Absoluteness of Intuitionistic Logic
Daniel Leivant
Abstract:
The present treatise is a corrected version of the author's Ph.D. dissertation, written at the University of Amsterdam in 1974/75 under the direction of Professor A.S. Troelstra. My research, as well as the production of the dissertation, were generously supported by the Mathematical Center, and as is customary for such dissertations, it now appears as a Mathematical Centre Tract.
The text that follows is divided into two parts. Part A deals with theories whose arithmetical fragment is part of IA* := Heytings arithmetic IA extended with transfinite induction over all recursive well-orderings. Such theories (as well as some others closely related to them) are named "regular". It is shown that fairly strong intuitionistic theories, and - in particular - the intuitionistic impredicative theory of types, are regular.
In part B we treat maximality (or "absoluteness") properties of intuitionistic (Heyting's) propositional andpredicate logic IL0 and IL] for regular theories. Here, L is said to be maximal (or "absolute") for T if
\forall L F[P1,...,Pk] => \forall T F[A1,...,Ak]
for some arithmetical relations Ai of the same arity as Pi (i=l,...,k). The maximality is uniform if the Ai's are independent of F. We are also interested in having the substituted relations Ai as low as possible in the arithmetical hierarchy.
Refined versions of the results of part A are incorporated in LEIVANT [A], while the theorems of part B together with several other maximality results are proven in LEIVANT [B]. Nevertheless, the present exposition might still be useful to the interested reader. In contrast to the aforementioned papers, we use here natural deduction systems, and the proofs, especially in part B, illustrate the convenience of using natural deduction to straightforwardly formalize one's intuitive ideas. The sections in part B motivating the proofs are particularly relevant here. The sequential calculi used in our [A] and [B] allow more succinct presentations, but at the cost of concealing to some extent the motivating ideas. Also, our exposition here is more leisurely, so that, in conjunction with the use made of natural deduction, the effect is to reduce the effort required from the reader.
To do justice to the reader, we should place the maximality results proven here amongst other similar results.
Since we prove "absoluteness" of IL, the interest in treating propositional logic ILO lies only in reducing the complexity of the substituted sentences (0-ary relations). D.H.J. de JONGH and C.SMORYNSKI [73] have proved that there exist uniform arithmetical substitutions, and also - locally - \Sigma_1^0 substitutions for IL0 and T=IA. Theorem I of part B below improves this by making the Sigma_1^0 substitutions depend only on the number of propositional letters in the schema F. However, by a uniformization lemma proved in LEIVANT [B]§l.2, already the local Sigma_1^0-absoluteness implies completely uniform absoluteness with Sigma_1^0 substitutions. From this,using metamathematical properties of IL0, one easily derives uniform absoluteness also with (binary) disjunctions of Pi_1^0 sentences as substitutions (idem, §1.6). Similar statements are also true when IA is replaced by any regular T, and also for T extended with either Church's Thesis CT0 or the Independence-of-Premiss Principle IPO (cf. TROELSTRA [73] for their statement). When Markov's Principle M is added, Sigma_1^0 absoluteness fails (since ⊢_{IA+M} ¬¬A->A for any Sigma_1^0 sentence A), but IL0 is uniformly Sigma_2^0 absolute for T + IPO+ M (cf. idem).
Turning to Intuitionistic Predicate Logic IL1, weshould start by mentioning a proof ofDE.JONGH [73] of a relativized version of absoluteness. Theorem II of part B below states the uniform Pi_2^0 absoluteness of IL1 for any regular theory T. IL1 is also uniformly \Sigma_2^0 absoluteness for such T (LEIVANT [B] §2.4) but not even locally Sigma_1^0 absolute for IA (LEIVANT [76]; this was also proved in §B.6 of the original version of the dissertation). Nevertheless, Sigma_1^0-absoluteness does hold for certain fragments of IL1 (LEIVANT [B] thm.2.VII) .
All theories mentioned above, for which IL1 is proved absolute, are r.e., and the "regular" onesare in IA*. Allowing for more complex substitutions, one obtainsin one stroke maximality of IL1, for all regular theories, with the substitutions independent also of the theory; namely - one proves the uniform maximality of IL for IA* (LEIVANT [B] thm.2.VIII). The complexity of the substitutionsmay be somewhat reduced for ILO.
For logic with equality, we have the \Pi_2^0 (and Sigma_2^0) uniform absoluteness, for regular theories, of IL1 extended with the following axioms:
\forall x \forall y y (x=y V ¬x=y)
\exists x_1 \exists x_2 ... \exists x_n [for 0