Economics Dictionary of ArgumentsHome
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| Kurt Gödel: Kurt Gödel (1906 – 1978) was a logician, mathematician, and philosopher. He is best known for his incompleteness theorems, which show that within any axiomatic system powerful enough to express basic arithmetic, there will always be statements that can neither be proven nor disproven within that system. Major works are "On Formally Undecidable Propositions of Principia Mathematica and Related Systems" (1931), "Consistency-Proof for the Generally Covariant Gravitational Field Equations" (1939), "What is Cantor's Continuum Problem?" (1947), "Russell's Mathematical Logic" (1951), "On Undecidable Propositions of Formal Mathematical Systems" (1956). See also Incompleteness, Completeness, Proofs, Provability.
_____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. | |||
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W.V.O. Quine on Goedel - Dictionary of Arguments
XIII 82 Goedel/Goedel Theorem/Quine: Evidence/Self-Evidence/Quine: it is too much to ask that a proof should be self-evident. E.g. Euclid's parallel axiom is not self-evident. E.g. set theory is also not self-evident because it is shaken by paradoxes. Self-Evidence/Quine: we find it in a small number of axioms of number theory. They are the axioms of Dedekind, which are called the axioms of Peano. Elementary Number Theory/Quine: there was always the question whether there were still valid laws that could not be derived from the axioms. They existed! That was a question of adequacy. Laws/Quine: the question of further, still undiscovered laws seemed to be a problem of all branches of mathematics. By supplementing the axioms, perhaps this could be remedied? But Goedel proved in 1931 that this cannot be done! Goedel/Quine: proved that there can be no complete deductive system for even the smallest fragment of mathematics, such as Elementary Number Theory. XIII 82 Tendency: Goedel/Quine: proved that there can be no complete deductive system for even the smallest fragment of mathematics, such as the elementary number theory. Def Elementary Number Theory/Quine: includes digits, notation for plus, times, power and equality. >Numbers/Quine. XIII 83 Sentence operators: for "not", "and" and "or" and the quantifiers "Each number x is such that..." and "there is a number x so that...". The numbers are the positive integers and the zero. With this you can express e.g. Fermat's last theorem. Goedel/Quine: Thesis: No axiom system or other deductive apparatus can cover all truths that can be expressed even in this most moderate notation. Any valid proof procedure will disregard some true sentences, even infinitely many of them. Self-Evidence/Mathematics/Goedel/Quine: therefore we must drop the requirement of self-evidence. Wrong solution/Quine: could one not simply take all discovered truths as axioms? Vs: this is not impossible because there could be no axiom system with infinitely many axioms - which exist. Rather, it is the case that a proof must be able to be examined in finite time. Goedel/Goedel's Theorem/Quine: is related to the reflexive paradoxes. The point is that the notation of the elementary number theory must be able to speak about itself. ((s) Self-Reference). Goedel Numbering/Goedel Number/Quine: ...+... XIII 84 Mention/Use/Goedel/Quine: Goedel's evidence also requires this distinction. For example, the digit "6" names the number 6 and has the Goedel number 47. We can say that the Goedel number 47 names the number 6. Syntax/Arithmetic/Goedel/Quine: after all expressions have their naming by Goedel numbers, the syntactic operations can be mirrored by expressions, by arithmetic operations via numbers. Quote/Goedel/Quine: Problem: the corresponding notation is not part of symbolic logic and arithmetic. Quotation marks cannot be simply named by Goedel numbers. Quote/Quine: of an expression: names this expression. Goedel Numbers/Goedel number/Quine: 47 names 6, furthermore 5361 names 47 if 53 and 61 are randomly the Goedel numbers of the digits "4" and "7". ((s) Quotation marks sic). Quote/Goedel/Quine: the quote relation is represented as by the arithmetic relation that has 5361 to 47 and 47 to 6. The general relation can be expressed in the notation of the elementary number theory, though not easily. The arithmetic reconstruction of syntactic concepts like this was a substantial part of Goedel's work. Liar/Liar's Paradox/Goedel/Quine: is useful in one of the two parts where Goedel's proof can be