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Topic: layer logic: a new dimension to logic?
Replies: 12   Last Post: May 26, 2013 10:56 AM

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Oskar Trestone

Posts: 22
From: Germany
Registered: 9/17/11
Re: layer logic: a new dimension to logic?
Posted: Sep 19, 2011 5:28 PM
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The definition of natural numbers is done by the successor function M+:

To every set M we define a successor M+:

W(x e M+, t+1) := W(x e M, t) v W(x=M,1) ( for t=0 without W(x e M, t) )

The (adjusted) Peano axioms hold for M+.
We can define 0, 0+, 0++ etc. this way.

1=0+ : W(x e 0+, t+1) = W(x e 0, t) v W(x=0,1) = W(x=0,1)
For t>0 there is just one element in ´1´ and this is ´0´.

2=1+ : W(x e 1+, t+1) = W(x e 1, t) v W(x=1,1)
For t=1 there is just one element in ´2´ and this is ´1´.
For t>1 there are just two elements in ´2´ and these are ´0´ and ´1´.

For t=1 there is just one element in ´n+´ and this is ´n´.
For t>n there are n+1 elements in n+ and these are n, n-1, ?,1,0.

This numbers are very similar to natural numbers,
but only for large layers t.

It is possible to define the set of all natural numbers
and the laws of addition and multiplication.

But some usual characteristics of natural numbers we do not get:
The prime factorisation is not necessarily unique
and the square root of 2 has not to be irrational.

But more than in mathematical methods an playings
I am interested in interpretations of the main idea, the logical layer.


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