Talk:Gödel's ontological proof

Criticism Section

The criticism section for this article aren't really appropriate because they are for the ontological proof in general and not specific to Gödel's proof. Are there any published criticisms of Gödel's proof? Tweisbach (talk) 11:02, 6 January 2011 (UTC)

I agree that, as it stands, the criticism section does not belong here, because it applies to the ontological argument in general rather than Gödel's more sophisticated formalization. In particular, the C. Anthony Anderson quote does not refer to Gödel's proof, but rather to Anderson's own restatement of it, making its relevance here questionable. In the paper from which the Anderson quote is extracted, Christopher Small says:

"I should remind the reader again that the version of Gödel's argument that I have presented is not Gödel's original argument. When I first read Gödel's notes on his ontological argument, I, like many others, found the axioms confusing and difficult to accept. So I turned to the emended version of the argument due to Anderson. This version has now been widely discussed. But the more I fuss around with this version the more uncomfortable I become with it. I went back to Gödel's original version and found that it was far better than I had originally understood. If you are interested in studying this argument, and the reasons why I prefer it to Anderson's emendation, you can click to obtain a PDF file entitled Reflections on Gödel's Ontological Argument."4granite (talk) 23:19, 8 April 2011 (UTC)

THE DAMN PROOF

What's the damn proof in plain language? Sn't Anselm's ontological argument was stupid but Gödel is an intelligent man, here the article is abotu his proof but we are nto given his proof only a logical representation of it, he has to have made it in words, where is it?Undead Herle King (talk) 22:22, 7 January 2011 (UTC)

Why would he have made it in words? He was a logician. --Gwern (contribs) 18:04 17 January 2011 (GMT)

As an example of Gödel's logic, he starved himself to death because he thought that someone was trying to poison him.Lestrade (talk) 16:22, 19 January 2011 (UTC)Lestrade

Can someone *translate* the proof into plain language, then? Or is this against some rule of Wikipedia?72.213.177.26 (talk) 04:21, 1 May 2011 (UTC)

I'll look at doing this at some point. I don't know when I'll get round to it; if someone else wants to do it, I recommend using this source, which outlines the argument in plain English. ItsZippy ^{(talk • contributions)} 20:18, 9 December 2011 (UTC)

It just prove that very smart people can be also very dumb when it comes to religion. It's basically the same dumb Anselm argument disguised with math — Preceding unsigned comment added by 201.150.158.34 (talk) 19:56, 27 September 2020 (UTC)

Seems to me the Wiki Wangs are belling the cat

I have a real problem with nonsense such as this discussion page, and with reference [sic] works where no authorship is attributed. How do I know anyone posting here knows their own ass from a hole in the ground? Certainly, based on the stunningly inept commentary here, I can only conclude that this encyclopedia [sic] is the result of some centralized planning committee's anointed vision of the "People's Encyclopedia," in which all knowledge must be rendered to the lowest common denominator of humanity. Thus, attempting to explain any of Kurt Gödel's work by showcasing it through the distorted and defective lens that is the ignorance of a bunch of nearsighted Wiki Wangs strikes me as just trying to bell the cat. I'm sure Kurt Gödel and God have shared a laugh more than once while pondering this dreck. Just something to think about. — Preceding unsigned comment added by 209.169.203.158 (talk) 03:52, 2 November 2011 (UTC)

Lead rewrite

I've placed a lead rewrite tag on the article because it does not comply with WP:LEAD. The lead should summarise the entire article - nothing should be in the lead that is not in the article. At the moment, it is being used as an extra section. I recommend the current lead (copyedited) is moved into its own section and a new lead is written, summarising the entire article. I'll do this at some point; if someone gets to it first, that is what I'd recommend. ItsZippy ^{(talk • contributions)} 20:21, 9 December 2011 (UTC)

You mean you recommend it be moved into its own section? --Trovatore (talk) 21:22, 9 December 2011 (UTC)

