957:
289:
952:{\displaystyle {\begin{array}{rl}{\text{Ax. 1.}}&\left(P(\varphi )\;\wedge \;\Box \;\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 \;\Diamond \;\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.}}&\Diamond \;\exists x\;G(x)\\{\text{Df. 2.}}&\varphi {\text{ ess }}x\;\Leftrightarrow \;\varphi (x)\wedge \forall \psi \left(\psi (x)\Rightarrow \Box \;\forall y(\varphi (y)\Rightarrow \psi (y))\right)\\{\text{Ax. 4.}}&P(\varphi )\;\Rightarrow \;\Box \;P(\varphi )\\{\text{Th. 3.}}&G(x)\;\Rightarrow \;G{\text{ ess }}x\\{\text{Df. 3.}}&E(x)\;\Leftrightarrow \;\forall \varphi (\varphi {\text{ ess }}x\Rightarrow \Box \;\exists y\;\varphi (y))\\{\text{Ax. 5.}}&P(E)\\{\text{Th. 4.}}&\Box \;\exists x\;G(x)\end{array}}}
1151:
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".
272:(axiom 5), it must be a property of every Godlike object, as every Godlike object has all the positive properties (definition 1). 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 (theorem 4). Besides axiom 1-5 and definition 1-3, a few other axioms from modal logic were tacitly used in the proof.
96:, 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." 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." In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is
1150:
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
271:
Hence, it must follow from
Godlikeness. Moreover, Godlikeness is an essence of God, since it entails all positive properties, and any non-positive property is the negation of some positive property, so God cannot have any non-positive properties. Since necessary existence is also a positive property
84:
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
200:
is positive, too (axiom 1). Gödel then argues that each positive property is "possibly exemplified", i.e. applies at least to some object in some world (theorem 1). Defining an object to be
Godlike if it has all positive properties (definition 1), and requiring that property to be positive itself
279:: two or more objects are identical (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.
966:
Most criticism of Gödel's proof is aimed at its axioms: as with any proof in any logical system, if the axioms the proof depends on are doubted, then the conclusions can be doubted. It is particularly applicable to Gödel's proof – because it rests on five axioms, some of which are considered
970:
Many philosophers have called the axioms into question. The first layer of criticism is simply that there are no arguments presented that give reasons why the axioms are true. A second layer is that these particular axioms lead to unwelcome conclusions. This line of thought was argued by
1011:
There are, however, many more criticisms, most of them focusing on the 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
1021:(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 a theological task. It is this task which decides which religion's god has been proven to exist.
1001:, asking whether many other almost-gods would also be "proven" through Gödel's axioms. This counter-argument has been questioned by Gettings, who agrees that the axioms might be questioned, but disagrees that Oppy's particular counter-example can be shown from Gödel's axioms.
1048:
version). They also proved that this version's axioms are consistent, but imply modal collapse, thus confirming Sobel's 1987 argument. In the same paper, they suspected Gödel's original version of the axioms to be inconsistent, as they failed to prove their consistency.
85:
proof with classical assumptions (completeness, etc.) correspondingly axiomatized, is possible)." 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.
1082:. Moreover, they gave an argument that this version is inconsistent in every logic at all, but failed to duplicate it by automated provers. However, they were able to verify Melvin Fitting's reformulation of the argument and guarantee its consistency.
42:, in its most succinct form, is as follows: "God, by definition, is that for which no greater can be conceived. God exists in the understanding. If God exists in the understanding, we could imagine Him to be greater by existing in
1016:
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
75:
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
1188:
As a profane example, if the property of being green is positive, that of not being red is, too (by axiom 1), hence that of being red is negative (by axiom 2). More generally, at most one color can be considered
2537:
1959:
1313:
1916:
1849:
244:
has in that world (definition 2). Requiring positive properties being positive in every possible world (axiom 4), Gödel can show that
Godlikeness is an essence of a Godlike object (theorem 3). Now,
2021:
1471:, but Gödel's proof does not, Koons suggests to forbid this property-construction operation as the "most conservative" measure, before "rejecting or emending ... axioms (as Anderson does)".
2244:
1452:
1343:)", i.e. "There exists an object which has all positive, but no negative properties". Nothing more than axioms 1-3, definition 1, and theorems 1-2 needs to be considered for this result.
1412:
1254:
2060:
1379:
1076:
1781:
Kurt Gödel (Mar 1995). "Texts
Relating to the Ontological Proof (App. B)". In Solomon Feferman; John W. Dawson jr.; Warren Goldfarb; Charles Parsons; Robert M. Solovay (eds.).
