1. Prove in either of the proof systems we studied that x[VyGy→Hx] | [VYGY → 3xHx] And that x[ Hx VyGy] | [VxHx→ VyGy] 2. A torsion group G is a group all of whose elements are of finite order. Vg & G: ne gn= e. Let T be a theory in First Order Logic such that all of its models are torsion groups. Prove that there must be a bound to the torsion number. Hint: Deny, and run a compactness argument to get a model of T which is not Torsion. 3. Prove that there exists a nonstandard model of the Natural Numbers <N, +, ., 0, 1> with an infinite element. What do the non-standard models look like? 4. Sketch Henkin's proof of Completeness. (We did this in class) Just put it in your words. | 5. Define the member in the interval [0,∞) relation in structure of the Reals <R, +, > (equality in the language of course) 6. Prove that any automorphism of the Reals <R, +, ·, <, 0, 1> is trivial (identity map only) Extra Credit: Show that the least upper bound property of the Reals cannot be first order logic definable. Hint: try a proof by contradiction and Use Downward Lowenheim Skolem Theorem to get a countable model.