Titlebook: Logic and Foundations of Mathematics; Selected Contributed Andrea Cantini,Ettore Casari,Pierluigi Minari Book 1999 Springer Science+Busines

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Logic and Foundations of Mathematics978-94-017-2109-7Series ISSN 0166-6991 Series E-ISSN 2542-8292

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https://doi.org/10.1007/978-94-017-2109-7Philosophy of Science; Rudolf Carnap; compactness theorem; logic; modal logic; model theory; predicate log

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The Use of a Logical Principle of Uniform Boundedness in Analysis,This paper is part of a sequence of papers ([9–12]) resulting from our Habilitation-thesis [8] adressing the following question:.What is the impact on the growth of extractable uniform bounds the use of various analytical principles Γ in a given proof of an ∀∃-sentence might have?

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Epsilon-Substitution Method for the Ramified Language and ,-Comprehension Rule,The epsilon substitution method is based on the language introduced by Hilbert [7]. The main non-Boolean construction of this language is .[.], read as “an . satisfying the condition .[.]”. When . is numerical, it is interpreted as the least . satisfying .[.]. Existential and universal quantifiers become explicitly definable by .

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On Applicative Theories,ey soon turned out to be very important for the proof-theoretic analysis of subsystems of second order arithmetic and set theory. Moreover, systems of explicit mathematics provide a logical framework for functional programming languages.

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On Predicate Logic as Modal Logic,. and . assignments in the universe ., i.e., elements of .., and . = .. means that for all variables . except possibly .(.) = .(.). A modal perspective on predicate logic starts from the observation that (1) is similar to the clause for diamonds in Kripke semantics:

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