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Topic: Two Finite Arithmetics
Replies: 19   Last Post: Apr 9, 2014 9:27 PM

 Messages: [ Previous | Next ]
 William Elliot Posts: 2,637 Registered: 1/8/12
Re: Two Finite Arithmetics
Posted: Apr 8, 2014 11:03 PM

On Tue, 8 Apr 2014, Dan Christensen wrote:

> Here, I have translated your axioms into DC Proof format. Download my software, enter these axioms (the "@" is mapped to epsilon) and see what you can do with them. You have to construct your own proofs one line at a time, but the software will not let you make mistakes. Instant feedback is provided with each line you enter it. You might start by proving the associativity and commutativity of + and * as you have defined them. You can get started using with the tutorial or user manual.

I will not.

> 1 Set(n)
> Axiom
>
> 2 0 @ n
> Axiom
>
> 3 m @ n
> Axiom
>
> 4 ALL(a):[a @ n => s(a) @ n]
> Axiom
>
> 5 s(m)=m
> Axiom
>
> 6 ALL(a):~s(a)=0
> Axiom
>
> 7 ALL(a):ALL(b):[a @ n & b @ n & ~a=m & ~b=m => [s(x)=s(y) => x=y]]
> Axiom
>
> 8 ALL(a):[Set(a) & ALL(b):[b @ a => b @ n]
> => [0 @ a & ALL(b):[b @ a => s(b) @ a]
> => ALL(b):[b @ n => b @ a]]]
> Axiom
>
> 9 ALL(a):ALL(b):[a @ n & b @ n => a+b @ n]
> Axiom
>
> 10 ALL(a):[a @ n => 0+a=a]
> Axiom
>
> 11 ALL(a):ALL(b):[a @ n & b @ n => s(a)+b=s(a+b)]
> Axiom
>
> 12 ALL(a):ALL(b):[a @ n & b @ n => a*b @ n]
> Axiom
>
> 13 ALL(a):[a @ n => 0*a=0]
> Axiom
>
> 14 ALL(a):[a @ n & b @ n => s(a)*b=a*b+b]
> Axiom
>
>