[permalink] [id link]
Tarski's undefinability theorem ( general form ): Let ( L, N ) be any interpreted formal language which includes negation and has a Gödel numbering g ( x ) such that for every L-formula A ( x ) there is a formula B such that B ↔ A ( g ( B )) holds.
from
Wikipedia
Some Related Sentences
Tarski's and undefinability
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.
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.
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.
* Tarski's undefinability theorem ( Gödel and Tarski in the 1930s )
* Tarski's undefinability theorem
Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1936, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics.
Gödel also discovered the undefinability theorem in 1930, while proving his incompleteness theorems published in 1931, and well before the 1936 publication of Tarski's work ( Murawski 1998 ).
Tarski's undefinability theorem: There is no L-formula True ( n ) which defines T *.
The proof of Tarski's undefinability theorem in this form is again by reductio ad absurdum.
Smullyan ( 1991, 2001 ) has argued forcefully that Tarski's undefinability theorem deserves much of the attention garnered by Gödel's incompleteness theorems.
# REDIRECT Tarski's undefinability theorem
In the course of this he made several metamathematical discoveries, most notably Tarski's undefinability theorem using the same formal technique as Kurt Gödel used in his incompleteness theorems.
# REDIRECT Tarski's undefinability theorem
# REDIRECT Tarski's undefinability theorem
# REDIRECT Tarski's undefinability theorem
Tarski's and theorem
** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.
Examples of early results from model theory applied to fields are Tarski's elimination of quantifiers for real closed fields, Ax's theorem on pseudo-finite fields, and Robinson's development of non-standard analysis.
This is closely related to Tarski's indefinability theorem.
It was Tarski who stated the result in its most general form, and so the theorem is often known as Tarski's fixed point theorem.
* Tarski's indefinability theorem
Ultimately, the reason for this weakening is that whereas the model-theoretic satisfaction relation can be defined, truth itself cannot, due to Tarski's theorem.
In mathematics, the Tarski's theorem, proved by Alfred Tarski, states that in ZF the theorem " For every infinite set A, there is a bijective map between the sets A and A × A " implies the axiom of choice.
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 indefinability theorem.
* Tarski's theorem
We will first state a simplified version of Tarski's theorem, then state and prove in the next section the theorem Tarski actually proved in 1936.
A semantic proof of Tarski's theorem from Post's theorem is obtained by reductio ad absurdum as follows.
Tarski's and form
A variant of redundancy theory is the disquotational theory which uses a modified form of Tarski's schema: To say that '" P " is true ' is to say that P. A version of this theory was defended by C. J. F. Williams in his book What is Truth ?.
") Tarski's material adequacy condition, or Convention T, is: a definition of truth for an object language implies all instances of the sentential form
Tarski's material adequacy condition, also known as Convention T, holds that any viable theory of truth must entail, for every sentence P of a language, a sentence of the form ( T ):
Tarski's system has the unusual property that all sentences can be written in universal-existential form, a special case of the prenex normal form.
A variant of redundancy theory is the disquotational theory, which uses a modified form of Tarski's schema: To say that "' P ' is true " is to say that P. Yet another version of deflationism is the prosentential theory of truth, first developed by Dorothy Grover, Joseph Camp, and Nuel Belnap as an elaboration of Ramsey's claims.
Tarski's and Let
* Let us use the term “ deductive system ” as a set of sentences closed under consequence ( for defining notion of consequence, let us use e. g. Tarski's algebraic approach ).
Tarski's and L
For a language L containing ¬ (" not "), ∧ (" and "), ∨ (" or ") and quantifiers (∀ " for all " and ∃ " there exists "), Tarski's inductive definition of truth looks like this:
Tarski's and be
* Tarski's axioms: Alfred Tarski ( 1902 – 1983 ) and his students defined elementary Euclidean geometry as the geometry that can be expressed in first-order logic and does not depend on set theory for its logical basis, in contrast to Hilbert's axioms, which involve point sets.
However, the methods developed by Frege and Tarski for the study of mathematical language have been extended greatly by Tarski's student Richard Montague and other linguists working in formal semantics to show that the distinction between mathematical language and natural language may not be as great as it seems.
Unlike Tarski's approach, however, Kripke's lets " truth " be the union of all of these definition-stages ; after a denumerable infinity of steps the language reaches a " fixed point " such that using Kripke's method to expand the truth-predicate does not change the language any further.
Unlike the generalized solution to Tarski's circle-squaring problem, the axiom of choice is not required for the proof, and the decomposition and reassembly can actually be carried out " physically "; the pieces can, in theory, be cut with scissors from paper and reassembled by hand.
Tarski's theorem, on the other hand, is not directly about mathematics but about the inherent limitations of any formal language sufficiently expressive to be of real interest.
Tarski's basic goal was to provide a rigorously logical definition of the expression " true sentence " within a specific formal language and to clarify the fundamental conditions of material adequacy that would have to be met by any definition of the truth-predicate.
It is a rather controversial matter whether Tarski's semantic theory should be counted as either a correspondence theory or as a deflationary theory.
Hilbert's axioms, unlike Tarski's axioms, do not constitute a first-order theory because the axioms V. 1 – 2 cannot be expressed in first-order logic.
Another type of logics where Tarski's method is inapplicable are relevance logics, because given two theorems an implication from one to the other may not itself be a theorem in a relevance logic.
The T-schema or truth schema ( not to be confused with ' Convention T ') is used to give an inductive definition of truth which lies at the heart of any realisation of Alfred Tarski's semantic theory of truth.
A consequence of this axiom is that Alfred Tarski's proof of the decidability of the ordered real field could be seen as an algorithm to solve any problem in Euclidean geometry.
In fact they only needed the real version of the conjecture, defined below, to prove this result, which would be a positive solution to Tarski's exponential function problem.
Satisfaction is inspired by Tarski's truth definition, but can in fact be any binary relation.
This schema is indispensable ; Euclidean geometry in Tarski's ( or equivalent ) language cannot be finitely axiomatized as a first-order theory.
0.171 seconds.