# hw3 570 - MATH 570 MATHEMATICAL LOGIC HOMEWORK 3 Due date...

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MATH 570: MATHEMATICAL LOGIC HOMEWORK 3 Due date: Sep 17 (Wed) 1. Let A B and assume that for any finite S A and b B , there exists an automorphism f of B that fixes S pointwise (i.e. f ( a ) = a for all a S ) and f ( b ) A . Show that A B . 2. Show that ( Q , < ) ( R , < ) . Conclude that ( Q , < ) ( R , < ) , but ( Q , < ) ( R , < ) . Hint : Use the previous problem. 3. Conclude from the L¨ owenheim-Skolem theorem that any satisfiable theory T has a model of cardinality at most max { τ , 0 } . In particular, if ZFC is satisfiable, then it has a countable model (although that model M would believe it contains sets of uncountable cardinality, e.g. R M ). Explain why this DOES NOT imply that ZFC is not satisfiable. 4. Prove the Constant Substitution lemma. 5. Show the following: (a) (Associativity of + ) PA x y z [( x + y ) + z = x + ( y + z )] , (b) PA x ( 0 + x = x ) , (c) (Commutativity of + ) PA x y ( x + y = y + x ) . 6. Show that a τ -theory T is semantically complete if and only if for any A , B T , A B .

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• Fall '08
• Solecki,S
• Logic, Topology, Empty set, Open set, Topological space, Closed set, Clopen set

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