A coding theorem for isols

Extract

In [1] it is shown that for every sequence x = 〈x_n : n ∈ ω〉 ∈ X^ω Λ there is an isol x_ω (essentially an immunized product) such that

Here we have used the notation: Λ = the isols, ω = the nonnegative integers, p_n is the nth prime rational integer starting with p₀ = 2, ∣ denotes divisibility and ∤ its negation. If p is an arbitrary prime, p^y ∣ x, p^z ∣ x, and y < z then p^y+1 ∣ x. In particular since y ∈ ω is comparable with every element of Λ, the conditions p^y ∣ x and p^y+1 ∤ x uniquely determine y. Thus every sequence x ∈ X^ωω is uniquely determined by an x_ω 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 x ∈ X^ωΛ, 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.