A12153W1
SECOND PUBLIC EXAMINATION
Honour School of Computer Science Part B: Paper B1.2
Honour School of Computer Science and Philosophy Part B: Paper B1.2
Honour School of Mathematics Part B: Paper B1.2
Honour School of Mathematics and Computer Science Part B: Paper B1.2
Honour School of Mathematics and Philosophy Part B: Paper B1.2
Honour School of Mathematics and Statistics Part B: Paper B1.2
Honour School of Philosophy, Politics and Economics: Paper B1.2
SET THEORY
TRINITY TERM 2016
FRIDAY, 3 JUNE 2016, 9.30am to 11am
You may submit answers to as many questions as you wish but only the best two will count for
the total mark.
You must start a new booklet for each question which you attempt. Indicate on the front sheet the
numbers of the questions attempted. A booklet with the front sheet completed must be handed in
even if no question has been attempted.
Do not turn this page until you are told that you may do so
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1. (a) [8 marks] (i) Let X, Y be sets. Define what is meant by the notations X ∼ Y , X Y
and X ≺ Y .
(ii) State the Cantor–Schröder–Bernstein Theorem.
(iii) If A, B, U, V are sets with A ∼ U, B ∼ V and A ≺ B, prove that U ≺ V .
(b) [7 marks] (i) Let ω1 denote the smallest uncountable ordinal. Define the ordinal exponentiation ω1α for an ordinal α.
(ii) Prove that there exists an ordinal β such that ω1β = β.
+
[You may assume that if α 6 β then ω1α 6 ω1β , and that ω1α < ω1α for any α.]
(c) [10 marks] A set Z is said to be cartesian closed if, for every x, y ∈ Z, the ordered pair
(x, y) ∈ Z. Let X be a set.
(i) Prove that there exists a unique set Z = Z(X) which is cartesian closed, with X ⊆
Z(X) and such that Z(X) ⊆ Y for every cartesian closed set Y with X ⊆ Y .
(ii) The transitive closure of a set X is the unique set T = T (X) such that T is transitive,
X ⊆ T , and T ⊆ Y for every transitive set Y with X ⊆ Y . Every set has a transitive
closure. Sketch a proof that, for a set X, there exists a unique set W (X) with
X ⊆ W (X), which is transitive and cartesian closed, and such that W (X) is a subset
of Y for every set Y which is transitive and cartesian closed with X ⊆ Y .
2. (a) [12 marks] (i) State and prove Hartogs’s Theorem.
(ii) State the Well-Ordering principle (WO) and the Cardinal Comparability principle
(CC).
(iii) Let EO be the statement that every set is equinumerous with an ordinal. Prove that
WO, EO, and CC are all equivalent, in the presence of the Zermelo–Fraenkel (ZF)
axioms (without the Axiom of Choice).
(b) [8 marks] (i) Show that the collection of sets which are ordered pairs is a class, and that
it is a proper class.
(ii) State the Axiom of Foundation.
(iii) Let X be a set. Prove that P(X) ∈
/ X.
(iv) Let X be a set. Prove that X contains no element of the form (X, y) for any set y.
(c) [5 marks] A set S is called a strict partial order set if every element of S is an ordered
pair, and moreover the following hold for all sets x, y: (i) if (x, y) ∈ S then (y, x) ∈
/ S; (ii)
if (x, y) ∈ S and (y, z) ∈ S then (x, z) ∈ S; and (iii) no set (x, x) belongs to S.
Let S be a strict partial order set. Prove that there exists a set X such that S is a
strict partial order on X. Is the order S on X always total? Sketch a proof or give a
counterexample.
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3. (a) [8 marks] (i) Define what it means for a set to be an ordinal .
(ii) Let α, β be ordinals. Suppose that α+ = β + . Prove that α = β.
(iii) Give an example of a subset X ⊂ Q such that the order induced on X by the usual
order on Q is order isomorphic to ω + ω + 1.
(b) [10 marks] Sketch a proof or give a counterexample to each of the following statements
about ordinals:
(i) if ω 6 α then 1 + α = α;
(ii) if ω 6 α then 2 · α = α;
(iii) if α, β are ordinals then α ∪ β is an ordinal;
(iv) if ω 6 α, β and α 6= β then α + β 6= β + α;
(v) ω + ω · ω = ω · ω.
(c) [7 marks] (i) Are there infinite cardinals κ, λ with κλ = κ? Give an example or sketch a
proof that none exists.
(ii) State and prove Cantor’s Theorem.
(iii) Let κ be a cardinal number. Prove that 2κ 6= ℵ0 .
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