30,856 research outputs found
Some observations on the logical foundations of inductive theorem proving
In this paper we study the logical foundations of automated inductive theorem
proving. To that aim we first develop a theoretical model that is centered
around the difficulty of finding induction axioms which are sufficient for
proving a goal.
Based on this model, we then analyze the following aspects: the choice of a
proof shape, the choice of an induction rule and the language of the induction
formula. In particular, using model-theoretic techniques, we clarify the
relationship between notions of inductiveness that have been considered in the
literature on automated inductive theorem proving. This is a corrected version
of the paper arXiv:1704.01930v5 published originally on Nov.~16, 2017
A Direct Version of Veldman's Proof of Open Induction on Cantor Space via Delimited Control Operators
First, we reconstruct Wim Veldman's result that Open Induction on Cantor
space can be derived from Double-negation Shift and Markov's Principle. In
doing this, we notice that one has to use a countable choice axiom in the proof
and that Markov's Principle is replaceable by slightly strengthening the
Double-negation Shift schema. We show that this strengthened version of
Double-negation Shift can nonetheless be derived in a constructive intermediate
logic based on delimited control operators, extended with axioms for
higher-type Heyting Arithmetic. We formalize the argument and thus obtain a
proof term that directly derives Open Induction on Cantor space by the shift
and reset delimited control operators of Danvy and Filinski
Reverse mathematics and equivalents of the axiom of choice
We study the reverse mathematics of countable analogues of several maximality
principles that are equivalent to the axiom of choice in set theory. Among
these are the principle asserting that every family of sets has a
-maximal subfamily with the finite intersection property and the
principle asserting that if is a property of finite character then every
set has a -maximal subset of which holds. We show that these
principles and their variations have a wide range of strengths in the context
of second-order arithmetic, from being equivalent to to being
weaker than and incomparable with . In
particular, we identify a choice principle that, modulo induction,
lies strictly below the atomic model theorem principle and
implies the omitting partial types principle
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