They were followed by Tarski's undefinability theorem on the formal undefinability of truth, Church's proof that Hilbert's Entscheidungsproblem is unsolvable, and Turing's theorem that there is no algorithm to solve the halting problem.
|
|
The same applies to definability; see for example Tarski's undefinability theorem.
|
|
Gödel also discovered the undefinability theorem in 1930, while proving his incompleteness theorems published in 1931, and well before the 1933 publication of Tarski's work (Murawski 1998). While Gödel never published anything bearing on his independent discovery of undefinability, he did describe it in a 1931 letter to John von Neumann. Tarski had obtained almost all results of his 1933 monograph "Pojęcie Prawdy w Językach Nauk Dedukcyjnych" ("The Concept of Truth in the Languages of the Deductive Sciences") between 1929 and 1931, and spoke about them to Polish audiences. However, as he emphasized in the paper, the undefinability theorem was the only result he did not obtain earlier.
|
|
This result, known as Tarski's undefinability theorem, was discovered independently both by Gödel, when he was working on the proof of the incompleteness theorem, and by the theorem's namesake, Alfred Tarski.
|
|
But then Alfred Tarski's undefinability theorem of 1934 made it hopeless.Hintikka, "Logicism", in Philosophy of Mathematics (North Holland, 2009), pp 283–84. Some, including logical empiricist Carl Hempel, argued for its possibility, anyway. After all, nonEuclidean geometry had shown that even geometry's truth via axioms occurs among postulates, by definition unproved.
|
|
Yet Kurt Gödel's incompleteness theorem showed this impossible except in trivial cases, and Alfred Tarski's undefinability theorem shattered all hopes of reducing mathematics to logic. Thus, a universal language failed to stem from Carnap's 1934 work Logische Syntax der Sprache (Logical Syntax of Language). Still, some logical positivists, including Carl Hempel, continued support of logicism.
|
|
Uncertainty and indeterminacy are words for essentially the same concept in both quantum mechanics. Unquantifiability, and undefinability (or indefinability), can also sometimes be synonymous with indeterminacy. In science, indeterminacy can sometimes be interchangeable with unprovability or unpredictability. Also, anything entirely inobservable can be said to be indeterminate in that it cannot be precisely characterized.
|
|
Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that arithmetical truth cannot be defined in arithmetic. The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system.
|
|
For various syntactic properties (such as being a formula, being a sentence, etc.), these sets are computable. Moreover, any computable set of numbers can be defined by some arithmetical formula. For example, there are formulas in the language of arithmetic defining the set of codes for arithmetic sentences, and for provable arithmetic sentences. The undefinability theorem shows that this encoding cannot be done for semantic concepts such as truth.
|
|
It shows that no sufficiently rich interpreted language can represent its own semantics. A corollary is that any metalanguage capable of expressing the semantics of some object language must have expressive power exceeding that of the object language. The metalanguage includes primitive notions, axioms, and rules absent from the object language, so that there are theorems provable in the metalanguage not provable in the object language. The undefinability theorem is conventionally attributed to Alfred Tarski.
|
|
This last chapter is about a more ambitious project that Hofstadter started with student Gary McGraw. The microdomain used is that of grid fonts: typographic alphabets constructed using a rigid system of small rigid components. The goal is to construct a program that, given only a few or just one letter from the grid font, can generate the whole alphabet in the same style. The difficulty lies in the ambiguity and undefinability of style.
|
|
In mathematical logic, the diagonal lemma (also known as diagonalization lemma, self-reference lemma or fixed point theorem) establishes the existence of self-referential sentences in certain formal theories of the natural numbers—specifically those theories that are strong enough to represent all computable functions. The sentences whose existence is secured by the diagonal lemma can then, in turn, be used to prove fundamental limitative results such as Gödel's incompleteness theorems and Tarski's undefinability theorem.See Boolos and Jeffrey (2002, sec. 15) and Mendelson (1997, Prop.
|
|
If there were such a number then the theory would be inconsistent, contrary to the consistency hypothesis. So, under the assumption that the theory is consistent, there is no such number. It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "Q is the Gödel number of a false formula" cannot be represented as a formula of arithmetic. This result, known as Tarski's undefinability theorem, was discovered independently by Gödel (when he was working on the proof of the incompleteness theorem) and by Alfred Tarski.