Godel's definition of necessary existence seems absurd

Let's say the essence of venomous dogs are the propositions is_a_venomous-animal(y) and dog(y). Let's assume necessary existence of venomous animals and dogs. ${\begin{array}{l}E({\text{venomous dog}})\iff [\operatorname {\text{venomous}} {\text{ ess }}{\text{venomous dog}}\to \square \;\exists y\operatorname {\text{venomous}} (y)]{\text{ and }}[\operatorname {\text{dog}} {\text{ ess }}{\text{venomous dog}}\to \square \;\exists y\operatorname {\text{dog}} (y)]\\\square \;\exists y\operatorname {\text{dog}} (y)\\\square \;\exists y\operatorname {\text{venomous}} (y)\\E({\text{venomous dog}})\end{array}}$

I suppose it's not true that there is necessary existence of venomous animals, and necessary existence of dogs. But I think it brings up a flaw in Godel's definition. Why should necessary existence be defined as having all its essences manifested in at least one thing, when the assertion of the manifestations makes no logical connection to them all being connected together? — Preceding unsigned comment added by 149.144.132.180 (talk) 05:04, 6 February 2015 (UTC)

Edit: Ahhh, my tex screwed up. I'll fix it later. — Preceding unsigned comment added by 149.144.132.180 (talk) 05:11, 6 February 2015 (UTC) Edit: Fixed now — Preceding unsigned comment added by 149.144.132.180 (talk) 04:04, 13 February 2015 (UTC)

Hi 149. Per the talk page guidelines, this page is supposed to be used for discussing what should appear in the article. It is not for discussing the subject matter of the article per se, except insofar as it relates to how the article should or should not change.

You might ask this question on one of the Wikipedia reference desks, perhaps WP:RD/Humanities. --Trovatore (talk) 05:31, 13 February 2015 (UTC)

"Few other axioms"

If the article goes into detail of what the proof supposedly proves, given that the subject matter is modal logic, the article really should specifically state the "few other axioms from modal logic", since they have a very significant effect on proofs. Modal logic isn't a single thing but essentially a framework of separate logics that behave very differently depending on the axioms used. -- 88.192.19.110 (talk) 16:35, 19 May 2015 (UTC)

Criticism

Oops, I seem to have stumbled into a wasps nest of argument here.

I added a criticism section. Nobody seems to have pointed out the main flaw in this "proof": it depends on its axioms and we don't know whether the axioms are "true" or not.

The article as it was was misleading: It presents axioms and conclusions based on them. However, unlike for other logical systems, these axioms are questionable: it's basically disguised assumptions as axioms, and it needs to be clear that philosophers acknowledge that they're not necessarily true, and in fact attempts have been made to show that they are actively false, rather than just unknown. H123b wiki (talk) 12:14, 8 November 2015 (UTC)

Results by automatic theorem provers

What about these results:

• http://page.mi.fu-berlin.de/cbenzmueller/papers/C55.pdf
• http://page.mi.fu-berlin.de/cbenzmueller/papers/C40.pdf

Should we mention them?

--Pokipsy76 (talk) 12:19, 8 May 2016 (UTC)

The article dishonestly only mentions C55 claiming that Gödel is wrong, and does not include C40 which claims that Gödel is right. Actually, both papers prove Gödel is right. C55's only claim is that there is a "bug" in Gödel's proof if you introduce the bug yourself, so it only serves to clarify that there is one definition provided by Leibniz and Scott that cannot be ommitted from the final proof. 5.151.198.248 (talk) 17:07, 18 December 2016 (UTC)

Ultrafilter

Note 3 currently says:

If one considers the partial order $\preceq$ defined by $\varphi \preceq \psi$ iff $\square \forall y(\varphi (y)\to \psi (y))$, then Axioms 1-3 can be summarized by saying that positive properties form an ultrafilter on this ordering. Definition 1 and Axiom 4 are needed to establish the Godlike property as principal element of the ultrafilter.

