Re: Heap-Set Theory H-S

On Tue, 29 Jan 2008 23:45:42 +0100, G. Frege <nomail@invalid> wrote:


x = [x],


Here some "thoughts" I once formulated in connection with this property
of heaps:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


Then we would have for any /a/:

a = [a].

Suggestion: Let's call those objects /heaps/.

[...]

It seems that for heaps there's no difference between "e" and "c".

That's indeed the case, as Akhiro Kanamori proves in his paper "The
empty set, the singleton and the ordered pair." (Though Kanamori's
proof refers to /classes/.)

I'll just quote the proof:

"THEOREM. Aa(a = {a}) is equivalent to:

(*) AaAb(a c b <-> a e b).

PROOF. Suppose first that a = {a}. Then for any b, a c b if and only
if {a} c b. But {a} c b if and only if a e b, and hence we can
conclude that a c b if and only if a e b.
For the converse, first note that since a e {a}, (*) implies that a
c {a}. But also, since a c a, (*) implies that a e a, so that {a} c a.
Hence we can conclude that a = {a}. []

In other words, the identification of classes with their unit class
is equivalent to the very dissolution of the inclusion vs. membership
distinction [...]."


So let's stick with "c" (and forget about "e") for heaps:

a c A

"a is a constituent of the heap A"


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F.

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