HOMEWORK FOR FOUNDATIONS STUDENTS

Math 4000: Foundations, Fall 2006

Assignment	Assigned	Due	Problems
HW1 Solutions	08/30/06	09/06/06	1. Show that m+1=1+m=S(m) for all natural numbers m. 2. Show that m+n=n+m for all natural numbers m, n. (This may require more than one inductive proof.) 3. Use Theorem 0B to prove that the set of 1-variable polynomials with rational coefficients is countable.
HW2	09/06/06	09/13/06	Section 1.1: 1, 4, 5
HW3	09/13/06	09/20/06	Section 1.2: Try #1, but do not turn it in. Turn in #4, 5, 6.
nonHW4	09/24/06	---	I forgot to assign these problems on 09/20/06, so they are not assigned as HW. But try to solve them! Section 1.2: 10, Section 1.3: 2, 7
	09/2706		Midterm 1, no HW.
HW5	10/04/06	10/11/06	Section 1.5: 2, 3, 4
HW6	10/11/06	10/18/06	Section 2.1: 1 Section 2.2: 1, 3
HW7	10/18/06	10/25/06	Section 2.2: 9, 11(b)(d), 12 (except modify the statement of 12(c) to ``Give a formula to define [0,1] union [2,7].''
HW8	10/25/06	11/1/06	Section 2.2: 18, 21, 28
	11/01/06		Midterm 2, no HW.
HW9	11/08/06	11/15/06	Section 2.4: 2(a)(c), 5, 6
HW10	11/29/05	12/6/06	A set of sentences closed under entailment is called a theory. That is, T is a theory if whenever T entails a sentence phi, then phi is in T. If A is an L-structure, then the theory of A, written Th(A), is the set of all sentences true in A. 1. Prove that Th(A) is a theory. 2. Prove that Th(A) is a maximal consistent set of sentences . 3. Show that if A is an L-structure with the property that every element of A is named by a constant in L, then Th(A) is a theory with witnesses.