Redigerer Gödels ontologiske bevis for Gud

{{Språkvask|Dårlig oversettelse fra engelsk}}
{{opprydding}}
'''Gödels ontologiske bevis for Gud''' er et formelt [[argument]] for [[Gud#Guds eksistens|Guds eksistens]] som ble formulert av den østerriksk-amerikanske matematikeren [[Kurt Gödel]]. [[Gudsbevis]]et bygger på en lang tradisjon som begynte med [[Anselm av Canterbury]] (1033–1109). Den hellige Anselms [[Det ontologiske gudsbevis|ontologiske gudsbevis]] kan sammenfattes slik: «[[Gud]] er per [[definisjon]] det som forestilt som størst i tenkningen overhodet, gitt Guds rolle. Gud eksisterer i [[forståelse]]n. Hvis Gud eksisterer i forståelsen, så kan vi ikke tenke oss Gud større i [[virkeligheten]], gitt denne forståelsen som del av virkeligheten. Derfor, så må Gud eksistere, ved nødvendighet.» [[Gottfried Leibniz]] (1646-1716) utarbeidet en viderutviklet versjon som Gödel studerte og forsøkte å avklare med sitt eget ontologiske bevis.

Kurt Gödel sammenfattet sin filosofi i 14 punkter. Av disse har følgende betydning for Gödels ontologiske gudsbevis:
<blockquote>
4. Det finnes andre verdener [(tenkbare)] og intelligente vesener [(mennesker f.eks.)] av en forskjellig og høyere rang.<br />
5. Den verdenen vi lever i er ikke bare den vi skal leve i eller ha levet i, [dvs. i hensikt inkarnasjon eller himmelen e.l.]<br />
13. Det finnes en (eksakt) filosofi og teologi, som har å gjøre med de konsepter av høyeste abstraksjon; og det er også denne som er mest fruktbar for vitenskap.<br />
14. Religioner er, for det meste, dårlige, men religion er ikke, [dvs. den rette].<ref>Sitert i Wang 1996, p. 316. [http://cs.nyu.edu/kandathi/goedel_viewpoint.html «My philosophical viewpoint»] {{Wayback|url=http://cs.nyu.edu/kandathi/goedel_viewpoint.html |date=20120911135341 }}, c. 1960, upublisert.</ref>
</blockquote>

<!-- == Publikasjon ==
The first version of the ontological proof in Gödel's papers is dated «around 1941". Gödel is not known to have told anyone about his work on the proof until 1970, when he thought he was dying. In February, he allowed [[Dana Scott]] to copy out a version of the proof, which circulated privately. In August 1970, Gödel told [[Oskar Morgenstern]] that he was "satisfied" with the proof, but Morgenstern recorded in his diary entry for 29 August 1970, that Gödel would not publish because he was afraid that others might think "that he actually believes in God, whereas he is only engaged in a logical investigation (that is, in showing that such a proof with classical assumptions (completeness, etc.) correspondingly axiomatized, is possible)."<ref>Quoted in Gödel 1995, p. 388. The German original is quoted in Dawson 1997, p. 307. The nested parentheses are in Morgenstern's original diary entry, as quoted by Dawson.</ref> Gödel died January 14, 1978. Another version, slightly different from Scott's, was found in his papers. It was finally published, together with Scott's version, in 1987.<ref>The publication history of the proof in this paragraph is from Gödel 1995, p. 388</ref> -->

