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 symbol c L, h(cA ) = cB ; (b) for every n-ary function
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 function f : Y X with rng(f ) = X . 2. Show that the set of a
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 class of all limit ordinals is a proper class. 3. Show tha
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 false, give a counterexample. (a) If x is a transitive
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 subset S N there is an ultralter G F such that S G . 3
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 for any quantier-free L-formula (x1 , . . . , xn ) and a1
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 = f A . Proof () Suppose A B. Then by denition |A| |B|
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 of this handout we consider the language L = cfw_< where
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 cardinality of a set X , denoted by |X |e , is the e -equi
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 ; (iii) if is a nonzero limit ordinal and C for all < ,
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 There exists a set. 1. The empty set exists and is unique
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 (u, v1 , . . . , vk ) is a formula in the language of s
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 & u z (u = x u = y ) ] We denote the pairing set by cfw_x
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 over S is an ultralter if it is a maximal lter, i.e.,