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A coding theorem for isols

  • Erik Ellentuck (a1)

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In [1] it is shown that for every sequence x = 〈xn : nω〉 ∈ Xω Λ there is an isol xω (essentially an immunized product) such that

Here we have used the notation: Λ = the isols, ω = the nonnegative integers, pn is the nth prime rational integer starting with p0 = 2, ∣ denotes divisibility and ∤ its negation. If p is an arbitrary prime, pyx, pzx, and y < z then py+1x. In particular since yω is comparable with every element of Λ, the conditions pyx and py+1x uniquely determine y. Thus every sequence xXωω is uniquely determined by an xω satisfying (1) and consequently may be used as a “code” for that sequence. In Theorem 1 it is shown that (1) does not uniquely determine the values of an arbitrary sequence xXωΛ, however in Theorem 3 we find a different scheme which does. At the very end of the paper we give some reasons why coding theorems are useful. It should also be mentioned that for a coding theorem to be meaningful it is necessary to restrict the operations by which a sequence can be recaptured from its code. Otherwise a triviality results. Our coding theorem will allow all operations which are first order definable in Λ with respect to addition, multiplication, and exponentiation. We conjecture that the latter operation is really necessary.

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[1]Dekker, J. C. E. and Myhill, J., The divisibility of isols by powers of primes, Mathematische Zeitschrift, vol. 73 (1960), pp. 127133.
[2]Dekker, J. C. E. and Myhill, J., Recursive equivalence types, University of California Publications in Mathematics (N.S.), vol. 3 (1960), pp. 67214.
[3]Ellentuck, E., Universal isols, Mathematische Zeitschrift, vol. 98 (1967), pp. 18.
[4]Fuhrken, G., Skolem-type normal forms for first order languages with a generalized quantifier, Fundamenta mathematicae, vol. 54 (1964), pp. 291302.
[5]Nerode, A., Extensions to isols, Annals of mathematics, vol. 73 (1961), pp. 362403.
[6]Nerode, A., Non-linear combinatorial functions of isols, Mathematische Zeitschrift, vol. 86 (1965), pp. 410424.

A coding theorem for isols

  • Erik Ellentuck (a1)

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