|
|
However, that argument appears not to acknowledge the distinction between theorems of first-order logic and theorems of higher- order logic. The former can be proven using finistic methods, while the latter — in general — cannot. Tarski's undefinability theorem shows that Gödel numbering can be used to prove syntactical constructs, but not semantic assertions. Therefore, the claim that logicism remains a valid programme may commit one to holding that a system of proof based on the existence and properties of the natural numbers is less convincing than one based on some particular formal system.
|
|
The semantic conception of truth, which is related in different ways to both the correspondence and deflationary conceptions, is due to work by Polish logician Alfred Tarski. Tarski, in "On the Concept of Truth in Formal Languages" (1935), attempted to formulate a new theory of truth in order to resolve the liar paradox. In the course of this he made several metamathematical discoveries, most notably Tarski's undefinability theorem using the same formal technique Kurt Gödel used in his incompleteness theorems. Roughly, this states that a truth-predicate satisfying Convention T for the sentences of a given language cannot be defined within that language.
|
|
According to the footnote to the undefinability theorem (Twierdzenie I) of the 1933 monograph, the theorem and the sketch of the proof were added to the monograph only after the manuscript was sent to the printer in 1931. Tarski reports there that, when he presented the content of his monograph to the Warsaw Academy of Science on March 21, 1931, he expressed at this place only some conjectures, based partly on his own investigations and partly on Gödel's short report on the incompleteness theorems "Einige metamathematische Resultate über Entscheidungsdefinitheit und Widerspruchsfreiheit", Akademie der Wissenschaften in Wien, 1930.
|
|
Zero sharp was defined by Silver and Solovay as follows. Consider the language of set theory with extra constant symbols c1, c2, ... for each positive integer. Then 0# is defined to be the set of Gödel numbers of the true sentences about the constructible universe, with ci interpreted as the uncountable cardinal ℵi. (Here ℵi means ℵi in the full universe, not the constructible universe.) There is a subtlety about this definition: by Tarski's undefinability theorem it is not, in general, possible to define the truth of a formula of set theory in the language of set theory.
|
|
In September 1931, Ernst Zermelo wrote to Gödel to announce what he described as an "essential gap" in Gödel's argument (Dawson:76). In October, Gödel replied with a 10-page letter (Dawson:76, Grattan- Guinness:512-513), where he pointed out that Zermelo mistakenly assumed that the notion of truth in a system is definable in that system (which is not true in general by Tarski's undefinability theorem). But Zermelo did not relent and published his criticisms in print with "a rather scathing paragraph on his young competitor" (Grattan-Guinness:513). Gödel decided that to pursue the matter further was pointless, and Carnap agreed (Dawson:77).
|
|
Tarski's 1936 article "On the concept of logical consequence" argued that the conclusion of an argument will follow logically from its premises if and only if every model of the premises is a model of the conclusion. In 1937, he published a paper presenting clearly his views on the nature and purpose of the deductive method, and the role of logic in scientific studies. His high school and undergraduate teaching on logic and axiomatics culminated in a classic short text, published first in Polish, then in German translation, and finally in a 1941 English translation as Introduction to Logic and to the Methodology of Deductive Sciences. Tarski's 1969 "Truth and proof" considered both Gödel's incompleteness theorems and Tarski's undefinability theorem, and mulled over their consequences for the axiomatic method in mathematics.
|
|
Solovay constructed his model in two steps, starting with a model M of ZFC containing an inaccessible cardinal κ. The first step is to take a Levy collapse M[G] of M by adding a generic set G for the notion of forcing that collapses all cardinals less than κ to ω. Then M[G] is a model of ZFC with the property that every set of reals that is definable over a countable sequence of ordinals is Lebesgue measurable, and has the Baire and perfect set properties. (This includes all definable and projective sets of reals; however for reasons related to Tarski's undefinability theorem the notion of a definable set of reals cannot be defined in the language of set theory, while the notion of a set of reals definable over a countable sequence of ordinals can be.) The second step is to construct Solovay's model N as the class of all sets in M[G] that are hereditarily definable over a countable sequence of ordinals.
|
|