2116:
994:, but argued to be refutable by Anderson and Michael Gettings. Sobel's proof of modal collapse has been questioned by Koons, but a counter-defence by Sobel has been given.
1604:
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.
1542:
1198:
Continuing the color example, a godlike object must have the unique color that is considered positive, or no color at all; both alternatives may seem counter-intuitive.
1228:
63:
13. There is a scientific (exact) philosophy and theology, which deals with concepts of the highest abstractness; and this is also most highly fruitful for science.
2188:
1934:
1869:
2154:
979:" where every statement that is true is necessarily true, i.e. the sets of necessary, of contingent, and of possible truths all coincide (provided there are
2568:
967:
questionable. A proof does not necessitate that the conclusion be correct, but rather that by accepting the axioms, the conclusion follows logically.
1699:
152:
if it is true in all possible worlds. By contrast, if a statement happens to be true in our world, but is false in another world, then it is a
2435:
2397:
205:
possible world a Godlike object exists (theorem 2), called "God" in the following. Gödel proceeds to prove that a Godlike object exists in
2349:
Frode Alfson Bjørdal, "All Properties are Divine, or God Exists", in Logic and Logical Philosophy, Vol. 27 No. 3, 2018, pp. 329–350.
2522:
2424:— See Chapter "Ontological Proof", pp. 403–404, and Appendix B "Texts Relating to the Ontological Proof", pp. 429–437.
2219:
1765:
2193:
1332:
By removing all modal operators from axioms, definitions, proofs, and theorems, a modified version of theorem 2 is obtained saying "∃
2495:
2476:
2417:
2389:
2372:
1795:
1674:
1660:
Kurt Gödel (Mar 1995). Solomon Feferman and John W. Dawson jr. and Warren Goldfarb and Charles Parsons and Robert M. Solovay (ed.).
2100:
53:
Gödel left a fourteen-point outline of his philosophical beliefs in his papers. Points relevant to the ontological proof include:
2544:
2552:
2430:
Hazen, A. P. "On Gödel's Ontological Proof", Australasian Journal of Philosophy, Vol. 76, No 3, pp. 361–377, September 1998
2249:
1259:
2578:
2309:
2074:
1874:
1807:
2352:
Bromand, Joachim. "Gödels ontologischer Beweis und andere modallogische Gottesbeweise", in J. Bromand und G. Kreis (Hg.),
2403:
Kurt Gödel (Mar 1995). Solomon Feferman; John W. Dawson jr.; Warren Goldfarb; Charles parsons; Robert M. Solovay (eds.).
2404:
1782:
1661:
1008:
accepted Gödel's proof, calling it "an improvement over the Anselmian Ontological Argument (which does not work)."
276:
275:
From these hypotheses, it is also possible to prove that there is only one God in each world by Leibniz's law, the
1425:
1037:. The effort made headlines in German newspapers. According to the authors of this effort, they were inspired by
1030:
107:
1480:
Lines "T3" in Fig.2, and item 3 in section 4 ("Main findings"). Their theorem "T3" corresponds to "Th.4" shown
134:
1982:
Proc. Gödel '96: Logical Foundations of Mathematics, Computer Science and Physics — Kurt Gödel's Legacy
1388:
2339:
1930:
1096:
1005:
50:(1646–1716); this is the version that Gödel studied and attempted to clarify with his ontological argument.
2530:
1233:
1029:
Christoph Benzmüller and Bruno Woltzenlogel-Paleo formalized Gödel's proof to a level that is suitable for
2442:
1121:
1079:
980:
1355:
1055:
2427:
Goldman, Randolph R. "Gödel's Ontological Argument", PhD Diss., University of California, Berkeley 2000.
2054:
1745:
268:
in every possible world (definition 3). Axiom 5 requires necessary existence to be a positive property.
92:, Gödel argued at length for a belief in an afterlife. He did the same in an interview with a skeptical
2360:
2267:
2573:
2181:"Automating Gödel's Ontological Proof of God's Existence with Higher-Order Automated Theorem Provers"
1942:
1706:
1131:
39:
35:
1790:. Collected Works. Vol. III (1st ed.). Oxford: Oxford University Press. pp. 429–437.
2464:
2192:. Frontiers in Artificial Intelligence and Applications. Vol. 263. IOS Press. pp. 93–98.
