We show that the following relations are both in N.
[s DIV 2t ] M OD 2 = 1
[s DIV 2t ] M OD 2 = 1
where a DIV b gives the quotient of a b and a M OD b gives the remainder
of a b.
First we should recall what we already know: we know that exponentiation exp
0.1. INTRO TO LOGIC
0.1
1
Intro to Logic
Denition 0.1.1. We dene a sequence of sets R0 , R1 , . . . and R as follows:
0.2
0.2.1
Representability for Set Theory
Decidability
Denition 0.2.1. A formula of LS is said to be 0 if it contains no functions
or con
More problems! PHIL 350
December 3, 2014
1. Write out a formula of LS for 0 -truth. That is, a formula True0 (x)
such that HF |= True0 (t) i and only if tR is an instance of a 0
formula and HF |= tR .
2. Write out a formula of LS for 1 -truth. That is, a
Answer each question to the best of your ability. You have two hours
to complete this exam. You may assume that all theories include all logical
truths including those for equality.
1. Let T be a 0 complete theory in LSthat is, every true (in HF)
closed i
More problems! PHIL 350
November 12, 2014
1. Write out a formula of LS for 0 -truth. That is, a formula True0 (x)
such that HF |= True0 (t) i and only if tR is an instance of a 0
formula and HF |= tR .
2. Write out a formula of LS for 1 -truth. That is, a
We showed in class how to use the Fixed-point theorem to generate sentences that even strong theories of arithmetic and set theory cant decide
that is, they can neither prove nor refute these sentences. Here is some
further discussion of how these argumen
PHIL 350: More Exercises!
Sean C. Ebels-Duggan
December 3, 2014
1. Chapter 5, Exercise 8.1: Show there is no formula R(x, y) such that R
represents the relation x is a formula representing the set y.
Answer: One might think (as I did) to do this by showin