Tuesday, 27 August 2013

Mathematical curiosities: Gödel's ontological proof of the existence of God

As a curiosity, i had to share this !!

Famous logician, mathematician, and philosopher Kurt F. Gödel was considered with Aristotle and Frege to be one of the most significant logicians in human history, Gödel made an immense impact upon scientific and philosophical thinking in the 20 th century, a time when others were pioneering the use of logic and set theory to understand the foundations of mathematics.
Calling Godel interesting is an understatement. A man who was Einstein's only reason to go to the university, in his later years; just to have the privilege of walking back home with Godel !

He used Leibniz "positive and negative properties" concept to formulate an "ontological proof" for the existence of God

Definition 1: x is God-like if and only if x has as essential properties those and only those properties which are positive

Definition 2: A is an essence of x if and only if for every property B, x has B necessarily if and only if A entails B

Definition 3: x necessarily exists if and only if every essence of x is necessarily exemplified

Axiom 1: Any property entailed by—i.e., strictly implied by—a positive property is positive
Axiom 2: If a property is positive, then its negation is not positive
Axiom 3: The property of being God-like is positive
Axiom 4: If a property is positive, then it is necessarily positive
Axiom 5: Necessary existence is a positive property
Axiom 6: For any property P, if P is positive, then being necessarily P is positive

Axiom 1 assumes that it is possible to single out positive properties from among all properties. Gödel comments that "Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world)... It may also mean pure attribution as opposed to privation (or containing privation)." (Gödel 1995). Axioms 2, 3 and 4 can be summarized by saying that positive properties form a principal ultrafilter.
From these axioms and definitions and a few other axioms from modal logic, the following theorems can be proved:

Theorem 1: If a property is positive, then it is consistent, i.e., possibly exemplified.
Corollary 1: The property of being God-like is consistent.
Theorem 2: If something is God-like, then the property of being God-like is an essence of that thing.
Theorem 3: Necessarily, the property of being God-like is exemplified.

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