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Course: MATH 502, Fall 2008
School: Ill. Chicago
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502 Math Metamathematics I Problem Set 1 Due: Friday September 12 1) Suppose M is an L-structure, : V M is an assignment, is an L-formula and M |= v1 v2 . Prove that M |= v2 v1 . 2)1 a) Suppose that 1 , . . . , n are L-formulas and is a Boolean combination of 1 , . . . , n (i.e., is built from 1 , . . . , n using only , and ). Prove there is S P({1, . . . , n}) such that |= XS iX i iX i . b) Show...

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502 Math Metamathematics I Problem Set 1 Due: Friday September 12 1) Suppose M is an L-structure, : V M is an assignment, is an L-formula and M |= v1 v2 . Prove that M |= v2 v1 . 2)1 a) Suppose that 1 , . . . , n are L-formulas and is a Boolean combination of 1 , . . . , n (i.e., is built from 1 , . . . , n using only , and ). Prove there is S P({1, . . . , n}) such that |= XS iX i iX i . b) Show that every formula is equivalent to one of the form Q1 v1 . . . Qm vm , where is quantier-free and each Qi is either or . 3) Let L = {, e} be language the of groups. Show that there is a sentence such that M |= if and only if M Z/2Z Z/2Z. [Note: More generally, if = L is any nite language and M is a nite L-structure. There is an L-sentence such that N |= if and only if N M.] = 4) Let L be any countable language. Show that for any innite cardinal there are at mos...

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