[FOM] Quine Atom x={x}

[Axiomize@aol.com]

Define the following three functions from the set of functions to the set of 
sets of functions.  That is, each takes in one or two functions and returns a 
set of functions:

s1(x) = { x(x) }
s2(x,y) = { x(x,y) }
s3(x) = { x(x), 0 }

For example, s1 takes in function x, applies x to x, and returns the set 
containing the value of x applied to x.

1. Consider the following specific set:

s1(s1) = { s1(s1) }

Thus s1(s1) is a Quine atom, a set that contains only itself.

2. Consider the following function s4:

s4(x) = s2(s2,x) = { s2(s2,x) }

Thus for each x function s4 returns a set that is a Quine atom.  Is there 
more than one value in the range of s4?

3. Consider the following set:

s3(s3) = { s3(s3), 0 }

Then s3(s3) is a solution to the equation x={x,0}.

Comments?

Charlie Volkstorf
Cambridge, MA