book covering introduction to mathematical proofs

I am looking for some introductory books covering mathematical proofs, axioms, propositions, proof techniques etc in general.
1
vote
6answers
874 views

Is it possible to have a field without an additive identity?

If I drop the axiom that Zero is the identity of an addition what consequences does this entail? What do I need to change to my axiomatization? By definition it is not possible, but are there ...
0
votes
1answer
75 views

Question about a list of ZF axioms and, in particular, SEP

In A.G. Hamilton's Logic for mathematicians, eight axioms of ZF are given: EXT, NULL, PAIR, UNION, POW, REP, INF and REG. The Axiom Scheme of Replacement is formulated like this: $$ (\forall ...
0
votes
0answers
16 views

CC and one of its equivalent condition [duplicate]

How we can prove: CC is equivalent to for all sequence(Xn) of non empty sets there exist (xn) which meets infinitely many (Xn) One side assuming CC is obvious. But what about converse. How we can ...
0
votes
0answers
82 views

A simple question about rational numbers without a simple proof?

As in this question, study the quasigroup $(Q_+,/)$ of positive rational numbers under division. There are two obvious identities: $a/(b/c)=c/(b/a)$, for all $a,b,c\in Q_+$ $(a/b)/c=(a/c)/b$, for ...
0
votes
1answer
46 views

Given a collection of functions $f_i$ with the same domain, how to replace with values (w/o axiom replacement)

I know from a collection of ordered pairs we can project onto the first coordinate. I'm interested if there's a way (without using the axiom of replacement) to "project" a collection of functions onto ...