|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| '''Gödel's ontological proof''' is a formal argument for [[God]]'s existence by the mathematician [[Kurt Gödel]] (1906–1978).
| | If you have to speed up your PC then we have come to the appropriate region. I will show you, at the moment, five quick techniques to drastically enhance a computer's performance.<br><br>Many of the reliable companies can offer a full funds back guarantee. This means that you have the opportunity to get your cash back if you find the registry cleaning has not delivered what we expected.<br><br>H/w connected error handling - whenever hardware causes BSOD installing newest fixes for the hardware and/ or motherboard will enable. You could additionally add fresh hardware that is compatible with all the program.<br><br>There are strategies to make the slow computer work effective plus rapidly. In this article, I can tell we only 3 most effective strategies or techniques to avoid a computer of being slow and instead of which create it quicker and function even better than before.<br><br>In a word, to accelerate windows XP, Vista business, it's quite significant to disable some business products and clean and optimize the registry. You are able to follow the procedures above to disable unwanted programs. To optimize the registry, I suggest you utilize a [http://bestregistrycleanerfix.com/registry-reviver registry reviver] software. Because it's surprisingly dangerous for we to edit the registry by oneself.<br><br>Why this problem arises frequently? What are the causes of it? In truth, there are 3 major causes that will cause the PC freezing issue. To solve the issue, we have to take 3 procedures inside the following paragraphs.<br><br>Maybe we are asking why these windows XP error messages appear. Well, for we to be capable to understand the fix, you need to initially recognize where those errors come from. There is this software called registry. A registry is software that shops everything on a PC from a general configuration, setting, info, plus logs of activities from installing to UN-installing, saving to deleting, along with a lot more alterations you do inside a system pass through it plus gets 'tagged' and saved as a easy file for recovery purposes. Imagine it as a big recorder, a registrar, of all a records in your PC.<br><br>By changing the means we utilize the web you are able to have access more of the precious bandwidth. This will ultimately provide you a faster surfing experience. Here is a link to 3 techniques to personalize the PC speed found on the net. |
| | |
| It is in a line of development that goes back to [[Anselm of Canterbury]] (1033–1109). St. Anselm's [[ontological argument]], 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 [[reality]]. Therefore, God must exist." A more elaborate version was given by [[Gottfried Leibniz]] (1646–1716); this is the version that Gödel studied and attempted to clarify with his [[ontological argument]].
| |
| | |
| Gödel left a fourteen-point outline of his philosophical beliefs in his papers. Points relevant to the ontological proof include
| |
| | |
| :4. There are other worlds and rational beings of a different and higher kind.
| |
| :5. The world in which we live is not the only one in which we shall live or have lived.
| |
| :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.
| |
| :14. Religions are, for the most part, bad—but religion is not.<ref>Quoted in Wang 1996, p. 316. [http://cs.nyu.edu/kandathi/goedel_viewpoint.html "My philosophical viewpoint"], c. 1960, unpublished.</ref>
| |
| | |
| == History of Gödel's proof==
| |
| 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 [[Gottfried Wilhelm Leibniz|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".</ref>
| |
| | |
| ==Outline of Gödel's proof==
| |
| | |
| The proof uses [[modal logic]], which distinguishes between [[logical truth|''necessary'' truths]] and [[Contingency (philosophy)|''contingent'' truths]]. In the most common semantics for modal logic, many "possible worlds" are considered. 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 the planet is covered by water". If a statement happens to be true in our world, but is false in another world, 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.
| |
| | |
| 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 {{who?|date=June 2013}}.
| |
| | |
| To formalize the argument sketched above, the following definitions and axioms are needed:
| |
| | |
| * Definition 1: x is God-like if and only if x has as essential properties those and only those properties which are positive
| |
| * 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 [[logical consequence|entails]] B
| |
| * Definition 3: x necessarily exists if and only if every essence of x is necessarily exemplified
| |
| | |
| * Axiom 1: Any property entailed by—i.e., strictly implied by—a positive property is positive
| |
| * Axiom 2: If a property is positive, then its negation is not positive
| |
| * Axiom 3: The property of being God-like is positive
| |
| * Axiom 4: If a property is positive, then it is necessarily positive
| |
| * Axiom 5: Necessary existence is a positive property
| |
| | |
| Axiom 1 assumes that it is possible to single out ''positive'' properties from among all properties. Gödel comments that "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). Axioms 2, 3 and 4 can be summarized by saying that positive properties form a principal [[ultrafilter]].
