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Alfred Tarski (1901-1983)

Polish-American logician who created the field called "model theory" or semantics. Probably his best known work is "The concept of truth in formalized languages" (German version, 1936). In this, he tried to give "a materially adequate and formally correct definition of the term 'true sentence'", and obtained an important result closely related with Goedel's incompleteness theorem. That is, under very general assumptions, the set of sentences which are true in a model is not definable in that model. If you replace the concept "truth" by that of "provability", this result easily turns into Goedel's theorem. As is well known, these results are closely related with the old "liar paradox".

Tarski's work on truth gave rise to the discipline called "model theory", which relates formal symbols to set-theoretical structures and enables us to speak of the reference of a symbol, the truth-value of a sentence, or the meaning of formal expressions in general. Tarski taught at the University of California at Berkeley since 1942, and that was quite conducive to raising many logicians in the West-coast, and the model theory became one of the major fields of logic.

Tarski's theory of truth had a great impact on philosophy. For instance, philosophers such as Carnap and Popper were strongly influenced by that; and more recently, following Kripke's "Outline of a Theory of Truth" (1975) , various new theories of truth appeared, as an attempt to overcome some alleged "defects" contained in Tarski's original theory.

Recently, Banach-Tarski's Paradox is also discussed by mathematicians. This originates from their 1924 paper, and what is "paradoxical" is this. Suppose we have two spheres of different size (say, one is like a golf ball, and another is like the sun!). Then, by decomposing one sphere into finite parts, and re-assembling them in an appropriate manner, you can change them into another sphere. This is a theorem, however it may appear paradoxical!

Tarski's Theorem


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Last modified Dec. 15, 2008. (c) Soshichi Uchii