1741:
991:
972:
93:
2446:
2237:"The Inconsistency in Gödel's Ontological Argument: — A Success Story for AI in Metaphysics"
2551:
showing that if God exists (in the sense of Gödel), then Mathematics, as formalized by the usual
1518:
1468:
1013:
166:
158:
169:(modal) logic because the definition of God employs an explicit quantification over properties.
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2472:
2413:
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1761:
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1213:
1018:
81:
47:
2513:
2042:
2011:
1951:
1116:
987:, Sobel suggested in a 2005 conference paper that Gödel might have welcomed modal collapse.
31:
80:
to copy out a version of the proof, which circulated privately. In August 1970, Gödel told
2548:
2517:
2180:
1164:
properties from among all properties. Gödel comments that "Positive means positive in the
1107:
depicts the (fictional) rediscovery of Gödel's lost notebook about the ontological proof.
1034:
88:
In letters to his mother, who was not a churchgoer and had raised Kurt and his brother as
1571:
See the detailed discussion in section 4 "Intuitive Inconsistency Argument" (p. 939–941).
1090:
A humorous variant of Gödel's ontological proof is mentioned in Quentin Canterel's novel
1171:
sense (independently of the accidental structure of the world)... It may also mean pure
2381:
1854:
1419:
1038:
976:
60:
5. The world in which we live is not the only one in which we shall live or have lived.
2562:
2236:
1208:
127:
27:
1999:
156:
truth. A statement that is true in some world (not necessarily our own) is called a
2315:
1920:
The note might indicate that Gödel was aware of his axioms implying modal collapse.
1695:
1515:
is by Dana Scott. It differs from Gödel's original by omitting the first conjunct,
984:
23:
2412:. Collected Works. Vol. III (1st ed.). Oxford: Oxford University Press.
1977:
1669:. Collected Works. Vol. III (1st ed.). Oxford: Oxford University Press.
2384:, "Types, Tableaus, and Godel's God" Publisher: Dordrecht Kluwer Academic, 2002,
2210:
D. Scott (2004). "Appendix B: Notes in Dana Scott's Hand ". In J.H. Sobel (ed.).
1613:
The publication history of the proof in this paragraph is from Gödel 1995, p. 388
1995:
1316:
1179:(or containing privation)." (Gödel 1995), see also manuscript in (Gawlick 2012).
998:
172:
First, Gödel axiomatizes the notion of a "positive property": for each property
141:
123:
89:
1315:, then Axioms 1-3 can be summarized by saying that positive properties form an
294:
2133:
2015:
1955:
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77:
2279:
57:
4. There are other worlds and rational beings of a different and higher kind.
2046:
103:
1078:, i.e., is inconsistent in every modal logic with a reflexive or symmetric
2541:, Jan. 2012 — shows Gödel's original proof manuscript on p. 2-3
2103:[Existence and Necessity — Kurt Gödel's Axiomatic Theology]
2033:
Gettings Michael (1999). "Gödel's ontological argument: a reply to Oppy".
1165:
181:
1052:
In 2016, they gave an automated proof that the original version implies
2101:"Existenz und Notwendigkeit — Kurt Gödels axiomatische Theologie"
1319:
on this ordering. Definition 1 and Axiom 4 are needed to establish the
111:
43:
2342:, "Understanding Gödel's Ontological Argument", in T. Childers (ed.),
2155:"Scientists Use Computer to Mathematically Prove Gödel's God Theorem"
2138:
1126:
98:
2245:
Proc. 25th International Joint Conference on Artificial Intelligence
46:. Therefore, God must exist." A more elaborate version was given by
1984:. Lecture Notes in Logic. Vol. 6. Springer. pp. 167–172.
1705:(Unpublished Paper). University of Texas at Austin. Archived from
145:
2531:
Annotated bibliography of studies on Gödel's Ontological Argument
188:
must be positive, but not both (axiom 2). If a positive property
2235:
Christoph Benzmüller and Bruno Woltzenlogel-Paleo (Jul 2016).
66:
14. Religions are, for the most part, bad—but religion is not.
1044:
In 2014, they computationally verified Gödel's proof (in the
34:. The argument is in a line of development that goes back to
2214:. Cambridge: Cambridge University Press. pp. 145–146.
2115:] (in German). Heidelberg: Synchron. pp. 349–374.
1722:
The presentation below follows that in Koons (2005), p.3-7.
975:, showing that if the axioms are accepted, they lead to a "
2212:
Logic and Theism: Arguments for and Against Beliefs in God
2179:
Christoph Benzmüller and Bruno Woltzenlogel-Paleo (2014).
1802:
Here: p.435; probably, Sobel referred to Gödel's note 4.:
1756:. Cambridge/MA & London, England: MIT Press. pp.
