Some time prior to Church's paper " An Unsolvable Problem of Elementary Number Theory " ( 1936 ) a dialog occurred between G�del and Church as to whether or not ?-definability was sufficient for the definition of the notion of " algorithm " and " effective calculability ".
22.
Dudley has also written and edited straightforward mathematical works such as " Readings for Calculus " ( MAA 1993, ISBN 0-88385-087-7 ) and " Elementary Number Theory " ( W . H . Freeman 1978, ISBN 0-7167-0076-X ).
23.
It can be divided into elementary number theory ( where the integers are studied without the aid of techniques from other mathematical fields ); analytic number theory ( where calculus and complex analysis are used as tools ); algebraic number theory ( which studies the algebraic numbers-the roots of polynomials with integer coefficients ); geometric number theory; combinatorial number theory; transcendental number theory; and computational number theory.
24.
E . g ., learning " a little " algebraic number theory without knowing elementary number theory is possible if you stick to the abstract concepts such as Dedekind domains, DVR's etc . and know in your own mind what a prime ideal is without needing to motivate it from elementary number theory, but there will come a time when intuitions from elementary number theory play a role.
25.
E . g ., learning " a little " algebraic number theory without knowing elementary number theory is possible if you stick to the abstract concepts such as Dedekind domains, DVR's etc . and know in your own mind what a prime ideal is without needing to motivate it from elementary number theory, but there will come a time when intuitions from elementary number theory play a role.
26.
E . g ., learning " a little " algebraic number theory without knowing elementary number theory is possible if you stick to the abstract concepts such as Dedekind domains, DVR's etc . and know in your own mind what a prime ideal is without needing to motivate it from elementary number theory, but there will come a time when intuitions from elementary number theory play a role.
27.
Rosser's footnote # 5 references the work of ( 1 ) Church and Kleene and their definition of ?-definability, in particular Church's use of it in his " An Unsolvable Problem of Elementary Number Theory " ( 1936 ); ( 2 ) Herbrand and G�del and their use of recursion in particular G�del's use in his famous paper " On Formally Undecidable Propositions of Principia Mathematica and Related Systems I " ( 1931 ); and ( 3 ) Post ( 1936 ) and Turing ( 1936 7 ) in their mechanism-models of computation.
