SET THEORY: Take-home exam
IMPORTANT: Please type in capital letters on the first page of your
answers the following: your full name (both first name and family name),
and your program (MoL, BSc in Math etc).
DEADLINE AND PROCEDURE FOR RETURNING ANSWERS:
This exam is worth in total 100 points. It is due on the early morning of 3 June, at 9:00 a.m. (Amsterdam time). This deadline is
strict: no further extensions are possible. It should be returned electronically as pdf (obtained either from a typed text, or a scan of a clearly
handwritten text), sent to Hugo ([email protected], or also [email protected]), with CC to me ([email protected]) and Almudena ([email protected]).
IMPORTANT: No collaboration between students is allowed in the final
exam. If two exam papers are too similar, then they will both receive a failing
grade.
Question 1 (25 points in total).
1. (12 points) Prove the “Division with Remainder” Theorem (from slide
24 in Lecture Notes 5-6).
2. (13 points) Prove the Normal Form Theorem (from slide 9 in Lecture
Notes 7).
HINT: For part 1, let λ be the least ordinal s.t. β · λ ≥ α (-why does such an
ordinal exist? justify!); consider the two cases (a) β · λ = α and (b) β · λ > α;
case (a) is easy: the remainder is δ = 0 and the quotient is γ = λ; show
that in case (b), λ must be a successor ordinal λ = γ + 1 for some γ ∈ On;
use this, together with the “Ordinal Subtraction” Theorem (on slide 16 of
Lecture Notes 5-6) to prove the existence of the remainder and quotient; why
are they unique? For part 2, apply part 1 repeatedly and use the fact that
the class of ordinals is well-ordered to show that the process terminates.
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Question 2 (25 points in total)
1. (5 points) Show that the property “f is an injective function” is absolute.
2. (8 points) Show that the property “x is a cardinal” is absolute downwards.
3. (12 points) Prove the last result result on slide 25 of Lecture Notes 12,
on closure of Σ-functions under recursive iteration.
HINT: You may assume and use all the results about absoluteness proved in
Homework 6 (e.g. the fact that “f is a function” is ∆0 ), and the results in the
slides (concerning facts such as that ∆0 formulas are absolute, Σ formulas
are absolute upwards, their negations are absolute downwards etc). For the
first part, you need to express the property as a ∆0 formula (i.e. s.t. all the
quantifiers are bounded ones). For the next part, show that the negation of
“x is a cardinal” is (equivalent to) a Σ formula. For the last result (closure
of Σ functions under recursive iteration), you may of course use the previous
results on slide 25 of Lecture Notes 12 (or on previous slides), as well as the
following Observation: the statement “ F ∗ (α) = x ” is equivalent to “ α is an
ordinal and there exists some function f having dom(f ) = α + 1, satisfying
the given recursive conditions (from the definition of F ∗ ) for all ordinals in
its domain, and satisfying f (α) = x ”.
Question 3. (25 points in total) Let (A, ≤) be a well-ordered set. Let > be
the strict version of the converse of the relation ≤ (given by: x > y iff y ≤ x
and x 6= y). Let F be the Mostowski-Shepherdson mapping on (A, >) (as
defined in the Lecture Notes). Show that
1. (8 points) for every a ∈ A, we have F (a) = rk(a), where rk(a) is the
ordinal rank of a with respect to the well-order ≤ (defined recursively
as the least ordinal greater than the ordinal ranks of all the elements
b < a).
2. (8 points) F [A] is an ordinal.
3. (9 points) What is F [A] if we assume instead that A is a proper class
(satisfying in rest the same conditions as above)?
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HINT: Show the first part by induction on the well-founded relation ≤ (or
equivalently, by induction on the rank rk(a)); use part (1), together with the
fact that A is a set to prove part (2); give a very brief justification to your
answer to part (3).
Question 4 (25 points in total) Prove that, if M is a transitive class such
that Def (M ) ⊆ M , then OnM = On.
HINT: You may use all the definitions and results proven in the Lecture
Notes. Essentially, you need to abstract from the proof of OnL = On (given in
the slides), checking that the proof goes through using only the assumptions
that M is transitive and Def (M ) ⊆ M . If needed, you may also use the
results in Homework 7.
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