Embeddings of Heyting algebras, revised version

Abstract

It turns out that the fact that the Heyting algebra of Heyting's Arithmetic is RE, nonrecursive, is an immediate corollary of the results of the earlier preprint. This corollary is now worked into the text. A secondary improvement is that the proof of 3.7 is made more accessible.

In this paper we study embeddings of Heyting algebras (Ha's). It is pointed out that such embeddings are naturally connected with Derived Rules. We consider the Has embeddable in the Ha of the Intuitionistic Propositional Calculus (IPC), i.e. the free Ha on хo generators, those embeddable in the Ha of Heyting's arithmetic (HA) and those embeddable in the Ha of HA*, a 'natural' extension of HA. We prove the following theorems. The same Has on finitely many generators are embeddable in the Ha of IPC and in the Ha of Boolean (or: Brouwerean) combinations of Σ-sentences of HA. The Has on finitely many generators embeddable in the Ha of IPC are finitely axiomatizable. There is a non-recursive Ha on three generators, that can be embedded in the Ha of HA. Every recursively enumerable prime Ha is embeddable in the Ha of HA*.