1308:{\displaystyle \square \forall y(\varphi (y)\to \psi (y))}
990:
There are suggested amendments to the proof, presented by
1976:
Curtis Anthony Anderson and Michael Gettings (Aug 1996).
1911:{\displaystyle \varphi (x)\Rightarrow \Box \varphi (x)}
1844:{\displaystyle \varphi (x)\Rightarrow \Box \varphi (x)}
1553:
Lines "CO'" in Fig.2, and item 5 in section 4 (p. 97).
140:. In the most common semantics for modal logic, many "
2266:
Christoph Benzmüller and David Fuenmayor (May 2017).
1877:
1857:
1810:
1640:
Wang 1996, p. 317. The ellipsis is Knowledge (XXG)'s.
1521:
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1236:
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Was sind und was sollen mathematische Gottesbeweise?
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Proc. European Conference on Artificial Intelligence
1658:
Gödel's proof is reprinted on p.403-404,429-437 of:
1502:
Line "MC" in Fig.2, and item 6 in section 4 (p. 97).
1493:
Line "CO" in Fig.2, and item 1 in section 4 (p. 97).
1562:
Item 8 in section 4.1 "Informal argument" (p. 940).
1910:
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1752:On Being and Saying: Essays for Richard Cartwright
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1536:
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2365:Logical Dilemmas: The Life and Work of Kurt Godel
2268:"Types, Tableaus and Gödel's God in Isabelle/HOL"
2059:: CS1 maint: DOI inactive as of September 2024 (
1323:property as principal element of the ultrafilter.
1935:"Some Emendations of Gödel's Ontological Proof"
1918:should follow first from the existence of God."
1094:. The proof is also mentioned in the TV series
240:necessarily entails all other properties that
2436:"Reflections on Gödel's Ontological Argument"
1871:], ... but that is the inferior way. Rather,
1851:is assumed [as following from the essence of
1690:
1688:
1686:
1160:It assumes that it is possible to single out
8:
1744:(Nov 1987). "Gödel's ontological proof". In
1593:A Logical Journey: From Gödel to Philosophy.
1447:{\displaystyle \Box p\Rightarrow \Diamond p}
220:is an object in some world, then a property
2488:A Logical Journey: from Gödel to Philosophy
1467:Since Sobel's proof of modal collapse uses
1033:or at least computational verification via
2545:A Divine Consistency Proof for Mathematics
2075:"Godel's Theorem and the Existence of God"
997:Gödel's proof has also been questioned by
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16:Logical argument for the existence of God
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1407:{\displaystyle \Diamond p\Rightarrow p}
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2052:
1512:
1481:
1045:
2295:The Jolly Coroner: A Picaresque Novel
1978:"Gödel's ontological proof revisited"
1458:whenever there are accessible worlds.
1249:{\displaystyle \varphi \preceq \psi }
7:
1595:A Bradford Book, 1997. Print. p.316.
2523:Stanford Encyclopedia of Philosophy
2367:. Wellesley, Mass: AK Peters, Ltd.
1374:{\displaystyle p\Rightarrow \Box p}
1071:{\displaystyle \Diamond \Box \bot }
2569:Arguments for the existence of God
2354:Gottesbeweise von Anselm bis Gödel
1700:Sobel on Gödel's Ontological Proof
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516:
469:
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408:
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14:
2242:. In Subbarao Kambhampati (ed.).
2153:Knight, David (23 October 2013).
1025:Computationally verified versions
2255:from the original on 2016-11-13.
2248:. AAAI Press. pp. 936–942.
2199:from the original on 2014-07-14.
2122:from the original on 2016-05-18.
2000:"Godelian ontological arguments"
1965:from the original on 2015-06-04.
2406:Unpublished Essays and Lectures
1784:Unpublished Essays and Lectures
1663:Unpublished Essays and Lectures
236:) is true in that world and if
201:(axiom 3), Gödel shows that in
2490:. Cambridge, Mass: MIT Press.
2471:. Cambridge, Mass: MIT Press.
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983:worlds at all). According to
196:in each possible world, then
2547:— A submitted work by
1103:Jeffrey Kegler's 2007 novel
224:is said to be an essence of
165:Furthermore, the proof uses
1537:{\displaystyle \varphi (x)}
2597:
2297:. Acorn Independent Press.
2134:"FormalTheology/GoedelGod"
277:identity of indiscernibles
38:(1033–1109). St. Anselm's
2469:Reflections on Kurt Gödel
2293:Quentin Canterel (2015).
