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Books > Science & Mathematics > Mathematics > Mathematical foundations > Set theory
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Logic of Arithmetic (Hardcover)
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Logic of Arithmetic (Hardcover)
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For propositional logic it can be decided whether a formula has a
deduction from a finite set of other formulas. The present volume
begins with a method to decide this for the quantified formulas of
those fragments of arithmetic which express the properties of
order-plus-successor and or order-plus-addition (Presburger
arithmetic); it makes use of an algorithm eliminating quantifiers
which, in turn, is also applied to obtain consistency proofs for
these fragments. Stronger fragments of arithmetic, also containing
multiplication, are sufficiently rich to express a primitive
recursive encoding of terms, formulas and deductions, and this
leads to Godel's theorem exhibiting statements already undecidable
in these fragments. Its central idea, isolated in Tarski's fixpoint
lemma, has a certain analogy with Eubulides' antinomy of the Liar,
and in a non-technical chapter, accessible to a wider class of
readers, this analogy is exploited for an informal discussion of
undefinability and incompleteness. The technical tools required to
verify the hypotheses on arithmetical representability, on the
other hand, are collected in an independent presentation of
recursive functions and relations.
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