IΣ n
 andBΣ

 n
 are well known fragments of first-order arithmetic with induction and collection forΣ

 n
 formulas respectively;IΣ

 n
 0
 andBΣ

 n
 0
 are their second-order counterparts. RCA0 is the well known fragment of second-order arithmetic with recursive comprehension;WKL
0 isRCA
0 plus weak Konig's lemma. We first strengthen Harrington's conservation result by showing thatWKL
0 +BΣ

 0
 is Π
 1
 1
 -conservative overRCA
0 +BΣ

 0
 . Then we develop some model theory inWKL
0 and illustrate the use of formalized model theory by giving a relatively simple proof of the fact thatIΣ
1 provesBΣ

n+1 to be Π
n+2-conservative overIΣ

 n
 . Finally, we present a proof-theoretic proof of the stronger fact that theΠ

n+2 conservation result is provable already inIΔ
0 + superexp. ThusIΣ

n+1 proves 1-Con (BΣ

n+1) andIΔ
0 +superexp proves Con (IΣ

 n
 )↔Con(BΣ

n+1).

Proof Theory

On some formalized conservation results in arithmetic

We continue the earlier research of [1]. In particular, we work out a class of regular interstices and show that selective types are realized in regular interstices. We also show that, contrary to the situation above definable elements, the stabilizer of an element inside M(0) whose type is selective need not be maximal.

/pdf/on-regular-interstices-and-selective-types-in-countable-36s6948rg1.pdf

On regular interstices and selective types in countable arithmetically saturated models of Peano Arithmetic

In this paper we give an introduction to a technique for greatly increasing the efficiency of automated theorem provers for non-standard logics This technique takes advantage of the fact that while most important non-standard logics do not have finite characteristic models (in the sense that truth tables are a finite characteristic model for classical propositional logic), they do have finite models These models validate all the theorems of a given logic, though some non-theorems as well They invalidate the rest of the non-theorems Hence this technique involves using the models to direct a search by an automated theorem prover for a proof by filtering out or pruning the items of the search space which the models invalidate

Towards Efficient Knowledge-Based Automated Theorem Proving for Non-Standard Logics

This note arose out of my efforts to understand results of van den Dries, Denef, and Weispfenning on definable Skolem functions in the elementary theory of Q p . The first person to prove their existence was van den Dries, who devised and applied a model-theoretic criterion for theories, admitting elimination of quantifiers, which also admit definable Skolem functions [3]. The proof, though elegant, does not describe how one defines the Skolem functions. In the particular case of Q p , Denef found an ingenious, easily described method for writing out the definitions [2, pp. 14–15]. Unfortunately, his argument directly applies only in the following special case: if and there is a fixed m ≥ 1 such that for all , then can be given as a definable function of . While this special case includes many of interest, van den Dries' theorem seems more general. Weispfenning suggested how his results on primitive-recursive quantifier elimination could produce algorithms yielding definitions of Skolem functions in the specific theories van den Dries considered [10, pp. 470–471]. Though these algorithms provide a more concrete version of van den Dries' theorem, and do not suffer the lack of generality of Denef's result, Weispfenning's argument is extremely subtle and applies only to certain theories of valued fields.

A Note on Definable Skolem Functions

A type-theoretic framework for formal reasoning with different logical foundations is introduced and studied. With logic-enriched type theories formulated in a logical framework, it allows various logical systems such as classical logic as well as intuitionistic logic to be used effectively alongside inductive data types and type universes. This provides an adequate basis for wider applications of type theory based theorem proving technology. Two notions of set are introduced in the framework and used in two case studies of classical reasoning: a predicative one in the formalisation of Weyl's predicative mathematics and an impredicative one in the verification of security protocols.

/pdf/a-type-theoretic-framework-for-formal-reasoning-with-5906ykl36v.pdf

Proof Theory

Citations

On some formalized conservation results in arithmetic

On regular interstices and selective types in countable arithmetically saturated models of Peano Arithmetic

Towards Efficient Knowledge-Based Automated Theorem Proving for Non-Standard Logics

A Note on Definable Skolem Functions

A type-theoretic framework for formal reasoning with different logical foundations

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