![]() But now Tarski takes a further step: he offers a (partial) reconstruction of the notion of definability in purely mathematical terms. So Tarski sketches here, for the first time, a rigorous definition of definability in metamathematical terms - in particular, in terms of the metamathematical notion of satisfaction. So the object of Tarski's investigation is the system of reals with the primitive concepts 1, ≥, and +, or the system, the set of all positive reals, and the set of all reals x such that 0 ≤ x≤ 1 these sets are determined respectively by the sentential functions ' σ( x, x, x)', '¬ σ( x,x,x). The last three are special to the theory of real numbers, where the first-order variables range over reals, and ' v(x)' says that x =1, 'µ(x, y)' says th at ' x ≥ y', and '≥(x, y, z)' says that x + y = z. There are four primitive sentential functions denoted by '∈', ' v(x)', 'µ(x,y)'ε(x,y,z)'. The notion of a sentential function is introduced recursively. Tarski works within the framework of the simple theory of types (where the variables of the first order range over individuals, variables of the second order range over sets of individuals, and so on). Here I present an analogous problem concerning the term 'definable set of real numbers'. ![]() It was just such problems that the geometers solved when they established the meaning of the terms 'movement', 'line', 'surface', or 'dimension' for the first time. We then seek to construct a definition of this term which, while satisfying the requirements of methodological rigour, will also render adequately and precisely the actual meaning of the term. Our interest is directed towards a term of which we can given an account that is more or less precise in its intuitive content, but the meaning of which has not at present been rigorously established, at least in mathematics. ![]() "The problem set in this article belongs in principle to the type of problems which frequently occur in the course of mathematical investigations.
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