<!-- 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, who was not a churchgoer and had raised Kurt and his brother as freethinkers,<ref>Dawson 1997, pp. 6.</ref> Gödel argued at length for a belief in an afterlife.<ref>Dawson 1997, pp. 210-212.</ref> He did the same in an interview with a skeptical [[Hao Wang (academic)|Hao Wang]], who said: "I expressed my doubts as G spoke [...] Gödel smiled as he replied to my questions, obviously aware that his answers were not convincing me."<ref>Wang 1996, p. 317. The ellipsis is Wikipedia's.</ref> Wang reports that Gödel's wife, Adele, two days after Gödel's death, told Wang that "Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning."<ref>Wang 1996, p. 51.</ref> In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is ''[[Theism|theistic]]'', not [[Pantheism|pantheistic]], following [[Leibniz]] rather than [[Spinoza]]."<ref>Gödel's answer to a special questionnaire sent him by the sociologist Burke Grandjean. This answer is quoted directly in Wang 1987, p. 18, and indirectly in Wang 1996, p. 112. It's also quoted directly in Dawson 1997, p. 6,
who cites Wang 1987. -->

<!-- The Grandjean questionnaire is perhaps the most extended autobiographical item in Gödel's papers. Gödel filled it out in pencil and wrote a cover letter, but he never returned it. "Theistic" is italicized in both Wang 1987 and Wang 1996. It is possible that this italicization is Wang's and not Gödel's. -->

<!-- The quote follows Wang 1987, with two corrections taken from Wang 1996. Wang 1987 reads "Baptist Lutheran" where Wang 1996 has "baptized Lutheran". "Baptist Lutheran" makes no sense, especially in context, and was presumably a typo or mistranscription. Wang 1987 has "rel. cong.", which in Wang 1996 is expanded to "religious congregation". -->

== Beviset ==
Symbolsk:

<math>
\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 x[\varphi(x) \to \psi(x)]\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 x \; \varphi(x)] \\
			
\text{Ax. 5.} & P(E) \\
			
\text{Th. 4.} & \Box \; \exists x \; G(x)
  
\end{array}
</math>
:
Eller:

UoD], engelsk, «Diskusjonsunivers», definerer det som skal «diskuteres» i det oppsatte logiske argumentet: Alt.<br />
Gx: x er som-Gud<br />
Ex: x har essensielle egenskaper.<br />
Ax: x er en essens av A.<br />
Bx: x er en egenskap av B.<br />
Px: egenskap x er positiv.<br />
Nx: x er en Generell egenskap.<br />
Xx: x er Positiv eksistens.<br />
Cx: x er konsistent.
:
Fra:

<!-- Jeg forstår ikke meningen med å la den være på engelsk! Dette er Wikipedia på bokmål. Det finnes en engelsk Wikipedia. -->
Engelsk tekst følger for Gödels logiske argumentasjon, i sin helhet og slik han har historisk sett satt dem opp:<br />
Definition 1: x is God-like if and only if x has as essential properties those and only those properties which are positive<br />
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<br />
Definition 3: x necessarily exists if and only if every essence of x is necessarily exemplified<br />
Axiom 1: If a property is positive, then its negation is not positive.<br />
Axiom 2: Any property entailed by—i.e., strictly implied by—a positive property is positive<br />
Axiom 3: The property of being God-like is positive<br />
Axiom 4: If a property is positive, then it is necessarily positive<br />
Axiom 5: Necessary existence is positive<br />
Axiom 6: For any property P, if P is positive, then being necessarily P is positive.<br />
Theorem 1: If a property is positive, then it is consistent, i.e., possibly exemplified.<br />
Corollary 1: The property of being God-like is consistent.<br />
Theorem 2: If something is God-like, then the property of being God-like is an essence of that thing.<br />
Theorem 3: Necessarily, the property of being God-like is exemplified.

<!-- Engelsk oversettelse trengs og jeg er trett.
==Modal logic==
The proof uses [[modal logic]], which distinguishes between ''necessary'' truths and ''contingent'' truths.<br />

In the most common interpretation of modal logic, one considers "all possible worlds". A [[truth]] is ''necessary'' if its negation entails a contradiction, such as 2 + 2 = 3, and 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 the planet is covered by water". If a statement happens to be true in our world, but is false in some other worlds, then it is a contingent truth. A statement that is true in some world (not necessarily our own) is called a ''[[logically possible|possible]]'' truth.

