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

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14
votes
6answers
768 views

When does the set enter set theory?

I wonder about the foundations of set theory and my question can be stated in some related forms: If we base Zermelo–Fraenkel set theory on first order logic, does that mean first order logic is not ...
7
votes
2answers
487 views

How does (ZFC-Infinity+“There is no infinite set”) compare with PA?

How does (ZFC-Infinity+"There is no infinite set") compare with (first order) PA? Intuitively, neither theory should be more powerful than the other.
12
votes
2answers
727 views

Where is axiom of regularity actually used?

Where is axiom of regularity actually used? Why is it important? Are there some proofs, which are substantially simpler thanks to this axiom? This question was to some extent provoked by Dan ...
14
votes
1answer
839 views

Proofs given in undergrad degree that need Continuum hypothesis?

Or alternative you need to assume CH is false. I know several proofs that use axiom of choice. Heine Borel theorem is the best example I can think off. Zorns lemma is heavily used in the non ...
5
votes
2answers
694 views

What are the consequences if Axiom of Infinity is negated?

What mathematics can be built with standard ZFC with Axiom of Infinity replaced with its negation? Can the analysis be built? Is there special name for "ZFC without Infinity" set theory? I also ...
23
votes
6answers
2k views

What are natural numbers?

What are the natural numbers? Is it a valid question at all? My understanding is that a set satisfying Peano axioms is called "the natural numbers" and from that one builds integers, rational ...
3
votes
1answer
108 views

When do surjections split in ZF? Two surjections imply bijection?

We have that the Axiom of Choice is equivalent to the principle that every surjection has a right inverse. However, without the Axiom of Choice we can determine for some $X$ that $X\succeq Y\implies ...
7
votes
3answers
951 views

The existence of the empty set is an axiom of ZFC or not?

I found in the Wolfram MathWorld page of the Axiom of the Empty Set that this is one of the Zermelo-Fraenkel Axioms, however on the page about these ZFC Axioms I read that it is an axiom that can be ...
2
votes
2answers
666 views

what is the relationship between ZFC and first-order logic?

In Wikipedia, it says that ZFC is a one-sorted theory in first-order logic. However, I was not really able to comprehend the later parts that seem to elaborate on that point. Can anyone explain the ...
54
votes
10answers
13k views

How is a system of axioms different from a system of beliefs?

Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?
62
votes
7answers
11k views

Can you explain the “Axiom of choice” in simple terms?

As I'm sure many of you do, I read the XKCD webcomic regularly. The most recent one involves a joke about the Axiom of Choice, which I didn't get. I went to Wikipedia to see what the Axiom of ...
17
votes
5answers
1k views

What is the modern axiomatization of (Euclidean) plane geometry?

I have heard anecdotally that Euclid's Elements was an unsatisfactory development of geometry, because it was not rigorous, and that this spurred other people (including Hilbert) to create their own ...
14
votes
1answer
624 views

Can we prove the existence of $A\cup B$ without the union axiom?

If $A$ and $B$ are sets, then $A\cup B$ is defined as follows: $$A\cup B:=\{x: x\in A \, \, \,\text{ or } \, \, \, x\in B\}.$$ In ZFC, $A\cup B$ exists because $\bigcup\{A,B\}$ exists by union axiom, ...
7
votes
9answers
772 views

Motivating implications of the axiom of choice?

What are some motivating consequences of the axiom of choice (or its omission)? I know that weak forms of choice are sometimes required for interesting results like Banach-Tarski; what are some ...
3
votes
2answers
336 views

Axiom schema and the definition of natural numbers

An axiom schema is used to generate the axioms, which inductively define the natrual numbers using the empty set and the successor function $S$. I don't understand why you have to define this set as ...
5
votes
3answers
743 views

How do the separation axioms follow from the replacement axioms?

It has come to my attention that the pairing axiom and the separation axiom schema are rarely listed since they follows from the replacement axioms. I see how this works for the pairing axiom, since ...
3
votes
3answers
280 views

Equality of positive rational numbers.

I am reading the second article Rational Numbers of the book "A Treatise on Advanced Calculus" by Philip Franklin. I have mainly 3 questions regarding this article. I am writing all these $3$ ...
4
votes
1answer
401 views

Can it be shown that ZFC has statements which cannot be proven to be independent, but are?

