I have read that an axiom is defined as "an obvious truth." I have also heard that an axiom is a truth so obvious that no proof could make it more clear. My question is: why is one thing considered an axiom and not requiring of a proof, but another thing might be called a Proposition and thus requiring a proof? Consider this the following example as an illustration of my point:

Axiom for the Order of Integers
If m and n are positive integers, then m+n and mn are positive integers.

Proposition If m is an integer, then:
a. m ∈ Z+ iif m>0
b. m ∈ Z+ iif -m<0

What I don't get is why the proposition requires a proof but the axiom does not? Or another way, why isn't the proposition also an axiom? It seems just as obvious as the axiom and in fact the proof does not seem to make it more clear. If anything the opposite is true.

FYI: I am doing a first course in Discrete Math and this question is not for assignment or test purposes, just curiosity coming from having seen these definitions formally for the first time.

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    $\begingroup$ Once you choose your axioms, anything that can be proven from those axioms is a proposition. For example, in Peano Arithmetic (a set of axioms), the principle of induction is taken as an axiom; in ZFC (another set of axioms) it must be proven. $\endgroup$ – Akiva Weinberger Apr 12 '15 at 15:07
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    $\begingroup$ Another example, this time from geometry: Playfair's Axiom states that given a line $l$ and a point $P$ not on the line there exists a unique line through $P$ parallel to $l$. If you take Euclid's set of axioms, this is a proposition. However, if you take Euclid's axioms 1 through 4, plus Playfair's Axiom, as axioms, then it's an axiom. (It's called "Playfair's Axiom" because it's intended as a replacement for Euclid's complicated fifth axiom.) $\endgroup$ – Akiva Weinberger Apr 12 '15 at 15:11
  • $\begingroup$ Oh, so what is an axiom is just relative to the situation? What axioms would be axioms in the truest sense then? By that I mean the most fundamental axioms of mathematics. I ask that because my text book describes that as the Axioms for Addition and Multiplication. Is there another area where these would just be propositions following from other axioms? $\endgroup$ – 1west Apr 12 '15 at 15:12
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    $\begingroup$ People usually take Peano Arithmetic as a good set of axioms for math. (ZFC is more complicated.) $\endgroup$ – Akiva Weinberger Apr 12 '15 at 15:13
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    $\begingroup$ In somewhat basic language, you can think of the following: Axioms are the basic facts of the system under study. You cannot prove the axioms, they are just the assumed truths. Propositions are the facts and theorems that are the results and consequences of the axioms. You can prove the propositions from the axioms, but you cannot prove axioms. $\endgroup$ – Michael Burr Apr 12 '15 at 15:17

Basically, a set of axioms defines a mathematical structure. For example, the Peano axioms define what a natural number is, and the group axioms define what a group is. Propositions are true statements about the mathematical structure that can be derived from the axioms.

Now it may happen that different sets of axioms define the same mathematical structure, that is they are equivalent. Then an axiom of one axiom set may well be a proposition in the other.


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