split. The bomb explodes when the two parts are put together. The liar can be completely XIII 85 expressed by Goedel numbering with the exception of a single expression: "truth". If that could be done, we would have solved the paradox, but discredited the elementary number theory. Truth/Goedel Number/Goedel Number/Quine: truth is not definable by Goedel numbers, within the elementary number theory. >Goedel Numbers/Quine. Goedel's Theorem/Quine: formal: no formula in the notation of the elementary number theory is true of all and only the Goedel numbers of truths of the elementary number theory. (This is the one part). Other part/Quine: deals with every real evidence procedure, here it is about that every evidence must be testable. Formal: a given formula in the notation of the elementary number theory is true of all and only the Goedel numbers of provable formulas. Church/Quine: here I skip his thesis (Church-Thesis), (see recursion below). Goedel/Quine: the two parts together say that the provable formulas do not coincide with the truths of the elementary number theory. Either they contain some falsehoods, or they do not cover some truths. God forbids that. Goedel/Quine: his own proof was more direct. He showed that a given sentence, expressed in Goedel numbers, cannot be proved. Either it is false or provable, or true and not provable. Probably the latter. Wrong solution/Quine: one could add this lost truth as an axiom, but then again others remain unprovable. Goedel/N.B./Quine: ironically, it was implausible that there could be a proof procedure for all truths of the elementary number theory. This would clarify Fermat's theorem, and much more. XIII 86 On the other hand, Goedel's result hit him like a bomb. N.B.: these two shortcomings turned out to be equivalent! Because: Kleene/Quine: showed that if there is a complete evidence procedure, any statement could be tested as true or false as follows: a computer would have to be programmed to rewind any statement, in alphabetical order, the shortest first, then always longer. In the end, because of the completeness of the procedure, he will have proved or refuted every single sentence._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition. |
Quine I W.V.O. Quine Word and Object, Cambridge/MA 1960 German Edition: Wort und Gegenstand Stuttgart 1980 Quine II W.V.O. Quine Theories and Things, Cambridge/MA 1986 German Edition: Theorien und Dinge Frankfurt 1985 Quine III W.V.O. Quine Methods of Logic, 4th edition Cambridge/MA 1982 German Edition: Grundzüge der Logik Frankfurt 1978 Quine V W.V.O. Quine The Roots of Reference, La Salle/Illinois 1974 German Edition: Die Wurzeln der Referenz Frankfurt 1989 Quine VI W.V.O. Quine Pursuit of Truth, Cambridge/MA 1992 German Edition: Unterwegs zur Wahrheit Paderborn 1995 Quine VII W.V.O. Quine From a logical point of view Cambridge, Mass. 1953 Quine VII (a) W. V. A. Quine On what there is In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (b) W. V. A. Quine Two dogmas of empiricism In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (c) W. V. A. Quine The problem of meaning in linguistics In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (d) W. V. A. Quine Identity, ostension and hypostasis In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (e) W. V. A. Quine New foundations for mathematical logic In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (f) W. V. A. Quine Logic and the reification of universals In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (g) W. V. A. Quine Notes on the theory of reference In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (h) W. V. A. Quine Reference and modality In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (i) W. V. A. Quine Meaning and existential inference In From a Logical Point of View, , Cambridge, MA 1953 Quine VIII W.V.O. Quine Designation and Existence, in: The Journal of Philosophy 36 (1939) German Edition: Bezeichnung und Referenz In Zur Philosophie der idealen Sprache, J. Sinnreich (Hg), München 1982 Quine IX W.V.O. Quine Set Theory and its Logic, Cambridge/MA 1963 German Edition: Mengenlehre und ihre Logik Wiesbaden 1967 Quine X W.V.O. Quine The Philosophy of Logic, Cambridge/MA 1970, 1986 German Edition: Philosophie der Logik Bamberg 2005 Quine XII W.V.O. Quine Ontological Relativity and Other Essays, New York 1969 German Edition: Ontologische Relativität Frankfurt 2003 Quine XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |
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