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Step By Recursive Step: Church's Analysis Of Effective Calculability
 BULLETIN OF SYMBOLIC LOGIC
, 1997
"... Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, wa ..."
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Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of G odel's general recursiveness, not his own #definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of e#ective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the char...
ITERATED DEFINABILITY, LAWLESS SEQUENCES AND BROUWER’S CONTINUUM
"... Abstract. The research on which this article is based was motivated by the wish to find a model of Kreisel’s lawless sequence axioms in which the lawlike and lawless sequences form disjoint, inhabited, welldefined classes within Brouwer’s continuum. The original results, reported as they developed ..."
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Abstract. The research on which this article is based was motivated by the wish to find a model of Kreisel’s lawless sequence axioms in which the lawlike and lawless sequences form disjoint, inhabited, welldefined classes within Brouwer’s continuum. The original results, reported as they developed in four papers over a period of ten years from 1986 to 1996, have so far lacked a readerfriendly presentation. Since the question of absolute definability is related to the subject of these Bristol Workshops, I offer here a straightforward exposition of the final model and formal system with axioms for numbers, lawlike sequences, and arbitrary choice sequences. A choice sequence is defined to be lawless if it satisfies an extensional (un)predictability condition from which extensional versions of Kreisel’s axioms of open data and strong continuous choice follow. The law of excluded middle can be assumed for properties of lawlike and independent lawless sequences only, while Brouwer’s continuity principle applies to properties of all choice sequences. Iterating definability, quantifying over numbers and over lawlike and independent lawless sequences, yields a classical model of the lawlike sequences with a natural wellordering. Under the (classically consistent and intuitionistically plausible) assumption that the closure ordinal of the iteration is countable, a realizability interpretation establishes the consistency of a common extension FIRM(≺) of classical analysis R and Kleene’s intuitionistic analysis FIM. Lawlike sequences behave classically, while the lawless sequences form a disjoint, Baire comeager collection of choice sequences, of classical measure zero. Thus Brouwer’s continuum can be understood as a relatively chaotic expansion of a completely determined, wellordered classical continuum. 1.
Article Not Finitude but Countability: Implications of Imagination Positing Countability in Time
"... Abstract: In this article, we will show how imagination and time are two sides of the same coin. To explain this, we require that imagination posits countability of alternatives. A countable set of alternatives can be sequenced on a timeline, for example the thinking human’s past, or it can be expre ..."
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Abstract: In this article, we will show how imagination and time are two sides of the same coin. To explain this, we require that imagination posits countability of alternatives. A countable set of alternatives can be sequenced on a timeline, for example the thinking human’s past, or it can be expressed as a countably infinite set of cycles such as a Fourier transform gives us. At the heart of our discussion is a technical argument arising from Cantor’s diagonal method. A conclusion that we arrive at is that the finite/infinite opposition, in particular in philosophy, is confusing at best. Instead we propose a countable/uncountable opposition as being a far clearer basis for understanding human imagination and as a basis for the philosophy of time. We discuss Kant, Heidegger and Gödel in this light. We draw out implications for “machine imagination, ” and we propose a new basis for understanding human creativity and imagination.