Exploring the Relationship Between Axiom Count and Theory Validity

[Fabrizio]

I've been asking myself a few questions lately regarding the nature of a theory. It can be any type of theory. I hope someone can answer these to a degree. The questions are:

1. Can a theory have less than three axioms? Is three the minimum for a theory to make sense?
2. Is the statement "The less axioms, the more abstract the theory, the more facts will fit" true ?
3. How does the number of axioms relate to the number of contradictions that may appear within the theory?

Thanks in advance.

 

[pbuk]

1a. Can a theory have less than three axioms?

Yes, I present "the zero theory":

• Axiom 0: 0 is a natural number.

1b. Is three the minimum for a theory to make sense?

I don't think you could say that the zero theory doesn't make sense, but it is not very interesting. So perhaps you want to ask "is three the minimum for a theory to be interesting?". For that question to have any meaning you would have to define "interesting", and that is a matter of philosophy not mathematics. We don't discuss philosophy in these forums.

2. Is the statement "The less axioms, the more abstract the theory, the more facts will fit" true?

Again that requires a definition of "abstract"... however I think that the opposite of "the more facts will fit" is true - how many facts fit the zero theory?

3. How does the number of axioms relate to the number of contradictions that may appear within the theory?

If you can find any contradictions (i.e. a statement for which both the statement and its negation can be proved true from the axioms) then the axiomatic system is inconsistent and would not be called a theory.

 

[Fabrizio]

LOL, you're a robot. I guess i had to ask the questions on a philosophy forum because I meant a philosophical theory. Besides; The statement that zero is a natural number is a matter of dispute, even in such a "precise" field like mathematics.

 

[epenguin]

If you can find any contradictions (i.e. a statement for which both the statement and its negation can be proved true from the axioms) then the axiomatic system is inconsistent and would not be called a theory.

I think I heard that if you can find one contradiction then you can disprove all the statements you can prove in that theory - is this right?

 

[pbuk]

Besides; The statement that zero is a natural number is a matter of dispute, even in such a "precise" field like mathematics.

An axiom is a statement that is defined to be true, it is not something that can be disputed.

 

[Mark44]

LOL, you're a robot. I guess i had to ask the questions on a philosophy forum because I meant a philosophical theory.

Discussions about philosophy aren't permitted at this forum.

 

[epenguin]

Discussions about philosophy aren't permitted at this forum.

There we have very nice example of reductio ad absurdam.

 

[Ssnow]

I think we need a Logician that is an expert of Theories & Formal Systems to answer this question ... If the question is pertinent in a math forum I don't know ...

 

[pbuk]

I think I heard that if you can find one contradiction then you can disprove all the statements you can prove in that theory - is this right?

Once you have a contradiction, everything falls apart like this:

• Let the contradiction be expressed thus: (A is a theorem of S) (Theorem 1) and ((not A) is a theorem of S) (Theorem 2).
• Now for any well defined statement B, we have by Theorem 1 ((A or B) is a theorem of S) (Theorem 3) and also ((A or (not B) is a theorem of S) (Theorem 4).
  
• By Theorem 2 and Theorem 3, (B is a theorem of S) (Theorem 5).
• By Theorem 2 and Theorem 4 ((not B) is a theorem of S) (Theorem 6).

So if we have a contradiction, any statement and its inverse can both be proved.

 

[Curious3141]

There we have very nice example of reductio ad absurdam.

Have you just proven we don't exi...(poof)