Math 558 Homework #1
Due September 15, 2009
Solutions
1. A real number is said to be primitive recursive if the function f (n) =
the nth digit of is primitive recursive. A real number is said to be
algebraic if it is a root of a nonzero polynomial with in
Math 558 Homework #2
Due October 5, 2009
Solutions
1. Prove that the following sets are not recursive.
(1)
(a) T = cfw_e | e is total.
(1)
(b) E = cfw_e | e is empty.
(1)
(c) F = cfw_e | dom(e ) is nite.
(1)
(d) O = cfw_e | 1 rng(e ).
(1)
(1)
(e) I = cfw_
Copyright c 19952009 by Stephen G. Simpson
Foundations of Mathematics
Stephen G. Simpson
October 1, 2009
Department of Mathematics
The Pennsylvania State University
University Park, State College PA 16802
simpson@math.psu.edu
This is a set of lecture note
Math 558 Homework #3
Due October 29, 2009
1. Find a pair of numbers r, a such that (r, a, 0) = 11, (r, a, 1) = 19,
(r, a, 2) = 30, (r, a, 3) = 37, (r, a, 4) = 51.
Hint: First nd an appropriate a by hand. Then write a small computer
program to nd r by bru
Math 558 Homework #4
Due December 10, 2009
1. Dene ordinal exponentiation by transnite recursion as follows.
0
= 1,
+1 = ,
= sup<
for all limit ordinals . In this exercise we shall obtain a more explicit
representation of .
Let = type(A, R) and = type(B
Math 558 Homework #2
Due October 5, 2009
1. Prove that the following sets are not recursive.
(1)
(a) T = cfw_e | e is total.
(1)
(b) E = cfw_e | e is empty.
(1)
(c) F = cfw_e | dom(e ) is nite.
(1)
(d) O = cfw_e | 1 rng(e ).
(1)
(1)
(e) I = cfw_3i 5j | i
Math 558 Homework #1
Due September 15, 2009
1. A real number is said to be primitive recursive if the function f (n) =
the nth digit of is primitive recursive. A real number is said to be
algebraic if it is a root of a nonzero polynomial with integer coec
Math 558: Foundations of Mathematics
Fall 2009, Tue-Thu 9:4511:00 AM, 315 McAllister
Stephen G. Simpson, 305 McAllister, 863-0775, simpson@math.psu.edu
This course is suitable for all mathematics graduate students. The textbook
will consist of notes provi