What is the difference between an axiom and a property?

May 1, 2017

See explanation...

Explanation:

I could write for hours on this, but let me try to keep it short yet helpful.

An axiom is a statement that is assumed to be true in order to help provide a foundation from which other statements can be proved.

A property is a statement that is true in some particular context.

In your example, various "laws" of inequalities may be given, including the "addition axiom of inequality", e.g.:

$a < b \text{ " <=> " } a + c < b + c$

Then since subtraction is basically shorthand for adding an additive inverse, it follows that for any $a , b , c$:

$a < b \text{ " <=> " } a - c < b - c$

So if we assume as an axiom that the "addition axiom of inequality" holds, then it follows that the "subtraction property of inequality" holds.

We could have instead had an axiom called "the subtraction axiom of inequality" and deduced a property called "the addition property of inequality".

Generally speaking we try to find sets of axioms that are most natural, simplest and independent (i.e. no axiom can be proved from a subset of the others).