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Note that we could have written 0' ∈ s instead of [0', [0, 0']] ∈ s, since [1, [0, 1]] = (1+ ((0+1)2+0))2 + 1 = 5. e. ) The above definition of y = x! is an example of a definition by primitive recursion; we have a base clause 0! = 1 in which the function's value at zero is specified, and an induction clause (n+1)! ) in which the value at n+1 is specified in terms of its value at n. x. Here, the induction is carried out on the variable y; however the value of the function also depends on x, which is kept fixed while y varies.

Tarski's definition appeared when some logical positivists had expressed doubts about the possibility of a scientifically acceptable definition or theory of truth. Tarski showed that this way of defining satisfaction and hence truth existed, and that it had important uses in logic and mathematics. The second disjunct will be a formula true of two numbers a,s if a is an atomic formula of the form P13t1t2t3, where t1, t2 and t3 are terms of RE, and s is sufficient for and satisfies a: D2(a,s)=df.

N ≤ y)(∃p ≤ y)(y = [n, p] ∧ Digp (x, n, p)). x ∈ y is true of two numbers a,b if b codes a finite set S and a is a member of S. Note that Digp(m,n,p) and x ∈ y are formulae of Lim+. We could have carried out the construction in Lim, but it would have been more tedious, and would not have had any particular advantage for the purposes of this course. There are two special cases we should check to make sure our coding scheme works: namely, we should make sure that the sets {0} and Ø have codes. If y is not in the range of our pairing function, then x ∈ y will be false for all x; so y will code Ø.

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