# 哥德爾本體論證明

## 證明

### 聖安瑟倫的論點

11世紀的意大利僧侶聖安瑟倫，其論點用最簡潔的表達如下：「God, by definition, is that than which a greater cannot be thought（i.e.${\displaystyle G(x)\iff \forall \varphi [P(\varphi )\rightarrow \varphi (x)]}$）. God exists in the understanding（i.e.${\displaystyle \Diamond \;\exists x\;G(x)}$）. If God exists in the understanding, we could imagine Him to be greater by existing in reality. Therefore, God must exist.${\displaystyle \Box \;\exists x\;G(x)}$」。也就是：

：神是我們所能想象得到最偉大的存有。

：實際存在的物體比想像中的物體更偉大。

### 哥德爾的證明

${\displaystyle G}$ 解「像神特性」，${\displaystyle \varphi }$及${\displaystyle \psi }$為任一特性，${\displaystyle P(\varphi )}$ 解 「${\displaystyle \varphi }$ 為正（也可作「善」或「偉大」）特性」， ${\displaystyle \varphi (x)}$ 解「 x 擁有 ${\displaystyle \varphi }$ 特性」，${\displaystyle E}$ 解「必需存在」， ${\displaystyle \varphi \;\operatorname {ess} \;x}$ 解 「${\displaystyle \varphi }$ 是 x 的本質（essence）」，${\displaystyle \Box }$ 表示「必然性」，而 ${\displaystyle \Diamond }$ 表示「可能性」:


${\displaystyle {\begin{array}{rl}{\mbox{Ax. 0.}}&\Box \;\exists \varphi \;P(\varphi )\\{\mbox{Ax. 1.}}&\Box [(\;\forall x\lbrace [\varphi (x)\rightarrow \psi (x)]\land P(\varphi )\rbrace )\rightarrow P(\psi )]\\{\mbox{Ax. 2.}}&P(\neg \varphi )\leftrightarrow \neg P(\varphi )\\{\mbox{Th. 1.}}&P(\varphi )\rightarrow \Diamond \;\exists x\;[\varphi (x)]\\{\mbox{Df. 1.}}&G(x)\iff \forall \varphi [P(\varphi )\rightarrow \varphi (x)]\\{\mbox{Ax. 3.}}&P(G)\\{\mbox{Th. 2.}}&\Diamond \;\exists x\;G(x)\\{\mbox{Df. 2.}}&\varphi \;\operatorname {ess} \;x\iff \varphi (x)\land \forall \psi \lbrace \psi (x)\rightarrow \Box \;\forall y[\varphi (y)\rightarrow \psi (y)]\rbrace \\{\mbox{Ax. 4.}}&P(\varphi )\rightarrow \Box \;P(\varphi )\\{\mbox{Th. 3.}}&G(x)\rightarrow G\;\operatorname {ess} \;x\\{\mbox{Df. 3.}}&E(x)\iff \forall \varphi [\varphi \;\operatorname {ess} \;x\rightarrow \Box \;\exists y\;\varphi (y)]\\{\mbox{Ax. 5.}}&P(E)\\{\mbox{Th. 4.}}&\Box \;\exists x\;G(x)\end{array}}}$

## 批評

There are suggested amendments to the proof, presented by C. A. Anderson,[2] but argued to be refutable by C. A. Anderson and Michael Gettings.[3] Sobel's proof of modal collapse has been questioned by Koons,[4] but a counter-defence by Sobel has been given.[來源請求]

The proof has also been questioned by Oppy,[5] asking whether lots of other almost-gods would also be "proven" by Gödel's axioms. This counter-argument has been questioned by Gettings,[6] who agrees that the axioms might be questioned, but disagrees that Oppy's particular counter-example can be shown from Gödel's axioms.

There are many more criticisms, most focusing on the philosophically interesting question of whether these axioms must be rejected to avoid odd conclusions. The broader criticism is that even if the axioms cannot be shown to be false, that does not mean that they are true. Hilbert's famous remark about interchangeability of the primitives' names applies to those in Gödel's ontological axioms ("positive", "god-like", "essence") as well as to those in Hilbert's geometry axioms ("point", "line", "plane"). According to Fuhrmann (2005) it remains to show that the dazzling notion prescribed by traditions and often believed to be essentially mysterious satisfies Gödel's axioms. This is not a mathematical, but merely a theological task.[7]:364–366 It is this task which decides which religion's god has been proven to exist.

## 參考文獻

• C. Anthony Anderson, "Some Emendations of Gödel's Ontological Proof", Faith and Philosophy, Vol. 7, No 3, pp. 291–303, July 1990
• Kurt Gödel (1995). "Ontological Proof". Collected Works: Unpublished Essays & Lectures, Volume III. pp. 403–404. Oxford University Press. ISBN 0195147227
• A. P. Hazen, "On Gödel's Ontological Proof", Australasian Journal of Philosophy, Vol. 76, No 3, pp. 361–377, September 1998
• Jordan Howard Sobel, "Gödel's Ontological Proof" in On Being and Saying. Essays for Richard Cartwright, ed. Judith Jarvis Thomson (MIT press, 1987)
• Melvin Fitting, "Types, Tableaus, and Godel's God" Publisher: Dordrecht Kluwer Academic ©2002, ISBN 9781402006043

## 外部連結

• ^ Jordan Howard Sobel. Gödel's ontological proof. (编) Judith Jarvis Thomson. On Being and Saying: Essays for Richard Cartwright. Cambridge/MA & London, England: MIT Press. Nov 1987: 241–261. ISBN 978-0262200639.
• ^ C. A. Anderson. Some emendations of Gödel's ontological argument. Faith and Philosophy, 7:291–303, 1990.
• ^ Gόdel's Ontological Proof Revisited | C. Anthony Anderson and Michael Gettings, Gödel's Ontological Proof Revisited
• ^ Koons, Robert C. "Sobel on Gödel’s ontological proof." (2005)
• ^ Oppy, Graham. "Gödelian ontological arguments." Analysis 56.4 (1996): 226–230.
• ^ Gettings, Michael. "Gödel's ontological argument: a reply to Oppy." Analysis 59.264 (1999): 309–313.
• ^ André Fuhrmann. Existenz und Notwendigkeit — Kurt Gödels axiomatische Theologie [Existence and Necessity — Kurt Gödel's Axiomatic Theology] (PDF). (编) W. Spohn. Logik in der Philosophie [Logic in Philosophy]. Heidelberg: Synchron. 2005: 349—374 （德语）.