# FINAL EXAM SOLUTIONS Problem . Find the Cantor normal

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```					                             FINAL EXAM SOLUTIONS

MATH  (DAVID PIERCE)

Problem . Find the Cantor normal form of (ω2 + ω + 1) · (ω + 1) + ω in ordinal
arithmetic.
Solution. ω2 (ω + 1) + ω + 1 + ω = ω3 + ω2 + ω · 2.

Problem .     (a) Explain brieﬂy why ON (the class of ordinals) is a proper class.
(b) Explain brieﬂy why CN (the class of cardinals) is a proper class.
(c) Show ON = V.
Solution.     (a) If ON were a set, then ON (being transitive and well-ordered by mem-
bership) would be in ON, which would be absurd, since membership is irreﬂexive
on ON.
(b) ON CN because of the function α → ℵα .
(c) {{∅}} ∈ ON.
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Problem . Let α = ω + 320 and β = 2009.
(a) Compute the cardinal sum ℵα + ℵβ .
(b) Compute the cardinal product ℵα · ℵβ .
Solution. The answer in each case is ℵα , since α            β.
Remark. We did prove that the cardinal sum and product of two cardinals, at least one
of which is inﬁnite, is just the greater of the cardinals.

Problem . Write the following sets in order of cardinality, using                ,    , and ≈:
ℵ0 ,     ℵ1 ,       ω,   Q,         R,      2
ω,   ω
2,   ω
ω,      R
R,      P(ω).
Solution. ℵ0 ≈ ω ≈ Q ≈ 2 ω         ℵ1        R ≈ ω 2 ≈ ω ω ≈ P(ω)          R R.

Date: jUNE , .


Problem . Let (C, i, F ) and (D, j, G) be two iterative structures, and H : C → D.
(a) Deﬁne what it means if H is a homomorphism from (C, i, F ) to (D, j, G).
(b) Show that there is a homomorphism K from (ω · ω, 0, ) to (C, i, F ).
(c) Why does (b) not contradict the fact that (ω · ω, 0, ) is not an arithmetic struc-
ture?
Solution.     (a) H(i) = j and H(F (x)) = G(H(x)).
(b) ω · ω = {ω · k + : k, ∈ ω}, and the elements ω · k are limits; so by recursion
we deﬁne K by K(ω · k) = i and K(ω · k + + 1) = F (K(ω · k + )).
(c) An iterative structure is arithmetic if and only if there is a unique homomorphism
from it to an arbitrary iterative structure. The homomorphism K in (b) is not
unique, since the K(ω · k) where k = 0 can be chosen arbitrarily from C.

Problem . Let n be an arbitrary positive integer, and let + and · be as usual on N
and also on Z/nZ (the integers modulo n). Explain whether there always a function
(x, y) → x ∗ y given by
x ∗ 1 = x,                  x ∗ (y + 1) = (x ∗ y) · x
(a)   from   N × N to N;
(b)   from   Z/nZ × N to Z/nZ;
(c)   from   N × Z/nZ to N;
(d)   from   Z/nZ × Z/nZ to Z/nZ.
Solution.        (a) Yes, by recursion.
(b) Yes, by recursion.
(c) No, since if n = 2, we would have 1 = 3 in Z/nZ, but 2 ∗ 1 = 2, while 2 ∗ 3 =
2 · 2 · 2 = 8 = 2.
(d) No, since if n = 3, we would have 1 = 4 in Z/nZ, but 2 ∗ 1 = 2, while 2 ∗ 4 =
2 · 2 · 2 · 2 = 16 = 1 = 2.
Solution. The function in question is indeed exponentiation, as featured on the third
exam.

Problem . Assuming that there is a surjection f from a set b onto a set c, show that
there is an injection g from c to b.
Solution. g(x) = h(f −1 [ {x} ]), where h is a choice-function for b.

Problem . Suppose (a, <) is well-ordered, and b ∈ a. Show that there is no order-
preserving bijection between pred(b) and a.
Solution. If there is such a bijection for some b in a, then let b be least such that
there is, and let f be the corresponding bijection. Then f         pred(f −1 (b)) is also a
bijection between pred(f −1 (b)) and pred(b), so f ◦f pred(f −1 (b)) is a bijection between

pred(f −1 (b)) and a. But this contradicts the minimality of b, since f −1 (b) < b.
URL: http://arf.math.metu.edu.tr/~dpierce/courses/320/2008/