hw6.pdf - 21-300 F15 HW 6 IMPORTANT PLEASE EMAIL COMPLETED...

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21-300 F15 HW 6 IMPORTANT: PLEASE EMAIL COMPLETED HOMEWORKS TO: [email protected]om (1) Let M be a substructure of N . Suppose that for every formula ψ ( z, y 1 , . . . y n ) and all a 1 , . . . a n M , if there is b N such that N | = ψ ( b, a 1 , . . . a n ) then there is a M such that N | = ψ ( a, a 1 , . . . a n ). Prove that M is an elementary substructure of N , that is to say for all φ ( z 1 , . . . z n ) and all a 1 , . . . a n M , M | = φ ( a 1 , . . . a n ) ⇐⇒ N | = φ ( a 1 , . . . a n ). Hint: You know from the midterm (and may assume here) that the conclusion is true for all quantifier free formulae φ . Use the assumption to power an induction. Be careful, the conclusion talks about satisfaction in M and N but the assumption only mentions satisfaction in N . (2) Let L be a language with constant symbols 0 and 1, binary function sym- bols +, × , pow (for power) and binary relation symbols (the equality symbol) and < . In a slight abuse of notation, let R be the L -structure whose underlying set is the real numbers in which each symbol is given the natural interpretation; pow R
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