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Gödel’s ontological proof


Gödel's ontological proofFrom these hypotheses, it is also possible to prove that there is only one God in each world by Leibniz’s law, the identity of indiscernibles: two or more objects are identical (are one and the same) if they have all their properties in common, and so, there would only be one object in each world that possesses property G. Gödel did not attempt to do so however, as he purposely limited his proof to the issue of existence, rather than uniqueness. This was more to preserve the logical precision of the argument than due to a penchant for polytheism. This uniqueness proof will only work if one supposes that the positiveness of a property is independent of the object to which it is applied, a claim which some have considered to be suspect.

 

  • 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: A property is positive if and only if 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
  • Theorem 1: If a property is positive, then it is consistent, i.e., possibly exemplified.
  • Theorem 2: The property of being God-like is consistent.
  • Theorem 3: If something is God-like, then the property of being God-like is an essence of that thing.
  • Theorem 4: Necessarily, the property of being God-like is exemplified.

 

http://en.wikipedia.org/wiki/G%C3%B6del%27s_ontological_proof

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