1956:10.5840/faithphil19907325
1631:Dawson 1997, pp. 210–212.
1031:automated theorem proving
20:Gödel's ontological proof
2344:The Logica Yearbook 1998
2272:Archive of Formal Proofs
2109:Logik in der Philosophie
1223:{\displaystyle \preceq }
212:To this end, he defines
2514:"Ontological arguments"
2356:, Berlin 2011, 381-491.
2346:, Prague 1999, 214-217.
2308:Jeffrey Kegler (2007),
2099:André Fuhrmann (2005).
2047:10.1111/1467-8284.00184
2016:10.1093/analys/56.4.226
1980:. In Petr Hájek (ed.).
1931:Curtis Anthony Anderson
2443:University of Waterloo
2049:(inactive 2024-09-11).
1912:
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1845:
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1448:
1408:
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1309:
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1224:
1122:Philosophy of religion
1080:accessibility relation
1072:
953:
260:, there is an element
252:if, for every essence
2107:. In W. Spohn (ed.).
2022:Longer version (2005)
1913:
1866:
1846:
1746:Judith Jarvis Thomson
1731:Fitting, 2002, p. 139
1539:
1449:
1409:
1376:
1310:
1251:
1225:
1207:If one considers the
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1006:Fr. Robert J. Spitzer
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26:by the mathematician
2434:Small, Christopher.
2340:Frode Alfson Bjørdal
1943:Faith and Philosophy
1875:
1855:
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1389:
1356:
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1234:
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1132:Ontological argument
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290:
144:" are considered. A
40:ontological argument
36:Anselm of Canterbury
30:(1906–1978) for the
2579:Works by Kurt Gödel
2326:, full text online.
2113:Logic in Philosophy
1742:Jordan Howard Sobel
1622:Dawson 1997, pp. 6.
992:C. Anthony Anderson
973:Jordan Howard Sobel
192:implies a property
2486:Wang, Hao (1996).
1908:
1861:
1841:
1534:
1511:The version shown
1469:lambda abstraction
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1404:
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1246:
1220:
1068:
1004:Religious scholar
949:
947:
2398:978-1-4020-0604-3
2361:John W. Dawson Jr
1864:{\displaystyle x}
1649:Wang 1996, p. 51.
1092:The Jolly Coroner
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735:
629:
619:
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283:Symbolic notation
250:exist necessarily
82:Oskar Morgenstern
48:Gottfried Leibniz
2586:
2555:, is consistent.
2535:Thomas Gawlick,
2527:
2518:Zalta, Edward N.
2501:
2482:
2460:
2458:
2457:
2451:
2445:. Archived from
2440:
2423:
2411:
2378:
2327:
2325:
2324:
2323:
2314:, archived from
2305:
2299:
2298:
2290:
2284:
2283:
2263:
2257:
2256:
2254:
2241:
2232:
2226:
2225:
2207:
2201:
2200:
2198:
2185:
2176:
2170:
2169:
2167:
2165:
2150:
2144:
2143:
2130:
2124:
2123:
2121:
2106:
2096:
2090:
2089:
2087:
2086:
2071:
2065:
2064:
2058:
2050:
2041:(264): 309–313.
2030:
2024:
2019:
1992:
1986:
1985:
1973:
1967:
1966:
1964:
1939:
1927:
1921:
1917:
1915:
1914:
1909:
1870:
1868:
1867:
1862:
1850:
1848:
1847:
1842:
1801:
1789:
1778:
1772:
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1755:
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1723:
1720:
1714:
1713:
1711:
1704:
1692:
1681:
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1668:
1656:
1650:
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1629:
1623:
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1614:
1611:
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1602:
1596:
1589:
1572:
1569:
1563:
1560:
1554:
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1535:
1509:
1503:
1500:
1494:
1491:
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1478:
1472:
1465:
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1413:
1411:
1410:
1405:
1380:
1378:
1377:
1372:
1350:
1344:
1330:
1324:
1314:
1312:
1311:
1306:
1255:
1253:
1252:
1247:
1229:
1227:
1226:
1221:
1205:
1199:
1196:
1190:
1186:
1180:
1158:
1152:
1148:
1117:Existence of God
1077:
1075:
1074:
1069:
1035:proof assistants
958:
956:
955:
950:
948:
919:
916:
896:
893:
855:
852:
819:
816:
807:
804:
780:
777:
736:
733:
727:
723:
630:
627:
620:
617:
586:
583:
563:
560:
496:
493:
445:
442:
399:
396:
373:
369:
301:
298:
209:possible world.
