MHB Proving 0<1 with Axioms: A+B=B+A, A.B=B.A, and More!

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The discussion revolves around proving the inequality 0<1 using a set of axioms related to addition and multiplication. Participants question the clarity and implications of certain axioms, particularly regarding the properties of zero and one. There is a debate on the relevance of existing proofs and whether they rely on additional lemmas, suggesting that the axiomatization of integers may need refinement. The conversation highlights the complexity of proving such inequalities strictly within the given axiomatic framework. Ultimately, the challenge remains to establish 0<1 while adhering to the specified axioms.
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Given the following axioms:

For all A,B,C:

1) A+B=B+A

2) A+(B+C) =(A+B)=C

3) A.B=B.A

4) A.(B.C) = (A.B).C

5) A.(B+C)= A.B+A.C

6) A+0=A

7) A.1=A

8) A+(-A)=1

9) A.(-A)=0

10) Exactly one of the following:
A<B or B<A or A=B

11) A<B => A.C<B.C

12 [math] 1\neq 0 [/math]

Then prove using only the above axioms: 0<1
 
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solakis said:
9) A.(-A)=0
Do you mean $A\cdot0=0$? Also, don't you have an axiom that addition respects the order?

solakis said:
Then prove using only the above axioms: 0<1
This page has some proof. Also, several proof assistants have this theorem in their libraries, but they may use a number of lemmas, i.e., their proofs may not be the shortest.

Why are such problems interesting to you? After looking at several examples, they seem routine. It may be interesting to develop a new, somehow better axiomatization of integers, for example, but axiomatization of rings and fields seems good enough.
 
Evgeny.Makarov said:
Do you mean $A\cdot0=0$? Also, don't you have an axiom that addition respects the order?

This page has some proof. Also, several proof assistants have this theorem in their libraries, but they may use a number of lemmas, i.e., their proofs may not be the shortest.

Why are such problems interesting to you? After looking at several examples, they seem routine. It may be interesting to develop a new, somehow better axiomatization of integers, for example, but axiomatization of rings and fields seems good enough.

No.

A.(-A) =0 ,but you can prove : A.0 =0

The above is a mix of axiomatics.

And the question is :

Can we prove : 0<1 w.r.t the above axiomatics,since w.r.t the axiom 10 we can have i<0 ,o<1,1=0??
 
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