I suggest to change the definition of $\varphi \preceq \psi$ to $\forall y(\varphi (y)\to \psi (y))$ — i.e. without the "necessarily" square. With this definition, it is easy to prove that $P$ corresponds to an ultrafilter with principal element $G$ as follows:

• $G\preceq \varphi$ implies $P(\varphi )$ by Ax.3 and 1.
• Conversely, $P(\varphi )$ implies $G(x)\to \varphi (x)$ for each $x$ by Df.1, that is, $G\preceq \varphi$.

Hence, $P(\varphi )\iff G\preceq \varphi$.

This proof could be included in the note. Ax.4 isn't yet involved. This is as it should be, since Ax.4 shouldn't be mentioned before Th.2 and Df.2, to ease following the "Symbolic notation" outline.

I guess, $\varphi \preceq \psi$ defined by $\square \forall y(\varphi (y)\to \psi (y))$ — i.e., now including the square — can be shown to lead to an ultrafilter with principal element $G$ as well, using Ax.4 in an appropriate way. However, being unexperienced in modal reasoning, I'm unable to come up with a proof for this. Provided there is one, a corresponding note should be added at an appropriate later place (I suggest right after "Requiring positive properties being positive in every possible world (axiom 4),"), and Note 3 could be shortened or even omitted. - Jochen Burghardt (talk) 10:17, 3 June 2019 (UTC)

Modal collapse / ("Dubious")

Some remarks on modal collapse:

Benzmüller and Woltzenlogel have shown that Godel's proof (in Scott's version) implies modal collapse, that is, it implies that $\diamond p$ is equivalent to $p$, and in turn equivalent to $\Box p$. Using these equivalences (I'll call them MCE for short), the proof from section Gödel's_ontological_proof#Symbolic notation can be drastically simplified to:

${\begin{array}{rl}{\text{Ax.1'.}}&\left(P(\varphi )\;\wedge \;\forall x(\varphi (x)\Rightarrow \psi (x))\right)\;\Rightarrow \;P(\psi )\\{\text{Ax.2.}}&P(\neg \varphi )\;\Leftrightarrow \;\neg P(\varphi )\\{\text{Th.1'.}}&P(\varphi )\;\Rightarrow \;\exists x\;\varphi (x)\\{\text{Df.1.}}&G(x)\;\Leftrightarrow \;\forall \varphi (P(\varphi )\Rightarrow \varphi (x))\\{\text{Ax.3.}}&P(G)\\{\text{Th.2'.}}&\exists x\;G(x)\\\end{array}}$

Here, the dashed variants are obtained from the original ones by removing all modal operators per MCE. This proof is the one mentioned in [note 4].In presence of modal collapse, it is pointless to use modal logic.

Whoever likes to stick to modal logic, may apply MCE to Th.2'. to obtain the original Th.4. ("$\Box \;\exists x\;G(x)$"). This proof doesn't use Df.2,3 and Ax.4,5, but only some consequences of them (viz. MCE). That is, it needs weaker assumptions than Godel's/Scott's proof. Therefore, it would be preferred by logicians.

Many (or all?) philosophers would interpret modal collapse as determinism, including absence of free will ("If it is possible that I kill somebody, then it is necessary that I do that"). As discussed in footnote [11], Godel may have been aware that his axioms imply modal collapse, but he wanted it to follow from the existence of God.

As far as I saw, Benzmüller and Woltzenlogel never said that modal collapse is not necessarily a flaw. Neither did they say it is a flaw. They just pointed out some consequences along the above lines.

Maybe the remarks on modal collaps in the article should be collected in an own subsection, and maybe, some of the above remarks should be included there, too. - Jochen Burghardt (talk) 10:59, 30 June 2019 (UTC)

'This planet/our world' confusion

The following passage:

A truth is necessary if it is true in all possible worlds. By contrast, a truth is contingent if it just happens to be the case. For instance, "more than half of this planet is covered by water" is a contingent truth, that relies upon which planet "this planet" is. If a statement happens to be true in our world, but is false in another world, then it is a contingent truth.

is misleading at best, if not outright mistaken. Specifically, it's worded in such a way that many readers not familiar with modal logic will think that "our world" and "this planet" are being used synonymously. If they are being used synonymously, then "If a statement happens to be true in our world, but is false in another world, then it is a contingent truth" is a wrong explanation of the notion of contingent truth in modal logic. If they aren't meant to be synonymous here, then it ought to be made clear that "our world" means the actual world among the possible worlds, in the modal-logic technical sense, and not the planet Earth.