32:existence of God
2596:
2595:
2589:
2588:
2587:
2585:
2584:
2583:
2559:
2558:
2549:Harvey Friedman
2511:
2508:
2498:
2485:
2479:
2463:
2455:
2453:
2449:
2438:
2433:
2420:
2409:
2402:
2375:
2359:
2336:
2334:Further reading
2331:
2330:
2321:
2319:
2307:
2306:
2302:
2292:
2291:
2287:
2265:
2264:
2260:
2252:
2239:
2234:
2233:
2229:
2222:
2209:
2208:
2204:
2196:
2183:
2178:
2177:
2173:
2163:
2161:
2152:
2151:
2147:
2142:. 28 June 2021.
2132:
2131:
2127:
2119:
2104:
2098:
2097:
2093:
2084:
2082:
2073:
2072:
2068:
2051:
2032:
2031:
2027:
1994:
1993:
1989:
1975:
1974:
1970:
1962:
1937:
1929:
1928:
1924:
1873:
1872:
1853:
1852:
1806:
1805:
1798:
1787:
1780:
1779:
1775:
1768:
1740:
1739:
1735:
1730:
1726:
1721:
1717:
1709:
1702:
1696:Robert C. Koons
1694:
1693:
1684:
1677:
1666:
1659:
1657:
1653:
1648:
1644:
1639:
1635:
1630:
1626:
1621:
1617:
1612:
1608:
1603:
1599:
1591:In: Wang, Hao.
1590:
1586:
1581:
1576:
1575:
1570:
1566:
1561:
1557:
1552:
1548:
1517:
1516:
1510:
1506:
1501:
1497:
1492:
1488:
1479:
1475:
1466:
1462:
1424:
1423:
1387:
1386:
1354:
1353:
1351:
1347:
1331:
1327:
1258:
1257:
1232:
1231:
1212:
1211:
1206:
1202:
1197:
1193:
1187:
1183:
1159:
1155:
1149:
1145:
1140:
1113:
1088:
1054:
1053:
1027:
964:
946:
945:
920:
912:
911:
897:
889:
888:
853: ess
820:
812:
811:
805: ess
781:
773:
772:
737:
729:
728:
664:
660:
628: ess
621:
613:
612:
587:
579:
578:
564:
556:
555:
497:
489:
488:
446:
438:
437:
400:
392:
391:
308:
304:
302:
288:
287:
285:
142:possible worlds
122:The proof uses
120:
73:
24:formal argument
17:
12:
11:
5:
2594:
2593:
2590:
2582:
2581:
2576:
2571:
2561:
2560:
2557:
2556:
2542:
2533:
2528:
2512:Oppy, Graham.
2507:
2506:External links
2504:
2503:
2502:
2496:
2483:
2477:
2461:
2431:
2428:
2425:
2418:
2400:
2382:Melvin Fitting
2379:
2373:
2357:
2350:
2347:
2335:
2332:
2329:
2328:
2300:
2285:
2258:
2227:
2221:978-0511497988
2220:
2202:
2171:
2145:
2125:
2091:
2066:
2025:
2010:(4): 226–230.
1987:
1968:
1950:(3): 291–303.
1922:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1860:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1796:
1773:
1767:978-0262200639
1766:
1733:
1724:
1715:
1712:on 2020-08-02.