A ''property'' assigns to each [[object (philosophy)|object]], in every possible world, a [[truth value]] (either true or false). Note that not all worlds have the same objects: some objects exist in some worlds and not in others. A property has only to assign truth values to those objects that exist in a particular world. As an example, consider the property
:''P''(''s'') = ''s'' is pink

and consider the object
:''s'' = my shirt

In our world, ''P''(''s'') is true because my shirt happens to be pink; in some other world, ''P''(''s'') is false, while in still some other world, ''P''(''s'') wouldn't make sense because my shirt doesn't exist there.

We say that the property ''P'' ''entails'' the property ''Q'', if any object in any world that has the property ''P'' in that world also has the property ''Q'' in that same world. For example, the property
:''P''(''x'') = ''x'' is taller than 2 meters

entails the property
:''Q''(''x'') = ''x'' is taller than 1 meter.

The proof can be summarized as:

:IF it is possible for a rational omniscient being to exist THEN necessarily a rational omniscient being exists.<ref name="uwaterloo">{{cite web |url=http://www.stats.uwaterloo.ca/~cgsmall/ontology.html |title=Kurt Gödel's Ontological Argument |first=Christopher  |last=Small |page=1 |location=[[University of Waterloo]]}}</ref> -->

== Aksiomene ==
<!-- We first assume the following [[axiom]]:

:'''Axiom 1''': It is possible to single out ''positive'' properties from among all properties. Gödel defines a positive property thus: "Positive means positive in the [[moral]] [[aesthetics|aesthetic]] sense (independently of the accidental structure of the world)... It may also mean pure ''attribution'' as opposed to ''privation'' (or containing privation)." (Gödel 1995)

:If a property ''A'' entails a property ''B'' (ie in every possible world if an object has property A it must also have property B), and if A is positive, B must also be positive.

We then assume that the following three conditions hold for all positive properties (which can be summarized by saying "the positive properties form a principal [[ultrafilter]]"):

:'''Axiom 2''': For all properties ''A'', either ''A'' is positive or "not A" is positive.  Never both.
:'''Axiom 3''': The property of "being God-like", ''G'' is a positive property.
:'''Axiom 4''': If a property ''A'' is positive, then it is so in every possible world.

Finally, we assume:

:'''Axiom 5''': Necessary existence is a positive property (''Pos(NE)''). This mirrors the key assumption in Anselm's argument.

Now we '''define''' a new property ''G'': if ''x'' is an object in some possible world, then ''G''(''x'') is true if and only if ''P''(''x'') is true in that same world ''for all'' positive properties ''P''. ''G'' is called the "God-like" property. An object ''x'' that has the God-like property is called God. -->
<!--
==Criticisms of Ontological arguments in general==

[[St Thomas Aquinas]] rejected St. Anselm's ontological argument.<ref>[[Aquinas]], Thomas, Saint. [[Summa Theologica]], Part 1, Question 2, Article 1.</ref> Likewise, some Catholic theologians have rejected<ref>{{cite encyclopedia | first=P.J. | last=Toner | title=The Existence of God | encyclopedia=The Catholic Encyclopedia | url=http://www.newadvent.org/cathen/06608b.htm#IBf | accessdate=2007-01-19}}</ref> Gödel's revised version.<ref name="uwaterloo"/> [[Bertrand Russell]] noted: "The argument does not, to a modern mind, seem very convincing, but it is easier to feel convinced that it must be fallacious than it is to find out precisely where the fallacy lies."<ref>{{cite book | first=Bertrand | last=Russell | authorlink=Bertrand Russell | title=[[History of Western Philosophy (Russell)|History of Western Philosophy]] | publisher=[[Touchstone]] | year=1972 | isbn=0-671-20158-1 | page=536}} (Book 3, Part 1, Section 11)</ref> However, Russell was also known to say: "Great God in Boots! — the ontological argument is sound!" as a parody,<ref>Autobiography of Bertrand Russell, vol. 1, 1967.</ref> including many others that parodied it, such as [[Gaunilo_of_Marmoutiers#The_.22Lost_Island.22_refutation|Gaunilo's Island]]. In ''[[Critique of Pure Reason]]'', [[Immanuel Kant]] famously rejected existence as a [[property (philosophy)|property]].<ref>{{cite book | first=Immanuel | last=Kant | authorlink=Immanuel Kant | title=[[Critique of Pure Reason]] | year=1781/1787 | pages=A 592–602/B 620–630}}</ref>