| |
| | |
| From these axioms and definitions and a few other axioms from modal logic, the following theorems can be proved:
| |
| | |
| * Theorem 1: If a property is positive, then it is consistent, i.e., possibly exemplified.
| |
| * Corollary 1: The property of being God-like is consistent.
| |
| * Theorem 2: If something is God-like, then the property of being God-like is an essence of that thing.
| |
| * Theorem 3: Necessarily, the property of being God-like is exemplified.
| |
| | |
| Symbolically:
| |
| | |
| <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 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}
| |
| </math> | |
| | |
| There is an [https://github.com/FormalTheology/GoedelGod ongoing open-source effort to formalize Gödel's proof] using various theorem provers and proof assistants. The formalized proof of God's existence made headlines in German newspapers.<ref>{{cite news|last=Knight|first=David|title=Scientists Use Computer to Mathematically Prove Gödel's God Theorem|url=http://www.spiegel.de/international/germany/scientists-use-computer-to-mathematically-prove-goedel-god-theorem-a-928668.html|accessdate=28 October 2013|newspaper=Der Spiegel|date=23 October 2013}}</ref>
| |
| | |
| ==Criticism of Gödel's proof==
| |
| | |
| 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">{{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>
| |
| | |
| [[C. Anthony Anderson]], a contemporary American philosopher, 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.
| |
| | |
| ==See also==
| |
| * [[Absolute Infinite]]
| |
| * [[Existence of God]]
| |
| * [[Philosophy of religion]]
| |
| * [[Synthetic proposition]]
| |
| * [[Theism]]
| |
| * [[Ontological argument]]
| |
| | |
| ==Notes==
| |
| {{Reflist}}
| |
| | |
| ==References==
| |
| * {{cite book |author=John W. Dawson, Jr |title=Logical Dilemmas: The Life and Work of Kurt Godel |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. 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. 361–377, September 1998
| |
| * Jordan Howard Sobel, "Gödel's Ontological Proof" in ''On Being and Saying. Essays for [[Richard Cartwright (philosopher)|Richard Cartwright]],'' ed. [[Judith Jarvis Thomson]] (MIT press, 1987)
| |
| * {{cite book |author=Wang, Hao |title=Reflections on Kurt Gödel |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]]}}
| |
| | |
| ==External links==
| |
| * [http://www.stats.uwaterloo.ca/~cgsmall/ontology.html Kurt Gödel's Ontological Argument]
| |
| *{{sep entry|ontological-arguments|Ontological arguments|Graham Oppy}}
| |
| * [http://www.ontology.co/ontological-proof-contemporary.htm Annotated bibliography of studies on Gödel's Ontological Argument]
| |
| | |
| {{DEFAULTSORT:Godels Ontological Proof}}
| |
| [[Category:Christian philosophy]]
| |
| [[Category:Modal logic]]
| |
| [[Category:Arguments for the existence of God]]
| |
| [[Category:Theology]]
| |
If you have to speed up your PC then we have come to the appropriate region. I will show you, at the moment, five quick techniques to drastically enhance a computer's performance.
Many of the reliable companies can offer a full funds back guarantee. This means that you have the opportunity to get your cash back if you find the registry cleaning has not delivered what we expected.
H/w connected error handling - whenever hardware causes BSOD installing newest fixes for the hardware and/ or motherboard will enable. You could additionally add fresh hardware that is compatible with all the program.
There are strategies to make the slow computer work effective plus rapidly. In this article, I can tell we only 3 most effective strategies or techniques to avoid a computer of being slow and instead of which create it quicker and function even better than before.
In a word, to accelerate windows XP, Vista business, it's quite significant to disable some business products and clean and optimize the registry. You are able to follow the procedures above to disable unwanted programs. To optimize the registry, I suggest you utilize a registry reviver software. Because it's surprisingly dangerous for we to edit the registry by oneself.
Why this problem arises frequently? What are the causes of it? In truth, there are 3 major causes that will cause the PC freezing issue. To solve the issue, we have to take 3 procedures inside the following paragraphs.
Maybe we are asking why these windows XP error messages appear. Well, for we to be capable to understand the fix, you need to initially recognize where those errors come from. There is this software called registry. A registry is software that shops everything on a PC from a general configuration, setting, info, plus logs of activities from installing to UN-installing, saving to deleting, along with a lot more alterations you do inside a system pass through it plus gets 'tagged' and saved as a easy file for recovery purposes. Imagine it as a big recorder, a registrar, of all a records in your PC.
By changing the means we utilize the web you are able to have access more of the precious bandwidth. This will ultimately provide you a faster surfing experience. Here is a link to 3 techniques to personalize the PC speed found on the net.