Questions tagged [axioms]

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

An axiom is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy.

Axioms define and delimit the realm of analysis. In other words, an axiom is a formal statement that is assumed to be true. Therefore, its truth is taken for granted within the particular domain of analysis, and serves as a starting point for deducing and inferring other (theory and domain dependent) truths. An axiom is defined as a mathematical statement that is accepted as being true without a mathematical proof.

It should be mentioned that in modern times some statements receive a status of axioms, but they are still provable from weaker theories using other statements. One famous example is the axiom of choice, which is provable from ZF set theory if we assume Zorn's lemma. Generally, in modern foundations of mathematics, an axiom is just a statement in the base theory.

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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 is, but as often happens with things like this,…
Tyler
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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?
Gabriel
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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 run into a single one of these ZFC axioms. How can…
Imean
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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 "prove" the axioms? I can say, look, there are $3$…
user4951
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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 interesting is not really discussed there. I think…
Kasper
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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 $\forall n:0\neq S(n)$, or in English, that zero isn't the…
wythagoras
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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 so division by zero is undefined. Is there some…
Lorry Laurence mcLarry
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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. If $x, y \in F$, then $x = y = I$, so $x + y = I +…
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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 allowed to contain the notion of sets? The axioms…
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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 sets of axioms. I have two related questions: 1)…
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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 question, which might seem to be more philosophy than…
MGA
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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 of $+$. Associativity of $+$. Existence of an identity…
David Zhang
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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: $$\{1, 2, 3, 4, 5\}$$ Then one of $[1,2,3,4,5]$ is…
Jiew Meng
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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, real numbers, etc. But without any uniqueness…
ashpool
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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 through some axioms? Like: $$\sin 0 = 0, \cos 0 =…
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