Proving Axioms: An Explanation

[josdavi]

I know that a theorem can be deduced from the AXIOMS of a formal system,
but I do not know how to prove an axiom.
Would you please teach me ?

How to prove an axiom ?
How did the axiomatic rules become axiom ?

I think some statements or formulas are axiomatic, although they seem unbelievable, for example,

https://www.physicsforums.com/showthread.php?s=&threadid=3876
(Volumes of Regular Icosahedron and Regular Tetrahedron ¡V
in the Mathematics forum of Physics Forums.)

 

[chroot]

Axioms cannot be proved, by definition. They are "self-evident truths," the starting points of a model.

- Warren

 

[HallsofIvy]

As chroot says, you can't prove an axiom, however, I have a distaste for the phrase "self evident truths"- I don't believe there are such things. "Axioms" are accepted as part of the definition of the particular system we are working in.

In Euclidean geometry, "If a two lines are parallel, then a transversal cutting both has interior angles adding to one straight angle" is an axiom.

In hyperbolic geometry, "If two lines are paralle, then a transversal cutting both has interior angles adding to less than one straight angle" is an axiom.

Which of those is a "self evident truth"?

 

[mathman]

pedantry

From what I remember of high school geometry, there was a distinction made between axioms and postulates. Axioms are "self-evident truths". Postulates are assumptions made for the model. The "parallel" statements are considered postulates for Euclidean or non-Euclidean geometry, as the case may be.

My high school days were a long time ago. I don't know how things are done these days.

 

[selfAdjoint]

Nowadays I don't think mathematicians make that distinction. Mostly they too, are uncomfortable with the idea of "self evident truths" and don't think that describes what they define as sets of axioms.

Most axiom systems now are sets of minimal spanning properties that define whatever dingus you're thinking of. Like the axioms for a metric or a topology. The point is that every true dingus will have those properties, and anything that has those properties will be a dingus.

 

[chroot]

Okay, okay, I'm sorry for using the phrase "self-evident truth." I just wasn't sure how best to describe 'axiom,' and so consulted Webster, who felt that phrase appropriate. You're all correct though -- there is no such thing as objective truth. You can assemble any axioms you'd like into a system, and carry that system through to its logical conclusions, yet there are any infinite number of different systems with different conclusions. None of them can be said to be "right."

The example of Euclidean and non-Euclidean geometry is precient.

- Warren

 

[jeff]

Originally posted by chroot
Axioms cannot be proved, by definition.

Axioms may be decidable as propositions of some larger enveloping system.

 

[STAii]

There is something annoying about some axioms.
For example, take the following two statements (both in Euclidean geometry):
1-"Adding up the angles of a triangle makes 180 degrees"
2-"If a two lines are parallel, then a transversal cutting both has interior angles adding to one straight angle"

Now, you can proove each of these two using the other, but, which one do we consider an axiom, and why it (and not the other one) ?

 

[HallsofIvy]

There are many examples of "equivalent" axioms (one geometry text I recently read listed 12 different propositions that had been used, by different authors, in place of the parallel postulate). You can use anyone of equivalent statements as an axiom and prove the rest- the choice may be whichever you think is simpler or just more convenient.

Here are 7 equivalent "axioms" that distinguish the real numbers from the rational numbers:

Every Cauchy sequence converges.

If a set of real numbers has an upper bound then it has a least upper bound.

If a set of real numbers has a lower bound then it has a greatest lower bound.

If an increasing sequence of real numbers has an upper bound then it converges

If a decreasing sequence of real numbers has a lower bound then it converges.

The set of all real numbers, with the usual topology, is connected.

Any subset of the real numbers that is both closed and bounded (in the usual topology) is compact.

Given anyone of these, one can prove the others.

 

[HallsofIvy]

I might also point out that while anyone of these can be taken as an axiom for the real numbers, one can "go back" a level and define the real numbers in terms of rational numbers so that these becomes theorems.

For example, if you define the real numbers in terms of Dedekind Cuts, then the "least upper bound" property becomes easy to prove. If you define real numbers as "equivalence classes of Cauchy sequences" then the Cauchy Criterion is easy to prove.

What is an axiom and what is a theorem depends upon what "level" you want to work at and sometimes an arbitrary choice among equivalents.