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Questions tagged [axioms]

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

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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 Choice ...
86
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10answers
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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?
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7answers
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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 "...
77
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7answers
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Where are the axioms?

It is said that our current basis for mathematics are the ZFC-axioms. Question: Where are these axioms in our mathematics? When do we use them? I have now studied math for a year, and have yet to ...
76
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9answers
13k 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 ...
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9answers
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Has there ever been an application of dividing by $0$?

Regarding the expression $a/0$, according to Wikipedia: In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by $0$, gives $a$ (assuming $a\not= 0$), and ...
42
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11answers
7k views

What is exactly the difference between a definition and an axiom?

I am wondering what the difference between a definition and an axiom. Isn't an axiom something what we define to be true? For example, one of the axioms of Peano Arithmetic states that $\forall n:0\...
41
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2answers
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Can proof by contradiction 'fail'?

I am familiar with the mechanism of proof by contradiction: we want to prove $P$, so we assume $¬P$ and prove that this is false; hence $P$ must be true. I have the following devil's advocate ...
37
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6answers
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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. ...
37
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6answers
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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 ...
35
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8answers
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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 ...
35
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4answers
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Is every axiom in the definition of a vector space necessary?

Definition: A vector space over a field $K$ consists of a set $V$ and two binary operations $+: V \times V \to V$ and $\cdot: K \times V \to V$ satisfying the following axioms: Commutativity ...
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8answers
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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 numbers,...
31
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2answers
2k views

What axioms Gödel is using, if any?

Gödel states and proves his celebrated Incompleteness Theorem (which is a statement about all axiom systems). What is his own axiom system of choice? ZF, ZFC, Peano or what? He surely needs one, doesn'...
31
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4answers
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Axiomatic definition of sin and cos?

I look for possiblity to define sin/cos through algebraic relations without involving power series, integrals, differential equation and geometric intuition. Is it possible to define sin and cos ...
30
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7answers
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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 ...
28
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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). ...
26
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7answers
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Why does one have to check if axioms are true?

In Tao's book Analysis 1, he writes: Thus, from the point of view of logic, we can define equality on a [remark by myself: I think he forgot the word "type of object" here] however we please, so ...
25
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3answers
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Why is “points exist” not an axiom in geometry?

I am not sure why "points exist" is not an axiom in geometry, given that the other axioms are likewise primitive and seemingly as obvious.
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6answers
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Is consistency an axiom of mathematics?

I watched the numberphile video on Gödel's Incompleteness Theorem today, and I was wondering about something. It seems the key to accepting the truth of Gödel's Theorem is to demand that mathematics ...
23
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5answers
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Why does induction have to be an axiom?

I noticed that there is an axiom that says that if $S(n)\implies S(n+1)$, and $S(1)$ is true, then $\forall n \in \Bbb N, S(n).$ My question is why is this an axiom? why can't we derive this from the ...
23
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2answers
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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: $...
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2answers
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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 ...
21
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4answers
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Meaning of the word “axiom”

One usually describes an axiom to be a proposition regarded as self-evidently true without proof. Thus, axioms are propositions we assume to be true and we use them in an axiomatic theory as premises ...
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3answers
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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 ...
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2answers
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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 ...
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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, ...
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5answers
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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 ...
19
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3answers
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What are disasters with Axiom of Determinacy?

It is well-known that Axiom of Choice has several consequences which might be viewed as counter-intuitive or undesirable. For example, existence of non-measurable sets or Banach-Tarski Paradox. H. ...
18
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11answers
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What is an axiom in layman's terms? [closed]

I tried to explain to my kids what is an axiom in very nontechnical terminology (also known as layman's term) but I could not find anything that they could relate to easily. Do anyone have a good ...
18
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8answers
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Zero vector of a vector space

I know that every vector space needs to contain a zero vector. But all the vector spaces I've seen have the zero vector actually being zero (e.g. $\mathbf{0}=\langle0,0,\ldots,0\rangle$). Can't the "...
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6answers
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Do we have to prove how parentheses work in the Peano axioms?

One thing that has bothered me so far while learning about the Peano axioms is that the use of parentheses just comes out of nowhere and we automatically assume they are true in how they work. For ...
18
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3answers
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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 ...
18
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6answers
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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 ...
18
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2answers
587 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 ...
18
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2answers
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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.
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4answers
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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=\...
17
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2answers
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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, \...
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3answers
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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 ...
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3answers
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Was there ever an axiom rendered a theorem?

In the history of mathematics, are there notable examples of theorems which have been first considered axioms? Alternatively, was there any statement first considered an axiom that later has been ...
16
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3answers
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Why does what I've written fail to define truth?

I stumbled across a set of axioms for first order logic a bit ago. Intrigued, I decided to try to write it all down and organise what I read. After I did that, it seemed to me as though one could ...
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5answers
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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 ...
15
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2answers
363 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 ...
15
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3answers
244 views

Modeling numbers with vectors of vectors?

I stumbled upon the strange representation of integers where $$8=\langle\langle0,\langle0^{\infty}\rangle,0^{\infty}\rangle,0^{\infty}\rangle$$ I'll try explain the representation in a natural way. ...
15
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1answer
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Touch-axiomatization of point-set topology

This MathOverflow post presents an alternative axiomatization of point-set topology that closely aligns with the intuitive notion that a topology on a set tells you which points are "infinitesimally ...
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2answers
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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},\ \sqrt[5]{2+\...
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6answers
2k views

Multiplication by One

Throughout school we are taught that when something is multiplied by 1, it equals itself. But as I am learning about higher level mathematics, I am understanding that not everything is as black and ...
14
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6answers
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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: http://en....
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2answers
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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?
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3answers
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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 ...