1682:
1675:
1651:
1642:
1633:
1624:
1615:
1606:
1597:
1583:
1582:
1580:
1577:
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1573:
1564:
1555:
1546:
1533:
1530:
1527:
1524:
1504:
1495:
1486:
1473:
1460:
1454:holds for all
1443:
1440:
1437:
1434:
1431:
1420:indirect proof
1403:
1400:
1397:
1394:
1370:
1367:
1364:
1361:
1345:
1325:
1304:
1301:
1298:
1295:
1292:
1289:
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1283:
1280:
1277:
1274:
1271:
1268:
1265:
1245:
1242:
1239:
1219:
1200:
1191:
1181:
1175:as opposed to
1153:
1142:
1141:
1139:
1136:
1135:
1134:
1129:
1124:
1119:
1112:
1109:
1087:
1084:
1067:
1064:
1061:
1039:Melvin Fitting
1026:
1023:
1019:André Fuhrmann
977:modal collapse
963:
960:
944:
941:
938:
935:
931:
928:
924:
921:
914:
913:
910:
907:
904:
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416:
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410:
407:
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401:
394:
393:
390:
387:
384:
381:
377:
372:
368:
365:
362:
359:
356:
353:
350:
347:
344:
341:
338:
335:
332:
328:
324:
320:
317:
314:
311:
307:
303:
296:
295:
284:
281:
264:with property
119:
116:
72:
69:
68:
67:
64:
61:
58:
15:
13:
10:
9:
6:
4:
3:
2:
2592:
2591:
2580:
2577:
2575:
2572:
2570:
2567:
2566:
2564:
2554:
2550:
2546:
2543:
2540:
2539:
2534:
2532:
2529:
2525:
2524:
2519:
2515:
2510:
2509:
2505:
2499:
2497:0-262-23189-1
2493:
2489:
2484:
2480:
2478:0-262-23127-1
2474:
2470:
2466:
2462:
2452:on 2009-12-22
2448:
2444:
2437:
2432:
2429:
2426:
2421:
2419:0-19-507255-3
2415:
2408:
2407:
2401:
2399:
2395:
2391:
2390:1-4020-0604-7
2387:
2383:
2380:
2376:
2374:1-56881-025-3
2370:
2366:
2362:
2358:
2355:
2351:
2348:
2345:
2341:
2338:
2337:
2333:
2318:on 2021-07-28
2317:
2313:
2312:
2311:The God Proof
2304:
2301:
2296:
2289:
2286:
2281:
2277:
2273:
2269:
2262:
2259:
2251:
2247:
2246:
2238:
2231:
2228:
2223:
2217:
2213:
2206:
2203:
2195:
2191:
2190:
2182:
2175:
2172:
2160:
2156:
2149:
2146:
2141:
2140:
2135:
2129:
2126:
2118:
2114:
2110:
2102:
2095:
2092:
2080:
2076:
2070:
2067:
2062:
2056:
2048:
2044:
2040:
2036:
2029:
2026:
2023:
2017:
2013:
2009:
2005:
2001:
1997:
1991:
1988:
1983:
1979:
1972:
1969:
1961:
1957:
1953:
1949:
1945:
1944:
1936:
1932:
1926:
1923:
1919:
1902:
1896:
1893:
1884:
1878:
1858:
1835:
1829:
1826:
1817:
1811:
1799:
1797:0-19-507255-3
1793:
1786:
1785:
1777:
1774:
1769:
1763:
1759:
1754:
1753:
1747:
1743:
1737:
1734:
1728:
1725:
1719:
1716:
1708:
1701:
1697:
1691:
1689:
1687:
1683:
1678:
1676:0-19-507255-3
1672:
1665:
1664:
1655:
1652:
1646:
1643:
1637:
1634:
1628:
1625:
1619:
1616:
1610:
1607:
1601:
1598:
1594:
1588:
1585:
1578:
1568:
1565:
1559:
1556:
1550:
1547:
1528:
1522:
1514:
1508:
1505:
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1496:
1490:
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1457:
1441:
1438:
1432:
1429:
1421:
1417:
1401:
1395:
1392:
1384:
1368:
1365:
1359:
1349:
1346:
1342:
1338:
1335:
1329:
1326:
1322:
1318:
1296:
1290:
1281:
1275:
1269:
1263:
1243:
1240:
1237:
1217:
1210:
1209:partial order
1204:
1201:
1195:
1192:
1185:
1182:
1178:
1174:
1170:
1167:
1163:
1157:
1154:
1147:
1144:
1137:
1133:
1130:
1128:
1125:
1123:
1120:
1118:
1115:
1114:
1110:
1108:
1106:
1105:The God Proof
1101:
1099:
1098:
1093:
1086:In literature
1085:
1083:
1081:
1062:
1059:
1050:
1047:
1042:
1040:
1036:
1032:
1024:
1022:
1020:
1015:
1009:
1007:
1002:
1000:
995:
993:
988:
986:
982:
978:
974:
968:
961:
959:
939:
933:
929:
922:
905:
899:
879:
873:
869:
862:
856:
848:
842:
828:
822:
808:
800:
789:
783:
766:
760:
756:
745:
739:
724:
714:
708:
699:
693:
687:
680:
671:
665:
661:
657:
651:
645:
639:
631:
623:
606:
600:
596:
589:
572:
566:
546:
540:
531:
525:
519:
505:
499:
482:
476:
472:
465:
454:
448:
431:
425:
411:
402:
385:
379:
370:
360:
354:
345:
339:
333:
326:
322:
315:
309:
305:
282:
280:
278:
273:
269:
267:
263:
259:
255:
251:
247:
243:
239:
235:
231:
227:
223:
219:
215:
210:
208:
204:
199:
195:
191:
187:
183:
179:
175:
170:
168:
163:
161:
160:
155:
151:
147:
143:
139:
137:
132:
130:
125:
117:
115:
113:
109:
105:
101:
100:
95:
91:
86:
83:
79:
70:
65:
62:
59:
56:
55:
54:
51:
49:
45:
41:
37:
33:
29:
25:
21:
2536:
2521:
2487:
2468:
2454:. Retrieved
2447:the original
2405:
2364:
2353:
2343:
2320:, retrieved
2316:the original
2310:
2303:
2294:
2288:
2271:
2261:
2243:
2230:
2211:
2205:
2187:
2174:
2162:. Retrieved
2158:
2148:
2137:
2128:
2112:
2108:
2094:
2083:. Retrieved
2081:. 2017-04-26
2079:Magis Center
2078:
2069:
2055:cite journal
2038:
2034:
2028:
2007:
2003:
1998:(Oct 1996).
