# sher - The Logical Roots of Indeterminacy 101 The Logical...

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The Logical Roots of Indeterminacy GILA SHER I. Indeterminacy as Relativity to Logical Frameworks In 1915, Leopold Lowenheim proved a remarkable theorem: (L) If the domain is at least denumerably infinite, it is no longer the case that a first- order fleeing equation is satisfied for arbitrary values of the relative coefficients. (Lowenheim 1915, p. 235) In contemporary terminology the theorem says that if a formula @ of first-order logic with identity is finitely valid but not valid, then for every cardinal A 2 No, @ is not A-valid (i.e., if is satisfiable in an infinite model, then for every infinite cardinal A, is satisfiable in a model of cardinality A).' It follows from this theorem, Lowenheim pointed out, that "[all1 questions concerning the dependence or independence of Schroder's, Miiller's. or Huntington's class axioms are decidable (if at all) already in adenumerable domain." (19 15, p. 240). In a series of articles, Thoralf Skolem (1920, 1922, 1929, 1941, 1958) pre- sented a new version of Lowenheim's theorem and offered a new kind of proof for it. We can formulate Skolem's result as (LS) Let T be a countable 1st-order theory (where a theory is a set of 1st-order sentences). Then, if T has a model, T has a countable model; in particular: (i) T has a model in the natural numbers; (ii) If !?I is a rnodel of T, then there is a countable submodel 'LL' of 'LL, such that a' is a model of T. Skolem's theorem was extended by Tarski to (LST) Let T be a set of sentences in a language. L of cardinality K 2 No. Then, if T has an infinite model (a model with an infinite universe), T has a model of cardinality A for every A 1 K. Skolem regarded LS as signaling the'.unaioidable relativity of mathemati- cal notions to logical frameworks. Skolem's view is sometimes referred to as Skolem's paradox on the basis of passages such as this: So far as I know, no one has called attention to this peculiar and apparently paradoxical state of affairs. By virtue of the [set-theoretical] axioms we can prove the existence of higher cardinalities, of higher number classes, and so forth. How can it be, then, that the The Logical Roots of Indeterminacy 101 entire domain B [tk ~iverse of an "LS model" of set theory] can already beenumerated by means of the fin^;^ positive integers? (Skolem 1922, p. 295) However, the "pamdox" is swiftly explained away: The explanation is not difficult to find. In the axiomatization "set" does not mean an arbitrarily defined collection; the sets are nothing but objects that are connected with one another through certain relations expressed by the axioms. Hence there is no contradic- tion at all if a set M of the domain B is nondenumerable in the sense of the axiomatization; for this means merely that within B there occurs no one-to-one mapping 0 of M onto Zo (Zermelo's number sequence). Nevertheless there exists the possibility of numbering all objects in B, and therefore also the elements of M, by means of the positive integers; of course, such an enumeration too is a collection of certain pairs, but this collection is not a "set" (that is, it does not occur in the domain B). (Ibid.) LS

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sher - The Logical Roots of Indeterminacy 101 The Logical...

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