神の存在証明

 神の存在は信仰をもつ人には疑い得ないことで、無神論者にこそ必要なことと考えられがちである。そのためか、無神論者が多い日本人には割と関心の高いのが「神の存在証明」。むろん、神は信じる対象であって証明する対象ではないというのが通り相場の意見。
 一方、神学者や哲学者は神学や形而上学の研究の中で「神の存在証明」を長い間試みてきた。中でも有名なのがトマス・アクィナスによる「五つの道」で5通りの神の存在証明がまとめられている。また、かのデカルトも証明を試みている。

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(クルト・ゲーデル

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フォン・ノイマン

 クルト・ゲーデルと言えば、20世紀最大の数学基礎論の研究者。この胃弱の天才の成果は、第1階述語論理の完全性定理と不完全性定理(列挙すると矛盾するような名前の定理)、連続体仮説集合論の公理系の無矛盾性等々、実に偉大である。20世紀のモンスターであるフォン・ノイマンが基礎論研究を始めた頃にゲーデルを知り、自ら基礎論の分野を離れ、量子力学やコンピューターの研究へと向かったと言われている。
 さて、そのゲーデルプリンストン宇宙論、時間の哲学(J. Bellのエッセイを参照)に関心を寄せながら、「神の存在証明」をかつて研究した様相論理(modal logic)を使って証明している。その要旨を以下に述べてみよう(併せて数学の証明に使われる英語が中学生程度のものであることも実感してほしい)。

Kurt Gödel’s proof of the existence of God.
Gödel's argument is as follows:
1. First he tries to show that in some possible world there exists a being with the property Godlikeness. The property Godlikeness itself is positive as it is the conjunction of infinite positive properties.
2. Secondly he tries to define essence. If x is an object in some world, then the property P is said to be an essence of x if P(x) is true in that world and also that P entails all other properties that x has in that world. He also says that x necessarily exists, if for every essence P the following is true: in every possible world, there is an element x with P(x).
3. Now necessary existence is positive. Since Godlikeness is an essence of God, it entails all positive properties. Furthermore, God cannot have any non-positive property as it is the negation of some positive property. Since any Godlike object is necessarily existent, it follows that any Godlike object in one world is a Godlike object in all worlds, by the definition of necessary existence.
Gödel’s Formulation
Gödel formulated the argument with the following definitions and axioms.
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.

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.
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.