Math 5020 Handout 1 (corrected) Elementary Submodels and Extensions Denition Let L be a rst-order language and A, B be L-structures. A function h : |A| |B| is a homomorphism if (a) for every constant
Math 5020 Homework 5 due Friday, April 8, 2011 Assume ZFCRegularity. 1. Let X and Y be sets. Show that (a) |X | = |Y | i |X |e = |Y |e . (b) |X | |Y | i |X |e |Y |e . (c) |X | |Y | i there is a functi
Math 5020 Homework 4 due Wednesday, March 30, 2011 Assume ZFRegularity. 1. Show that there are arbitrarily large limit ordinals, i.e., for any ordinal there is a limit ordinal > . 2. Show that the cla
Math 5020 Homework 3 due Friday, March 4
1. Prove that (a, b) = (c, d) i a = c and b = d. 2. Determine whether each of the following statements is true or false. If it is true, give a proof; if it is
Math 5020 Homework 2 due Wednesday, February 23
1. Show that the following collection F is a lter over N: 1 F = XN : < . n+1
nX
2. Let F be the lter of all conite sets over N. Show that for any innite
Math 5020 Homework 1 due Friday, February 11
Throughout the homework assignment, unless otherwise specied, we x a rst-order language L and assume all structures are L-structures. 1. Show that A B i fo
Math 5020 Handout 1.2 Proofs of Theorems in Handout 1 Proposition A B i |A| |B| and for every constant symbol c L, n-ary relation symbol R L and function symbol f L, cB = cA , RB |A|n = RA , f B |A|n
Math 5020 Handout 1.1 Quantier Elimination and Elementary Submodels The aim of this handout is to show that (Q, <) (R, <). For this we rst prove a result of quantier elimination. Throughout the rest o
Math 5020 Handout 7 Cardinal Arithmetic and the Axiom of Choice
Cardinalities without AC 1. Dene the equinumerosity equivalence relation on V by X e Y there is a bijection between X and Y . 2. The car
Math 5020 Handout 6 Ordinal Arithmetic From now on we assume ZF Regularity, until further notice. 1. (Transnite Induction) Let C be a class of ordinals and assume that (i) 0 C ; (ii) if C then + 1 C ;
Math 5020 Handout 5 Basic Concepts of Axiomatic Set Theory (revised)
Throughout this handout we assume Extensionality, Pairing, Union, Power Set, Separation, Replacement, and the axiom of Existence Th
Math 5020 Handout 4 Basic Concepts of Axiomatic Set Theory
1. The language of set theory contains only one relation symbol whose intended interpretation is the membership relation between sets. 2. If
Math 5020 Handout 3 Axioms of Set Theory
Extensionality A set is uniquely determined by its elements. x y [ x = y z (z x z y ) ] Pairing The collection of a pair of sets is a set. x y z [ x z & y z &
Math 5020 Handout 2 Ultraproducts and Ultrapowers Denition Let S be a set and F P (S ). We say that F is a lter over S if (i) F ; (ii) if X F and X Y , then Y F ; (iii) if X, Y F then X Y F . A lter F