I'd also say that if you are going to define necessary truth in terms of possible worlds ("A truth is necessary if it is true in all possible worlds"), then the subsequent definition of non-necessary (contingent) truth should also be given in terms of possible worlds. "A truth is contingent if it just happens to be the case" doesn't do that, and is unnecessarily vague. JohnMason 16:38, 29 July 2020 (UTC)

I agree, but I am unable to come up with a good example to illustrate what "possible world" means without introducing the Kripke-semantics formalism. Do you have an idea for that? - Jochen Burghardt (talk) 19:17, 29 July 2020 (UTC)

Why not just link to the Wikipedia article Possible world? Anyway, even if an explanation isn't explicitly given, the definitions ought to be parallel, I'd say.JohnMason (talk) 21:13, 29 July 2020 (UTC)

I removed the misleading planet example and the "happens to be the case" definition, for now. However, I feel, a better example would be necessary here. - "Possible world" is already linked in that section, and I prefer its current target, the mathematical possible worlds, over the philosophical Possible world, since Godel's proof is a mathematical one. - Jochen Burghardt (talk) 08:08, 30 July 2020 (UTC)

I understand your preference for that "possible worlds" target (even if I'd regard Gödel's proof as more metaphysics than mathematics), but that level of technical detail is probably going to be impenetrable for a lot of readers of this article. The "philosophical" possible world target is (somewhat) more approachable, and is enough to give a general reader a sense of the relevant notion of "possible world." Thank you for making those changes.JohnMason (talk) 14:00, 30 July 2020 (UTC)

Also, as an example of the kind of parallel definition I had in mind, for the level this section is pitched at, I think it would be fine to say, e.g., "A truth is necessary if it is true in all possible worlds. By contrast, a truth is contingent if it is true in some, but not all, possible worlds." (including the link to the 'philosophical' Possible world page.) This looks fine to me as a general, not very technical explanation of the notion.JohnMason (talk) 14:09, 30 July 2020 (UTC)

I see that you've changed the passage -- thank you for that, and for correctly stating that a contingent truth has to be true in the actual world, not just "some, but not all, possible worlds" (as I carelessly got it wrong above).JohnMason (talk) 11:41, 31 July 2020 (UTC)

Adele, two days after Gödel's death, told Wang ... ?

Wang reports that Gödel's wife, Adele, two days after Gödel's death, told Wang that ... - didn't Adele die before Gödel ?? DamjanB52 (talk) 12:53, 25 October 2020 (UTC) DamjanB52

No, she survived him by several years. Quick web search indicates that she died in 1981. 2601:648:8200:970:0:0:0:1A5F (talk) 10:10, 25 March 2021 (UTC)

Friedman

Maybe add this?

https://u.osu.edu/friedman.8/files/2014/01/ConMathThe122512-1iea8ps.pdf

I don't know if it was formally published. It's entry #74 from Friedman's manuscripts page,[1] which says it was submitted in 2012. 2601:648:8200:970:0:0:0:1A5F (talk) 10:18, 25 March 2021 (UTC)

Requested move 1 May 2021

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: not moved. (closed by non-admin page mover) ~ Aseleste (t, e | c, l) 06:23, 19 May 2021 (UTC)

---

Gödel's ontological proof → Gödel's ontological argument – "Gödel's ontological proof"[2][3] is slightly more common on Google Scholar, but I would consider this to be a WP:POVTITLE since many people would not agree that this argument proves the existence of God. This title would also be WP:CONSISTENT with the more general article on ontological arguments. (t · c) buidhe 23:01, 1 May 2021 (UTC) —Relisting. ~ Aseleste (t, e | c, l) 03:57, 9 May 2021 (UTC)

