In 2000, FBI special agents lured Russian cyber criminals Vasiliy Gorshkov and Alexey Ivanov to the United States with a ruse involving a job interview. The FBI also downloaded evidence of their crimes from computers in Russia. Using information from your classroom, additional materials provided by your instructor, and your own independent research, discuss whether you think the FBI’s investigative techniques went too far. Did the FBI act under the color of any international laws or compacts, or did they “go off the reservation?” Do the FBI’s actions stand up to scrutiny under current US cyber laws? Have there been other, more recent examples of cases in which US law enforcement targeted non-US citizens in a similar manner? What, if any, are the positive or negative impacts of this case on subsequent cyber-crime investigations?

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Using nothing but Gödel’s system P, prove that LaTeX: a:veeneg aa ∨ ¬ a is true for all LaTeX:
aa.
(5 Points) Construct a complete formal system of proof and prove that it is complete. (This one is actually quite
easy, so when you discover the answer do not tell anyone!)
Prove or disprove the following propositions.
(10 Points) The set of all finite but unbounded sequences of the form LaTeX: leftlangle
x_1,x_2,ldots,x_nrightrangle⟨ x 1 , x 2 , … , x n ⟩ where LaTeX: x_iinmathbb{N}x i ∈ N and LaTeX:
ninmathbb{N}n ∈ N is countably infinite.
(15 Points) The set of all formal proof systems is countably infinite.
(10 Points) Given any consistent proof system, adding an undecidable proposition as an axiom still yields an
incomplete system of proof no matter how many iterations you perform. That is to say, consistent proof systems are recursively incomplete.

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