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ConcCalcMIT110409

# ConcCalcMIT110409 - CONCEPT CALCULUS by Harvey M Friedman...

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Unformatted text preview: CONCEPT CALCULUS by Harvey M. Friedman Ohio State University [email protected] http://www.math.ohio-state.edu/%7Efriedman/ Department of Philosophy MIT 1-3PM November 4, 2009 We have discovered an unexpectedly close connection between the logic of mathematical concepts and the logic of informal concepts from common sense thinking. Our results indicate that they are, in a certain precise sense, the same. This connection is new and there is the promise of establishing similar connections involving a very wide range of informal concepts. We call this development the Concept Calculus . We begin with some background concerning the crucial notion of interpretation between theories that is used to state results in Concept Calculus. We then give a survey of major results in Concept Calculus. In particular, we establish the mutual interpretability of formal systems for set theory and formal systems for a variety of informal concepts from common sense thinking . INTERPRETATION POWER The notion of interpretation plays a crucial role in Concept Calculus. Interpretability between formal systems was first precisely defined by Alfred Tarski. We work in the usual framework of first order predicate calculus with equality. An interpretation of S in T consists of • A one place relation defined in T which is meant to carve out the domain of objects that S is referring to, from the point of view of T. • A definition of the constants, relations, and functions in the language of S by formulas in the language of T, whose free variables are restricted to the domain of objects that S is referring to (in the sense of the previous bullet). • It is required that every axiom of S, when translated into the language of T by means of i,ii, becomes a theorem of T. In ii, we usually allow that the equality relation in S need not be interpreted as equality – but rather as an equivalence relation. INTERPRETATION POWER CAUTION : Interpretations do not necessarily preserve truth. They only preserve provability. We give two illustrative examples. Let S consist of the axioms for strict linear order together with “there is a least element”. • ¬(x < x) • x < y ∧ y < z 㱺 x < z. • x < y ∨ y < x ∨ x = y. • ( ∃ x)( ∀ y)(x < y ∨ x = y). Let T consist of the axioms for strict linear order together with “there is a greatest element”. I.e., • ¬(x < x) • x < y ∧ y < z 㱺 x < z. • x < y ∨ y < x ∨ x = y. • ( ∃ x)( ∀ y)(y < x ∨ y = x). INTERPRETATION POWER S • ¬(x < x) • x < y ∧ y < z 㱺 x < z. • x < y ∨ y < x ∨ x = y. • ( ∃ x)( ∀ y)(x < y ∨ x = y). T • ¬(x < x) • x < y ∧ y < z 㱺 x < z. • x < y ∨ y < x ∨ x = y....
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ConcCalcMIT110409 - CONCEPT CALCULUS by Harvey M Friedman...

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