leichter (05/02/83)
[I couldn't resist this, first because it's so amusing a note, second because
of the unique pairing of newsgroups it goes to.]
In the following; "+" denotes set union; < denotes set membership (epsilon);
and [ and ] are "subset" and "superset".
Theorem (due to Anslem, Aquinas, and others.)
The Axiom of Choice is equivalent to the existence of a unique God.
Proof:
==>: (Assuming the equivalence of the Axiom of Choice and Zorn's Lemma)
Partially order the set of subsets of the set of all properties of objects by
inclusion. This set has maximal elements. God is by definition (Anslem) one
of these maximal elements. Now
God [ God + {existence}
so
God = God + {existence}
Therefore God exists.
To prove uniqueness, let God and God' be two gods, then God + God' [ God
(Aquinas), therefore God ] God'; similarly, God' ] God; hence God = God'.
<==: Given a set {Ai | i < I} of sets, let the unique God pick xi < Ai
for each i < I. (He can do so by omnipotence, proved as for existence above.)
The (xi) < Product(Ai), i < I, as required.
QED
-- Jerry
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