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• W2013092827 abstract "In his paper, 'Constructivity, Consistency and Natural Languages Tucker seeks to define both the notion of a constructive procedure and the notion of a non-constructive procedure. 'Constructive procedures are defined as procedures which can, in some sense, be carried out whereas non-constructive procedures are those which, while they can (in some sense) be specified, cannot be carried out' (p. I52). Under this definition of constructive procedure Tucker claims that some procedures that have been regarded in the past as non-constructive are constructive according to the new definition (p. I53). It is the purpose of this discussion note to show that this claim is ill-founded. His first example is the diagonal procedure. He claims that this procedure has been regarded as non-constructive in the past but is in fact constructive. He supports his claim by giving an interpretation of the term 'indenumerable set'. 'Taken in the referential sense, the expression 'indenumerable set' means an actual infinity which is greater than an actual denumerable infinity' (p. I56). But there is another interpretation open to us, namely, 'a non-referential interpretation in which it means a set which contains an indenumerable element; where by an indenumerable element is meant an element which differs systematically from each element in an unending series whose generative recipe is given' (p. I56). It is difficult to make sense of this interpretation. What is needed is a clarification of the terms 'indenumerable element' and the term on which it depends, 'generative recipe'. From a later exposition that Tucker gives (p. I59) it appears that D is an indenumerable element of S if D belongs to S and is different from each element of S given by some law determining an initial element and the successor of any element. Under this definition it will turn out that many sets thought of as denumerable will be indenumerable. (Perhaps both denumerable and indenumerable, but Tucker does not define denumerable.) Even the set of natural numbers would become indenumerable. For the generative recipe-initial element 3, successor of an element x, x + i-will give two indenumerable elements, i and 2. i and 2 belong to the set of natural numbers and yet differ from each of the elements given by the generative recipe. That the set of natural numbers is non-denumerable is an absurd consequence and makes nonsense of the distinction initially brought in by Cantor. It might be said that my example ignores the fact that there is a generative recipe for the natural numbers and if I had taken this recipe then I should not have succeeded in obtaining an indenumerable element. In general this would mean that there is only one proper generative recipe" @default.
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• W2013092827 date "1974-01-01" @default.
• W2013092827 modified "2023-09-25" @default.
• W2013092827 title "Constructivity — A Defence and an Attack" @default.
• W2013092827 doi "https://doi.org/10.1093/mind/lxxxiii.330.263" @default.
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