• Weak don't move I don't care a whole lot which title it's under, but I don't think the move rationale is convincing. In this ambit calling something a "proof" is traditional, and doesn't necessarily mean that it proves what it purports to prove. --Trovatore (talk) 23:43, 1 May 2021 (UTC)
• Don't move The proof of Godel, as given in section "Symbolic notation", is formally correct, cf. the section about computer verification. Therefore, it deserves to be called "proof" more than most other philosophical arguments. (Of course, this does not mean that it proves any claim of any religion, as discussed in section "Criticism"). - Jochen Burghardt (talk) 14:06, 2 May 2021 (UTC)
• Oppose, per above comments and common name. Randy Kryn (talk) 11:16, 9 May 2021 (UTC)
• Oppose per above. It's a formal proof, so I don't see a POV problem. Srnec (talk) 16:10, 9 May 2021 (UTC)
• Oppose. Can be argued both ways but we need to go one way or another. So by default I think, no move. But see #Discussion, and also Andrew's Principle. Andrewa (talk) 08:39, 17 May 2021 (UTC)

Discussion

As per my !vote above, IMO this is something of a line call. It's POV either way, and both titles can claim to be the common name depending on who you consider reliable and how you compare the attestation counts.

Gödel's ontological proof will tend to be preferred among:

• Sources who believe in God (with an exception noted below)
• Sources who are particularly interested in proof theory and computer-assisted proof
• And as proof theory is relatively recent and computers even more so, recent sources may support it, but that brings up a can of worms... how do we compare the count of attestations (or occurrences) in publications in two different fields?

Gödel's ontological argument will tend to be preferred among:

• Sources who do not believe in God
• Sources who are particularly interested in the history of similar arguments
• Perhaps surprisingly, sources who are particularly concerned with the theology of faith (which I disclose is my personal POV, see this off-this-wiki essay, and note that my !vote goes against this personal bias but for other reasons)

Food for thought? Andrewa (talk) 08:39, 17 May 2021 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Capitalizing a theoretical deity

I was surprised by the use of orthography "God" in the proof section, expecting that form to be restricted to the Christian "God", rather than a theoretical single deity. This seems to be backed up by God_(word)#Capitalization which says

In the English language, capitalization is used for names by which a god is known, including 'God'. Consequently, its capitalized form is not used for multiple gods or when referring to the generic idea of a deity.

(my emphasis added). For me, the article would benefit from clearly distinguishing between "God" (the Abrahamic god in, eg, sentences about Gödel's personal beliefs) and "god" (a generic single deity, the subject of the proof). As nothing in the proof relates to any specific theology, it would be more accurate to use the uncapitalised form. Scarabocchio (talk) 15:23, 19 September 2021 (UTC) [continued] In the History section of the article we find

Morgenstern's diary is an important and usually reliable source for Gödel's later years, but the implication of the August 1970 diary entry—that Gödel did not believe in God—is not consistent with the other evidence. In letters to his mother, ... Gödel argued at length for a belief in an afterlife.

The juxtaposition of these sentences implies this is an inconsistency. There is none. The Kurt Gödel article includes this quote

"I am convinced of this [the afterlife], independently of any theology." It is "possible today to perceive, by pure reasoning" that it "is entirely consistent with known facts." "If the world is rationally constructed and has meaning, then there must be such a thing [as an afterlife]."

The words "independent of any theology" are critical here. Gödel was a theist. The introduction of the Abrahamic god into the proof is unwarranted. Scarabocchio (talk) 15:59, 19 September 2021 (UTC)

I think any time you use the word as the personal name of the deity, it has to be capitalized. That's just the basic rules of English; personal names are capitalized, whether the name is God or Fred. That's how I think it's being used here, not as specific to Abrahamic theology.

It would be possible to reword the arguments so that they don't use a personal name, but it would be awkward and would alter the flavor of them in a way I don't think Gödel would have approved. --Trovatore (talk) 17:50, 19 September 2021 (UTC)