First Session Downward LS Theorem

-Prove the Downward LS for 1st order logic theorem in a countable language via Skolem functions
-Talk about Mostowski collapse
-Talk about Skolem paradox
-Prove theorem for infinite languages (using Skolem functions)
-Introduce L_{\kappa,\gamma}. 
-Show you need to take into account the size of the formula (i.e. even in countable language there is a sentence of L_{kappa, omega} which only has sentences of size \kappa
-Use a downward LS for set theory to prove downward LS for sentences of L_{infty, omega}
-Show in L_{\gamma^+, \gamma} there is a sentence with just equality which only has models of size > \gamma. Also show there is a sentence which only has models of size cofinality > \gamma (if \gamma > omega).
-Use downward LS for set theory to prove (a form of) downward LS theorem for L_{kappa, gamma}.
-Define about absolute logics. 
-Give examples
-Use the same techniques to prove a downward LS theorem for any absolute logic. 

Second Upward LS Theorem

-Prove the upward LS theorem for 1st order logic using compactness
-Define the Hanf number of an abstract logic
-Show all (set sized) abstract logics have a Hanf number
-Show the Hanf number of omitting types is beth_{omega_1}
-Show the Hanf number of L_{omega_1, omega} is beth_{omega_1}