ATON is a hopefully evolving classification theory. It aspires to unify knowledge around numbers and prefers naive methods. Some of the older posts are wrong but I'll keep them for the sake of continuity.

Saturday, March 25, 2006

Exponential Set

This set should be called Factorial Set of order 2, in line with the updated definition of Mirror Set;
http://a-theory-of-nothingness.blogspot.com/2006/11/mirror-set-rev-1.html

E0=,
E1=EE0=E={,}={}
E2=EE1={,{}}={{},}
E3=EE2={,{},{,{}}}
E4=EE3={,{},{,{}},{,{},{,{}}}}
...
En=EE(n-1)

Where E is a reflection operator. It gathers the past under one roof. Although it has multiple content, it also has an overall duality like HSet. ESet looks like e^z since the self similarity in breadth. HSet is self similar in depth, for example H3={,{,{,{}}}}. Self similarity of exponential function is evident from differentiation, (d/dz)e^z=e^z. Taylor's expansion of e^z is an iterated process, like ESet.

The identity (E0=,) is a bit strange but my hand was forced by 0!=1. But it also makes some sense since the seperator is defined first. (E0=) is even stranger.

The number of states;

SE0=0!=1
SE1=1!0!=1
SE2=2!1!0!=2
SE3=3!2!1!0!=12
SE4=4!3!2!1!0!=288
SE5=5!4!3!2!1!0!=34560
SE6=6!5!4!3!2!1!0!=24883200
...
SEn=n!(n-1)!(n-2)!...2!1!0!

Self similarity also shows up in the state calculation. The factorial is calculated to depth 2.

Note the curious 'nineness' of 4-6;

2+8+8=18=2*9
3+4+5+6+0=18=2*9
2+4+8+8+3+2+0+0=27=3*9

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