Online research in philosophy

In this paper we generalize the pigeonhole principle by using isols as our fundamental counting tool.

We find necessary and sufficient conditions for compatibility of the Myhill and the Nerode extensions of a combinatorial function to the isols.

If T is an isol let D(T) be the least set of isols which contains T and is closed under predecessors and the application of almost recursive combinatorial functions. We find an infinite regressive isol T such that the universal theory (with respect to recursive relations and almost recursive combinatorial functions) of D(T) is the same as that of the nonnegative integers.

We give a direct mathematical proof of the Mathias-Silver theorem that every analytic set is Ramsey.

We examine the action of unary Δ 0 2 functions on the regressive isols. A manageable theory is produced and we find that such a function maps Λ R into Λ if and only if it is eventually $R\uparrow$ increasing and maps Λ R into Λ R if and only if it is eventually recursive increasing. Our paper concludes with a discussion of other methods for extending functions to Λ R.