Wednesday, December 18, 2013

Odd Logic, 3 of 5; Ultrasums

ULTRASUMS

An ultrasum tells us which infinities are odd and which are even. An ultrasum S sends all infinite sequences (b1, b2, b3,...) of 0's and 1's to {0,1}, subject to these rules:

S(0,0,0,0... )  = 0
S(1,1,1,1, ...) = 1

If b1, b2, ... equal 1 a finite and even number of times, then
S(b1, b2, b3, ...)  =  0
If b1, b2, ... equal 1 a finite and odd number of times, then
S(b1, b2, b3, ...)  =  1

S(a1+b1+c1, a2+b2+c2, a3+b3+c3, ... )
=    S(a1,a2,a3,...)+S(b1,b2,b3,...)+S(c1,c2,c3,...)
S(a1 xor b1, a2 xor b2, a3 xor b3, ... )
=    S(a1,a2,a3,...) xor  S(b1,b2,b3,...)
S(a1 iff b1, a2 iff b2, a3 iff b3, ... )
=    S(a1,a2,a3,...) iff  S(b1,b2,b3,...)
not S( a1, a2, a3, ...)
=   S( not a1, not a2, not a3, ... )
x and S( a1, a2, a3, ...)
=  S( x and a1,  x and a2,  x and a3, ...)
x or S( a1, a2, a3, ...)
=  S( x or a1,  x or a2,  x or a3, ...)

and in general
F(S( a1, a2, a3, ...))  = S( F(a1), F(a2), F(a3), ...)

You can construct an ultrasum via an ultrafilter. An ultrafilter U sends all infinite sequences of 0's and 1's to {0,1}, with these rules:

If b1, b2... equals 1 a finite number of times, then
U(b1, b2, b3,...)  =  0
If b1, b2... equals 0 a finite number of times, then
U(b1, b2, b3,...)  =  1

U(b1 and c1, b2 and c2, b3 and c1, ...)
=   U(b1, b2, b3 , ...) and U(c1, c2, c3 , ...)
U(b1 or c1, b2 or c2, b3 or c1, ...)
=   U(b1, b2, b3 , ...) or U(c1, c2, c3 , ...)
U tells 'large' from 'small' sets, as S tells 'even' from 'odd'.

Then you can define an ultrasum from an ultrafilter by:

S(b1, b2, b3,...)   =  U(b1, b1+b2+b3, b1+b2+b3+b4+b5, ...)

Note the odd sums! So an infinite sum of 0's and 1's is odd if odd partial sums of it are odd a 'large' number of times.

Given an ultrasum, then we can define infinite matrix logic, with dot products of 'and's or 'or's summed over infinite dimensions. You can also define infinite polynomials. Perhaps we could define arbitrary logic functions as such 'ultrapolynomials'.