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

Info icon This preview shows pages 1–2. Sign up to view the full content.

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 .
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Image of page 2
This is the end of the preview. Sign up to access the rest of the document.
  • Fall '08
  • Solecki,S
  • Logic, Topology, Empty set, Open set, Topological space, Closed set, Clopen set

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern