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homework01

# homework01 - A ⊆ B ⊆ C then A ⊆ C 3 Show that if A...

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Math 5020 Homework 1 due Friday, February 11 Throughout the homework assignment, unless otherwise speciﬁed, we ﬁx a ﬁrst-order language L and assume all structures are L -structures. 1. Show that A B iﬀ for any quantiﬁer-free L -formula φ ( x 1 , . . . , x n ) and a 1 , . . . , a n ∈ | A | , A | = φ [ a 1 , . . . , a n ] ⇐⇒ B | = φ [ a 1 , . . . , a n ] . 2. Show that if
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Unformatted text preview: A ⊆ B ⊆ C then A ⊆ C . 3. Show that if A ≺ B ≺ C then A ≺ C . 4. Finish the proof of the Upward L¨owenheim–Skolem Theorem. That is, assuming A is an inﬁnite L-structure and κ an inﬁnite cardinality, ﬁnd an elementary extension of A of cardinality exactly κ ....
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