In [[David Hume]]'s [[Dialogues Concerning Natural Religion]], the character Cleanthes argues that no being could ever be proven to exist through an ''a priori'' demonstration:<ref>{{cite web | url=http://www.philosophyofreligion.info/humeonaprioriproofs.html | first=Tim | last=Holt | title=The Ontological Argument: Hume on a priori Existential Proofs | accessdate=2012-10-05 | archiveurl=https://web.archive.org/web/20080410180808/http://www.philosophyofreligion.info/humeonaprioriproofs.html | url-status=dead }}</ref>

:[T]here is an evident absurdity in pretending to demonstrate a matter of fact, or to prove it by any arguments ''a priori''. Nothing is demonstrable, unless the contrary implies a contradiction. Nothing, that is distinctly conceivable, implies a contradiction. Whatever we conceive as existent, we can also conceive as non-existent. There is no being, therefore, whose non-existence implies a contradiction. Consequently there is no being, whose existence is demonstrable.

There have been many other arguments against ontological proofs such as: [[Ontological argument#Criticism by .22existence precedes essence.22|Existence precedes essence]]; [[Ontological argument#Gaunilo|Gaunilo's island]]; [[Ontological argument#Necessary nonexistence|Necessary nonexistence]]; [[Ontological argument#Kant: existence is not a predicate|Existence is not a predicate]]; and [[Ontological argument#Problem of incoherence|Problem of incoherence]]. -->
Formelt sett så er argumentet, Gödels ontologiske bevis for Gud, gjerne gitt (nå) som logisk gyldig, men at man stiller spørsmål ved argumentets «soundness<ref>Bergmann, M., et al., 2004. ''The Logic Book'', New York, NY: McGraw-Hill, s. 248.</ref>», troverdighet. «Soundness» er her en logisk term som man må forstå.<br />

Logisk gyldighet får man hvis linjen ved «Definisjon 1» blir dedusert ved «bikondisjonal eliminasjon» og at man, gitt tiden  Gödel har skrevet dette på, behandler argumentet litt «grovt», som ved moderne Fitch-struktur, så er konklusjonen relativt grei, og som tidligere nevnt, altså ''gyldig'', ved teorem 3, nødvendigvis så er egenskapen ved et vesen som er «som-Gud» eksemplifisert, dvs. ''eksisterer i virkeligheten'', gitt altså bare logikken da.

<!--==Criticism of Gödel's==

[[C. Anthony Anderson]] has said:<ref>{{cite journal|last=Anderso|first=C A|title=Some emendations of Gödel’s ontological proof|journal=Faith and Philosophy|year=1990|volume=7|issue=3|pages=291–303}}</ref>

:Consideration of the axioms, especially ... [Axiom 2], may tend to dampen one's confidence in ... [Axiom 3] and ... [Axiom 4] — that is, if one harbors any real doubt about self-consistency. I don't say that the argument begs the questions of ... [God's possible existence]; the charge is too difficult to establish. but observe that one cannot just tell by scrutinizing a property what it entails; one might be surprised at a consequence.

==Derivation==
From axioms 1 through 4, Gödel argued that in ''some'' possible world there exists God. He used a sort of modal [[plenitude principle]] to argue this from the logical consistency of Godlikeness. Note that this property is itself positive, since it is the conjunction of the (infinitely many) positive properties.

