For questions on axioms, mathematical statements that are accepted as being true without a mathematical proof.

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53
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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?
52
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 ...
52
votes
4answers
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 ...
46
votes
8answers
3k 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 ...
27
votes
7answers
2k views

Are there infinite sets of axioms?

I'm reading Behnke's Fundamentals of mathematics: If the number of axioms is finite, we can reduce the concept of a consequence to that of a tautology. I got curious on this: Are there infinite ...
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. ...
23
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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 ...
22
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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). ...
18
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5answers
691 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 ...
17
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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 ...
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, ...
15
votes
1answer
402 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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2answers
265 views

How much math do we need to prove all simple numeric identities?

Consider real numeric expressions build only from integers, operators $+,-,\times,/$ and taking a positive expression to a power (no variables involved), e.g. $$\frac{2}{7},\ 2^{1/2},\ ...
14
votes
6answers
766 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 ...
14
votes
1answer
833 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 ...
14
votes
1answer
603 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, ...
14
votes
2answers
313 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 ...
14
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2answers
193 views

How far is it true that statements dependent on Axiom of Choice are not constructive.

Axiom of Choice is often used in mathematics to construct various objects, such as basis of $\mathbb{R}$ as a vector space over $\mathbb{Q}$, unmeasurable subset of $\mathbb{R}$, or a non-principal ...
12
votes
2answers
722 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 ...
12
votes
3answers
768 views

In axiomatization of propositional logic, why can uniform substitution be applied only to axioms?

I'm reading an introductory book about mathematical logic for Computation (just for reference, the book is "Lógica para Computação", by Corrêa da Silva, Finger & Melo), and would like to ask a ...
11
votes
6answers
345 views

Axiomatic Foundations

I am trying to deduce how mathematicians decide on what axioms to use and why not other axioms, I mean surely there is an infinite amount of available axioms. What I am trying to get at is surely if ...
11
votes
2answers
636 views

Which axioms of ZFC or PA are known to not be derivable from the others?

Which, if any, axioms of ZFC are known to not be derivable from the other axioms? Which, if any, axioms of PA are known to not be derivable from the other axioms?
11
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3answers
4k views

Euclid / Hilbert: “Two lines parallel to a third line are parallel to each other.”

Background Many geometry books used to teach high-schoolers these days try to transfer Hilbert's reworking of Euclid's axioms into a (somewhat) palatable form for students. They don't usually seem to ...
11
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3answers
705 views

Proof that every circle has the ratio of $\pi$

We know that the ratio between the circumference's length and diameter is equal to $\pi$, but can this be proved for every circle? Or is this an axiom?
10
votes
5answers
574 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 ...
10
votes
3answers
734 views

What's so special about the group axioms?

I've only just begun studying group theory (up to Lagrange) following on from vector spaces and I am still finding them almost frustratingly arbitrary. I'm not sure what exactly it is about the ...
10
votes
3answers
541 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
591 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 ...
10
votes
2answers
226 views

When the mathematical community consider the inclusion of a new axiom?.

At first I was thinking about the axiom of choice, but let's keep it general. What motivates the inclusion of new axioms (or change the ones we already have in an already defined axiomatic theory?. It ...
10
votes
1answer
197 views

What is the smallest fragment of ZFC that has the same consistency strength as ZFC?

The question in the title is undoubtedly nonsensical, but I am not sure how to state this question properly. Perhaps some examples will help me explain it. Thanks to Godel and Cohen, we know that ...
9
votes
5answers
770 views

Are $x \cdot 0 = 0$, $x \cdot 1 = x$, and $-(-x) = x$ axioms?

Context: Rings. Are $x \cdot 0 = 0$ and $x \cdot 1 = x$ and $-(-x) = x$ axioms? Arguably three questions in one, but since they all are properties of the multiplication, I'll try my luck...
9
votes
2answers
301 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 ...
9
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0answers
106 views

Axiomatizing topology through continuous maps

Suppose we have some topological space $X$ and we somehow forgot about the topology. A friend of ours knows the topology and offers to tell us for any map $X\to Y$ into any topological space $Y$ ...
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 ...
8
votes
5answers
468 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 ...
8
votes
3answers
271 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 ...
8
votes
3answers
512 views

What constitutes an axiom - Spivak Calculus ch. 1

In chapter 1 of Spivak's Calculus text he lays out some fundamental axioms of the integers. For instance that: $a \cdot 1 = a$, for all $a$. However he doesn't list an axiom that for instance says: $a ...
8
votes
2answers
280 views

Axiomatic system and Hilbert's 2nd problem

Hilbert's second problem asks if the axioms of arithmetic are consistent. Has this problem been resolved? Shouldn't an axiomatic system ideally be consistent and complete(given that we have the ...
8
votes
3answers
529 views

Relationship between Category theory and Axiomatic set theory

I've recently started learning Category theory- and I have a pondering- wondering if anyone can help. Is it possible for two categories to satisfy two different set-axiom system. Namely- is it ...
7
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9answers
761 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 ...
7
votes
3answers
670 views

Has anyone ever proposed additional axioms?

According to Wikipedia, Godel's incompleteness theorem states: No consistent system of axioms whose theorems can be listed by an "effective procedure" (essentially, a computer program) is ...
7
votes
6answers
1k views

How to demystify the axioms of propositional logic?

How might I go about getting some intuition on the typical axiom schemes given for propositional logic? They seem rather mysterious at first glance. For example, these are taken from: ...
7
votes
3answers
934 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 ...
7
votes
2answers
482 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.
7
votes
2answers
183 views

Is $\mathrm{ZFC}^E$ outright inconsistent?

From $\mathrm{ZFC},$ define a new theory $\mathrm{ZFC}^E$ by adjoining a constant symbol $E$ together with axioms to the effect that: $E$ is countable and transitive $(E,\in)$ is an elementarily ...
7
votes
3answers
277 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 ...
7
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3answers
274 views

Why is zero important?

I am not sure whether this question is more appropriate here or in theoretical computer science. I leave it to the wisdom of moderators. On the computer science site I came across the following ...
7
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1answer
964 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?
7
votes
2answers
851 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 ...
7
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2answers
132 views

Is there a way to axiomatize the category of sets and relations?

The system of axioms known as ETCS axiomatizes the category of sets and functions. Does anyone know of a way to axiomatize the category (and/or allegory) of sets and relations?