Sound, totally sound, and unsound recursive equivalence types

• Published in: Annals of pure and applied logic Vol. 31; no. 1; pp. 1 - 20
• Main Author: Downey, R.G.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 1986; Elsevier; North-Holland
• Subjects: Exact sciences and technology; Logic and foundations; Mathematical logic, foundations, set theory; Mathematics; Recursion theory; Sciences and techniques of general use; Equivalence theorem; Mathematical logic; Set theory; Recursivity
• ISSN: 0168-0072
• DOI: 10.1016/0168-0072(86)90060-6

An independent subset I of V ∞ is called sound is I is contained in a r.e. independent set. If not, we shall call I unsound. We call an RET A totally sound if every independent set in A is sound. Clearly every RET containing an r.e. set is totally sound, and it had been suggested that the converse held: viz, every totally sound RET contained a recursive set. However, we show that there are 2 χ 0 sets { A π } such that if B ⩽ m A π then RET( B) is totally sound. In the co-r.e. case it is shown that if A is co-r.e. nonrecursive and nonisolic, then RET( A) is not totally sound. Indeed, RET( A) contains an independent set which is a basis of a subspace (of codimension 1 in V ∞), no basis of which is sound. This result is deduced from a construction of a new type of r.e. subspace of V ∞. That is, we show that if R is a fully co-r.e. nonrecursive subspace of V ∞ there existse an r.e. subspace V sich that V⊕ R = V ∞ and V has the property that if W is an r.e. subspace with W⊃ V, then dim( V ∞/ W) < ∞ implies W = V ∞. On the other hand , the co-simple isols provide a nice dichotomy. First, it is shown that (in each high r.e. degree) there is a co-simple totally sound RET. Second it is shown that if a is any nonzero r.e. degree, there exists a nonzero r.e. degree ⩽ T a such that b bounds no nontrivial totally sound co-r.e. RET. That is, if B is r.e. with Turing degree b and C is co-r.e. with C⩽ T B, then either (i) C is recursive or (ii) RET( C) is not totally sound. Finally, we extend our results to general algebraic settings; first to a class of Steinitz closure systems, and later to a general model-theoretic setting (without dependence).