APPENDIX TO CHAPTERS 1 5
We collect here some basic mathematical results, primarily from set theory,
which are used in the rst ve chapters of these notes.
Notations. The (cartesian) product of two sets A, B is the set of all
ordered pairs from A and B ,
INTRODUCTION TO THE THEORY OF PROOFS
In order to study proofs as mathematical objects, it is necessary to introduce deductive systems which are richer and model better the intuitive
proofs we give in mathematics than the Hilbert system of Chapte
SOME RESULTS FROM MODEL THEORY
Our (very limited) aim in this chapter is to introduce a few, basic methods
of constructing countable models of theories and analysing their properties.
2A. Elementary embeddings and substructures
The results in th
FIRST ORDER LOGIC
Our main aim in this st chapter is to introduce the basic notions of logic
and to prove Gdels Completeness Theorem 1I.1, which is the rst, funo
damental result of the subject. Along the way to motivating, formulating
Math 220a, Solutions to HW3
x1.32. Let A = N cfw_a, where a is any object not in N (that we think
of as being innitely large). We interpret 0 by 0, and the functions S, +,
as usual on N, and for a we set (for any n N):
1. S (a) = a.
2. a + n = n + a = a
Math 220a, Solutions to HW4
x1.52. For each n 1, let
en (d0 , . . . , dn ) : d0 > d1 & d1 > d2 & & dn1 > dn ,
where d0 , d1 , . . . are (fresh) constants.
(1) The class of wellorderings is not elementary. Suppose
(A, ) W (A, ) |= T,
S = T cfw_en (