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Gödel's ontological proof is a formal argument for God's existence by the mathematician Kurt Gödel. Can someone please explain what those symbols are, and explain the proof? Thanks.

$ \begin{array}{rl} \text{Ax. 1.} & \left\{P(\varphi) \wedge \Box \; \forall x[\varphi(x) \to \psi(x)]\right\} \to P(\psi) \\ \text{Ax. 2.} & P(\neg \varphi) \leftrightarrow \neg P(\varphi) \\ \text{Th. 1.} & P(\varphi) \to \Diamond \; \exists x[\varphi(x)] \\ \text{Df. 1.} & G(x) \iff \forall \varphi [P(\varphi) \to \varphi(x)] \\ \text{Ax. 3.} & P(G) \\ \text{Th. 2.} & \Diamond \; \exists x \; G(x) \\ \text{Df. 2.} & \varphi \text{ ess } x \iff \varphi(x) \wedge \forall \psi \left\{\psi(x) \to \Box \; \forall y[\varphi(y) \to \psi(y)]\right\} \\ \text{Ax. 4.} & P(\varphi) \to \Box \; P(\varphi) \\ \text{Th. 3.} & G(x) \to G \text{ ess } x \\ \text{Df. 3.} & E(x) \iff \forall \varphi[\varphi \text{ ess } x \to \Box \; \exists y \; \varphi(y)] \\ \text{Ax. 5.} & P(E) \\ \text{Th. 4.} & \Box \; \exists x \; G(x) \end{array} $

Does it prove both existence and uniqueness ?

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It would be useful if you point out where you found this proof and where the image is taken from. –  Asaf Karagila Dec 1 '12 at 11:45
    
I found it here en.wikipedia.org/wiki/G%C3%B6del's_ontological_proof –  Amr Dec 1 '12 at 11:47
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Have you read the Wikipedia article and the related articles? Can you point at a particular point that was not clear to you? –  Asaf Karagila Dec 1 '12 at 11:52
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@Amr: Where it is accompanied by a plain English paraphrase and full explanation. –  Brian M. Scott Dec 1 '12 at 11:53
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hahaha, it's funny, because 0x90 is assembly code for NOOP, and Asaf is trying to tell people to direct comments to the OP, but 0x90 means NOOP. Sorry I had my fun [= –  eazar001 Jan 12 at 5:16

3 Answers 3

up vote 22 down vote accepted

The modal operator $\square$ refers to necessity; its dual, $\lozenge$, refers to possibility. (A sentence is necessarily true iff it isn't possible for it to be false, and vice versa.) $P(\varphi)$ means that $\varphi$ is a positive (in the sense of "good") property; I'll just transcribe it as "$\varphi$ is good". I'll write out the argument colloquially, with the loss of precision that implies. In particular, the words "possible" and "necessary" are vague, and you need to understand modal logic somewhat to follow their precise usage in this argument.

  • Axiom $1$: If $\varphi$ is good, and $\varphi$ forces $\psi$ (that is, it's necessarily true that anything with property $\varphi$ has property $\psi$), then $\psi$ is also good.
  • Axiom $2$: For every property $\varphi$, exactly one of $\varphi$ and $\neg\varphi$ is good. (If $\neg\varphi$ is good, we may as well say that $\varphi$ is bad.)
  • Theorem $1$ (Good Things Happen): If $\varphi$ is good, then it's possible that something exists with property $\varphi$.

Proof of Theorem $1$: Suppose $\varphi$ were good, but necessarily nothing had property $\varphi$. Then property $\varphi$ would, vacuously, force every other property; in particular $\varphi$ would force $\neg\varphi$. By Axiom $1$, this would mean that $\neg\varphi$ was also good; but this would then contradict Axiom $2$.

  • Definition $1$: We call a thing godlike when it has every good property.
  • Axiom $3$: Being godlike is good.
  • Theorem $2$ (No Atheism): It's possible that something godlike exists.

Proof of Theorem $2$: This follows directly from Theorem $1$ applied to Axiom $3$.

  • Definition $2$: We call property $\varphi$ the essence of a thing $x$ when (1) $x$ has property $\varphi$, and (2) property $\varphi$ forces every property of $x$.
  • Axiom $4$: If $\varphi$ is good, then $\varphi$ is necessarily good.
  • Theorem $3$ (God Has No Hair): If a thing is godlike, then being godlike is its essence.

