By Peter Cholak

This paintings explores the relationship among the lattice of recursively enumerable (r.e.) units and the r.e. Turing levels. Cholak offers a degree-theoretic process for developing either automorphisms of the lattice of r.e. units and isomorphisms among a number of substructures of the lattice. as well as delivering one other facts of Soare's Extension Theorem, this system is used to end up a set of recent effects, together with: each non recursive r.e. set is automorphic to a excessive r.e. set; and for each non recursive r.e. set $A$ and for each excessive r.e. measure h there's an r.e. set $B$ in h such that $A$ and $B$ shape isomorphic central filters within the lattice of r.e. units.

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**Example text**

We would, that is, like to offer some formal theory which would enable us to tell, in a given inferential context, when a ticket is Ticket entailment (T_) §6 43 available only as a ticket, and when available as an embarkation point from which another ticket might take us to a destination; or, less fancifully, when an entailment is available as a minor premiss, and when only as a major. " Indeed, most natural deduction systems are of no more help. But if we look at the question from the point of view of Fitch's subproof formulations, then the nesting of subproofs itself provides a natural and obvious formal analogue to the "ticketfact" hierarchy, as follows: (I) Recall that the distinction is to be contextual; we identify "inferential context" with "subordinate proof," expecting that what is "ticket" in one context (subproof) may be "fact" in another.

We therefore consider only the cases (i), (ii), and (iv). Case (i) is by --+E, just as before. _lkI and A--+Bb. But for FT~ we need both forms of transitivity. For suppose that max(a-Ikll :0: max(b). H->B yields Ticket entailment (:1'_) 46 Ch. E. And from the latter, again conforming to the restriction, we get H---+BcaUb)-{kl as required. H---+Bb, and then The reader may verify that neither form of transitivity will do the job of the other; so both forms of transitivity are required for case (ii).

A->C and without further proof we state the following THEOREM. A formula is provable in FT_ just in case it is provable in T_. The differences between T_ (contained in E_), E_ (contained in R_), and R_, suggest some philosophical remarks about modal logic, which in turn suggest a couple of formal problems. Proponents of the view that the two-valued propositional calculus and its quantificational extensions are the only systems of'logic worth any sane person's attention, harbor as it seems to us, two confusions.