Then, Gödel defined ''essences'': 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 if ''P'' entails all other properties that ''x'' has in that world. We also say that ''x'' ''necessarily exists'' if for every essence ''P'' the following is true: in every possible world, there is an element ''y'' with ''P''(''y'').

Since necessary existence is positive, it must follow from Godlikeness. Moreover, Godlikeness is an essence of God, since it entails all positive properties, and any nonpositive property is the negation of some positive property, so God cannot have any nonpositive properties. 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. Given the existence of a Godlike object in one world, proven above, we may conclude that there is a Godlike object in every possible world, as required.

From 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. --> <!-- Jeg har ikke lyst til å bli "bannet" for dette. Jeg har bort-kommentert en del tekst. Dette kan også være kompromisset for å importere engelsk tekst i "stort tempo" for så å lage en betraktelig større Wikipedia (enn alle andres). -->

== Referanser ==
<references/>

== Litteratur ==
<!-- {{More footnotes|date=June 2012}} -->
* {{cite book |author=John W. Dawson, Jr |title=Logical Dilemmas: The Life and Work of Kurt Godel |url=https://archive.org/details/logicaldilemmasl0000daws |publisher=AK Peters, Ltd |location=Wellesley, Mass |year=1997 |pages= |isbn=1-56881-025-3 |oclc= |doi=}}
* Melvin Fitting, «Types, Tableaus, and Godel's God» Publisher: Dordrecht Kluwer Academic ©2002, ISBN 1-4020-0604-7, ISBN 978-1-4020-0604-3
* Kurt Gödel (1995). «Ontological Proof». ''Collected Works: Unpublished Essays & Lectures, Volume III''. pp.&nbsp;403–404. Oxford University Press. ISBN 0-19-514722-7
* A. P. Hazen, «On Gödel's Ontological Proof», Australasian Journal of Philosophy, Vol. 76, No 3, pp.&nbsp;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)
* {{cite book |author=Wang, Hao |title=Reflections on Kurt Gödel |url=https://archive.org/details/reflectionsonkur0000wang |publisher=MIT Press |location=Cambridge, Mass |year=1987 |pages= |isbn=0-262-23127-1 |oclc= |doi=}}
* {{cite book |author=Wang, Hao |title=A logical journey: from Gödel to philosophy |publisher=MIT Press |location=Cambridge, Mass |year=1996 |pages= |isbn=0-262-23189-1 |oclc= |doi=}}
* {{cite web |url=http://www.stats.uwaterloo.ca/~cgsmall/Godel.final.revision.PDF |title=Reflections on Gödel’s Ontological Argument |first=Christopher |last=Small |location=[[University of Waterloo]] |access-date=2012-10-05 |archive-date=2009-12-22 |archive-url=https://web.archive.org/web/20091222082410/http://www.stats.uwaterloo.ca/~cgsmall/Godel.final.revision.PDF |url-status=yes }}

== Se også ==
* [[Gudsbevis]]
* [[Det ontologiske gudsbevis]]
* [[Det teleologiske gudsbevis]] 
* [[Det kosmologiske gudsbevis]]
* [[Gud#Guds eksistens|Guds eksistens]]
* [[Syntetiske proposisjoner]], av Kant antakeligvis
* [[Det absolutt uendelige]]
* [[Religionsfilosofi]]
* [[Teisme]]

== Eksterne lenker ==
* [http://www.stats.uwaterloo.ca/~cgsmall/ontology.html Kurt Gödel's Ontological Argument] {{Wayback|url=http://www.stats.uwaterloo.ca/~cgsmall/ontology.html |date=20110410182503 }}
* http://plato.stanford.edu/entries/ontological-arguments. Besøkt 06.10.2012.
* [https://www.ontology.co/biblio/ontological-proof-contemporary-biblio.htm Annotated bibliography of studies on Gödel's Ontological Argument]
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