I am familiar with the concept that a statement can be proven indepent such as in the case of the continuum hypothesis where both ZFC+CH and ZFC+(CH is false) are both proven consistent, but I would ...
5
votes
2answers
867 views

A first order sentence such that the finite Spectrum of that sentence is the prime numbers

The finite spectrum of a theory $T$ is the set of natural numbers such that there exists a model of that size. That is $Fs(T):= \{n \in \mathbb{N} | \exists \mathcal{M}\models T : |\mathcal{M}| =n\}$ ...
48
votes
8answers
4k views

Does mathematics require axioms?

I just read this whole article: http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf which is also discussed over here: Infinite sets don't exist!? However, the paragraph which I found most ...
26
votes
6answers
1k views

Is $\{0\}$ a field?

Consider the set $F$ consisting of the single element $I$. Define addition and multiplication such that $I+I=I$ and $I \times I=I$ . This ring satisfies the field axioms: Closure under addition. ...
10
votes
3answers
555 views

How does the axiom of regularity forbid self containing sets?

The axiom of regularity basically says that a set must be disjoint from at least one element. I have heard this disproves self containing sets. I see how it could prevent $A=\{A\}$, but it would seem ...
10
votes
5answers
578 views

Axiomatic approach to polynomials?

I only know the "constructive" definition of $\mathbb K [x]$, via the space of finite sequences in $\mathbb K$. It essentially tells a polynomial is its coefficients. Is there a way to define ...
8
votes
5answers
475 views

A confusion about Axiom of Choice and existence of maximal ideals.

The proof of the theorem of statement that every ring has a maximal ideal uses zorn's lemma or the axiom of choice.Now, the defintion of ring as well as the definition of maximal ideal don't depend on ...
7
votes
3answers
281 views

Axiom of extensionality in ZF - pointless?

In every textbook on ZF set theory I come across the Axiom of extensionality, which basically says that if two sets have the same elements, then those two sets are equal: $$\forall x \forall y ...
14
votes
2answers
318 views

Non-constructive axiom of infinity

Looking at the ZFC axioms (in the Wikipedia version), the axiom of infinity stands out because it contains an extremely specific construction. This seems rather unelegant to me, therefore I've thought ...
5
votes
2answers
1k 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 ...
4
votes
3answers
618 views

A basic doubt on axiom of foundation of Zermelo-Fraenkel set theory

I have read in a book that the "axiom of foundation prevents anomalies such as a set being an element of itself". Now, axiom of foundation says that there exist an element in every set which is ...
5
votes
1answer
189 views

A weaker Axiom of Infinity?

As I understand, the Axiom of Infinity in ZFC theory gives us an infinite set from which the natural numbers can be extracted. (See "Axiom of Infinity" at ...
5
votes
4answers
289 views

Axiomatization of $\mathbb{Z}$

Though I've seen several cool axiomizations of $\mathbb{R}$, I've never seen any at all for $\mathbb{Z}$. My initial guess was that $\mathbb{Z}$ would be a ordered ring which is "weakly" well-ordered ...
2
votes
6answers
153 views

How do we define arc length?

In trying to write a nice proof of the derivatives of $\sin(x)$ and $\cos(x)$, I encountered a serious problem, namely that I have never seen a proper definition of the notion of arc length. Based on ...
1
vote
2answers
92 views

¿Can you help me with axiom of regularity? [duplicate]

I understand that if we define a set such that $A=\{A\}$ we have then a contradiction because of regularity axiom, thus, a set can not be a member of itself. My question is; how can it be so, if I ...
0
votes
1answer
46 views

Whitehead's axioms of projective geometry and a vector space over a field

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
-1
votes
1answer
33 views

Axiom of infinity: What is an inductive set?

This Axiom states that there exists an inductive set. But, what is the definition of an inductive set?
53
votes
6answers
5k views

In what sense are math axioms true?

Say I am explaining to a kid, $A +B$ is the same as $B+A$ for natural numbers. The kid asks: why? Well, it's an axiom. It's called commutativity (which is not even true for most groups). How do I ...
22
votes
6answers
1k views

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). ...
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
votes
5answers
593 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 ...
18
votes
5answers
699 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 ...
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 ...
5
votes
1answer
714 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 ...
6
votes
1answer
252 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 ...
7
votes
1answer
989 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?
5
votes
2answers
101 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
1answer
164 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
1answer
265 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
284 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
249 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
170 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
153 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 ...