1990:
1981:
1971:
1947:
1941:
1933:(Jul 1990).
1925:
1803:
1783:
1776:
1751:
1736:
1727:
1718:
1707:the original
1698:(Jul 2005).
1662:
1654:
1645:
1636:
1627:
1618:
1609:
1600:
1592:
1587:
1567:
1558:
1549:
1507:
1498:
1489:
1476:
1463:
1455:
1415:
1382:
1348:
1340:
1336:
1333:
1328:
1320:
1203:
1194:
1184:
1176:
1172:
1161:
1156:
1146:
1104:
1102:
1095:
1091:
1089:
1051:
1043:
1028:
1010:
1003:
996:
989:
985:Robert Koons
969:
965:
286:
274:
270:
265:
261:
257:
253:
249:
245:
241:
237:
233:
229:
225:
221:
217:
213:
211:
206:
202:
197:
193:
189:
185:
177:
173:
171:
167:higher-order
164:
157:
153:
149:
135:
128:
121:
110:rather than
106:, following
97:
90:freethinkers
87:
74:
52:
19:
18:
2574:Modal logic
2159:Der Spiegel
1996:Graham Oppy
1544:, in Df.2.
1317:ultrafilter
1230:defined by
1173:attribution
1097:Hand of God
999:Graham Oppy
248:is said to
124:modal logic
104:pantheistic
2563:Categories
2553:ZFC axioms
2456:2010-08-31
2322:2021-03-25
2164:28 October
2085:2018-05-23
1579:References
1352:Formally,
981:accessible
154:contingent
136:contingent
78:Dana Scott
28:Kurt Gödel
2465:Wang, Hao
2280:2150-914X
1897:φ
1894:◻
1891:⇒
1879:φ
1830:φ
1827:◻
1824:⇒
1812:φ
1523:φ
1439:◊
1436:⇒
1430:◻
1399:⇒
1393:◊
1366:◻
1363:⇒
1291:ψ
1288:→
1276:φ
1267:∀
1264:◻
1244:ψ
1241:⪯
1238:φ
1218:⪯
1189:positive.
1177:privation
1169:aesthetic
1066:⊥
1063:◻
1060:◊
1041:'s book.
962:Criticism
927:∃
923:◻
874:φ
867:∃
863:◻
860:⇒
849:φ
843:φ
840:∀
836:⇔
797:⇒
767:φ
757:◻
753:⇒
746:φ
709:ψ
706:⇒
694:φ
685:∀
681:◻
678:⇒
666:ψ
658:ψ
655:∀
652:∧
640:φ
636:⇔
624:φ
594:∃
590:◊
541:φ
538:⇒
532:φ
520:φ
517:∀
513:⇔
477:φ
470:∃
466:◊
462:⇒
455:φ
432:φ
423:¬
419:⇔
412:φ
409:¬
386:ψ
376:⇒
355:ψ
352:⇒
340:φ
331:∀
327:◻
323:∧
316:φ
176:, either
150:necessary
129:necessary
2467:(1987).
2363:(1997).
2250:Archived
2194:Archived
2117:Archived
2035:Analysis
2020:—
2004:Analysis
1960:Archived
1804:"... If
1414:for all
1385:implies
1381:for all
1162:positive
1111:See also
214:essences
182:negation
159:possible
99:theistic
94:Hao Wang
2520:(ed.).
1758:241–261
1748:(ed.).
1321:Godlike
180:or its
162:truth.
118:Outline
112:Spinoza
108:Leibniz
71:History
44:reality
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2139:GitHub
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1127:Theism
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