Proof of Theorem $3$: First note that if $x$ is godlike, it has all good properties (by definition) and no bad properties (by Axiom $2$). So any property that a godlike thing has is good, and is therefore necessarily good (by Axiom $4$), and is therefore necessarily possessed by anything godlike.

  • Definition $3$: We call a thing indispensable when something with its essence (if it has an essence) must exist.
  • Axiom $5$: Being indispensable is good.
  • Theorem $4$ (Yes, Virginia): Something godlike necessarily exists.

Proof of Theorem $4$: If something is godlike, it has every good property by definition. In particular, it's indispensable, since that's a good property (by Axiom $5$); so by definition something with its essence, which is just "being godlike" (by Theorem $3$), must exist. In other words, if something godlike exists, then it's necessary for something godlike to exist. But by Theorem $2$, it's possible that something godlike exists; so it's possible that it's necessary for something godlike to exist; and so it is, in fact, necessary for something godlike to exist. QED.

Convinced?

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I am convinced, though it proves the existence of a godlike entity, it doesn't prove the uniqueness, right? –  0x90 Oct 26 '13 at 15:53
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Well, if $x$ is a godlike thing, then "is the same as $x$" is a property that $x$ has. So it must be forced by $x$'s essence, which is the property of being godlike. So the property "is godlike" forces the property "is the same as $x$". We conclude that there can't be any godlike thing distinct from $x$, i.e., uniqueness also holds. –  mjqxxxx Oct 26 '13 at 22:55
    
Would you mind to add it to your answer as well ? –  0x90 Oct 26 '13 at 22:57
    
Is "good" assumed to exist? Or is there proof of the existence of "good" within the argument? –  Aakil Fernandes Nov 14 '13 at 15:18
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How are the axioms motivated? None of them really persuades me. –  Hagen von Eitzen Aug 20 at 17:15

I think dismissing an argument that stands foursquare in a 900 year tradition of logical discourse as a 'strange apparent aberration' is a little questionable. Anyway, it's worth noting that the argument's overly strong axioms and definitions lead to modal collapse. Modifications have been suggested that appear to solve this issue. For example see http://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0CC0QFjAB&url=http%3A%2F%2Fappearedtoblogly.files.wordpress.com%2F2011%2F05%2Fanderson-anthony-c-22some-emendations-of-gc3b6dels-ontological-proof22.pdf&ei=cNP0U-TnKafH7AbkwoEQ&usg=AFQjCNGBuNm20ZkEB12IkNgklPHZABw58A&sig2=MJyHYb7IbeDV9QmOLb8U0A&bvm=bv.73231344,d.ZGU.

Like any proof, Gödel's Ontological Proof depends on acceptance of the axioms, and I would suggest the only argument that can be made for them is one of 'reasonableness'. Recently, work has been published looking at the application of automated theorem provers to the ontological proof. Notwithstanding the modal collapse issue of Gödel's original, it seems to have stood up quite well as an exercise in higher order modal logic. See http://page.mi.fu-berlin.de/cbenzmueller/papers/C40.pdf

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The box is a modal operator (it is necessarily true that ...), with the diamond its dual (it is possibly true that ...). '$P(\varphi)$' holds when the property expressed by $\varphi$ is 'positive' (maybe better, is a perfection). Other novelties are defined. $G(x)$ says $x$ has all perfections (so is God). $\varphi$ ess $x$ says the property $\varphi$ is the essence of $x$. $E$ is the property of necessary existence (existence in virtue of your essence). [Don't blame me, I'm just reporting ....]

You'll find out just a little more about Gödel's very strange apparent aberration here: http://plato.stanford.edu/entries/ontological-arguments/#GodOntArg

Robert Adams introduction to Gödel's original note in Kurt Gödel: Collected Works. Vol III, Unpublished Essays and Lectures is well worth reading.

Petr Hajek has an amazingly patient exploration of the current state of play in investigations of Gödel's Ontological Proof and its variants in Matthias Baaz et al. (eds) Kurt Gödel and the Foundations of Mathematics (CUP 2011). But I can't say that this changed my impression that this is little more than a curious side note in logic.

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