Axioms in math?

Question

Axiom is a assumption accepted as ture without proof.



Why do we accept axioms in math? Shouldn't it be proven first before accepting it.

Answer 1

The problem is, you can't start with nothing. You have to have something to begin with - after all, a formal proof at each step cites an earlier proof or an axiom and a logic rule applied to it. If there are no axioms where does the first proof come from?

One of the challenges of rigorous math is to choose axioms as basic and simple as possible. For instance, in geometry for ages people debated the parallel postulate, that for any line and a point not on that line there is exactly one line through the point that does not intersect the given line. Many people thought this postulate was not as obvious as the other postulates, and thought it could be proven from the others. It was not until 100 years ago that mathematicians understood that this postulate not only is necessary, but distinguishes 3 different types of plane geometry: Euclidean, Lobachevskian, and Riemann geometries, where there are exactly 1 parallel, lots of parallels, and no parallels, respectively.

Selection of axioms is particularly important because of Godel's Incompleteness Theorem, which states that any axiomatic system is either inconsistent or incomplete. While mathematicians don't like to have to accept that their systems are incomplete, it is very important not to choose axioms that are inconsistent, since in this case everything can be proven true.

Good question.

Answer 2

In maths, an Axiom would not genuinely *have* to be 'self-obtrusive'; it only must be something that the mathematicians playing with the ensuing homes are arranged to agree among themselves is something that sounds as though it would produce something exciting. maximum axiomatic structures - working example, the Euclidean axioms or the integer axioms - do sound obtrusive on a typical foundation, yet non-euclidean axioms, as an instance, easily do no longer (parallel traces might meet, working example). think of of arithmetic as being a game: "howdy, only for argument sake, we could agree that that's actual and notice what occurs?"

Answer 3

Axioms: an established rule of mathematics principals with a step by step explanation of geometric proofs.

They have already been proven and it is not necessary to prove them on a daily basis.

Click on the URL below for additional information concerning Axioms

www.sparknotes.com/math/geometry3/axiomsandpostulates/section1.html

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Answer 4

what is accepted as an axiom in class 6 is proved in class 7 and so on.so nothing is accepted without proof.