Set Theory Final

How many natural numbers can be constructed from these two axioms? (a) One (b) Two (c) Infinitely many

(c) Infinitely many

Using just the Axiom of Existence and the Axiom of Pair (and assuming the Axiom of Extensionality), how many distinct sets can be constructed? (a) One (b) Two (c) Infinitely many

(c) Infinitely many

Transfinite Sequence

A function with an ordinal as its domain

Limit Ordinal

A limit ordinal is any ordinal that is not a successor ordinal

B

B

Lemma for Cardinal Addition

Lemma 1.2 ensures the addition of cardinals is well-defined. If |A| = |A'|, |B| = |B'|, and A ∩ B = ∅ (as well as A' ∩ B' = ∅), then |A ∪ B| = |A' ∪ B'|. This means the result of cardinal addition does not depend on the specific sets chosen, only their cardinalities.

Suppose n is a natural number. Which of the following (if any) hold: (a) n ⊆ P(n) (b) υn ⊂ n (c) ∩n = ∅ (d) n \ m is a natural number for all m ∈ ℕ

(a) (b) (c)

Suppose A is a non-empty set of natural numbers. Which of the following (if any) hold: (a) ∩A ∈ ℕ (b) ∩A ∈ A (c) υA ∈ ℕ (d) υA ∈ A (e) If A is infinite υA = ℕ

(a) (b) (e)

Lemma for Cumulative Hierarchy

(a) If x ∈ Vα and y ∈ x, then y ∈Vβ for some β < α​ (b) ​If β < α then Vβ ⊆ Vα (c) For all α, Vα is transitive and well-founded

Basic Properties of Ordinal Numbers

(a) If α < β and β < γ, then α < γ. (b) It is not possible for α < β and β < α to both hold. (c) Either α < β or α = β or β < α holds. (d) Every nonempty set of ordinal numbers has a least element in the ordering by <. (e) For every set of ordinal numbers X, there is an ordinal number α ∉ X. (i.e. the set of all ordinal numbers does not exist)

Corollary on Well-Ordered Sets

(a) No well-ordered set is isomorphic to an initial segment of itself (b) Each well-ordered set has only one automorphism i.e. the identity (c) If W1 and W2 are isomorphic well-ordered sets, then the isomorphism between W1 and W2 is unique

A

A

Cumulative Hierarchy (Vα)

A hierarchy where each level Vα is constructed from the powerset of the previous level Vα+1​ = P(Vα​) for all ordinals α, andVλ ​= ⋃Vβ (for β < λ)​ for limit ordinals λ Example:V0​ = ∅, V1​ = P(V0​) = {∅}, andV2​ = P(V1​)={∅, {∅}}

Definition of Finite Sets

A set S is finite if it is equipotent to some natural number n ∈ N. We denote this by ∣ S ∣ = n and say that S has n elements. A set is infinite if it is not finite. Example: The set {2, 4, 6, 8} is finite because it has 4 elements, corresponding to the natural number 4

Definition of Transitive Set

A set T is transitive if every element of T is also a subset of T. For every x ∈ T, x ⊆ T Example: The set {{∅}, ∅} is transitive because ∅ is a subset of {{∅}, ∅} and {∅} is also a subset since its only element, ∅, is in the set.

Well-Founded Set

A set X is well-founded if TC(X) is a transitive well-founded set A set X is well-founded if every non-empty subset of X has an ∈-minimal element Example: The set of natural numbers is well-founded because every non-empty subset has a least element

Theorem about Cumulative Hierarchy

A set is well-founded iff X ∈ Vα for some ordinal α

Definition of Ordinal Number

A set α is an ordinal if: (a) α is transitive and (b) α is well-ordered by ∈. Example: The set {∅, {∅}, {{∅}}} is an ordinal because it is transitive (each element is also a subset) and well-ordered by ∈.

Successor Ordinal

A successor ordinal is an ordinal number α that is the successor of another ordinal β, denoted by α = β + 1. Example: If β = {∅, {∅}}, then α = β ∪ {β} = {∅, {∅}, {∅, {∅}}} is a successor ordinal.

Axiom of Foundation (Axiom of Regularity)

All sets are well-founded according to this axiom. Ensures no set can be a member of itself, preventing infinite ∈-chains Example Given any non-empty set X under this axiom, there is an element in X that is disjoint from X, preventing any loop such as X ∈ X

Properties of Cardinal Addition

Cardinal addition is both associative and commutative: Associative: (κ + λ) + μ = κ + (λ + μ) Commutative: κ + λ = λ + κ Example: If κ = |A|, λ = |B|, and μ = |C| where A, B, and C are disjoint sets, then (|A| + |B|) + |C| equals |A| + (|B| + |C|), which equals |A ∪ B ∪ C|.

Cardinal and Ordinal Numbers

Cardinal numbers measure the "size" of sets, while ordinal numbers describe the position of an element in a well-ordered set Example: The cardinal number of the set of natural numbers is ℵ0​ (aleph-null), and the ordinal number of the set {a, b, c}, where a < b < c, is 3 since 'c' is the third element.

What is true about elements of an ordinal number?

Every element of an ordinal number is itself an ordinal number.

Well-Ordering Principle

Every set can be well-ordered, which means its elements can be arranged in a sequence such that every nonempty subset has a least element Example: The set of natural numbers N is well-ordered since for any nonempty subset of N, there is a smallest number. To clarify with an example: let's say we have a set A that is well-ordered. The elements of A can be written as a sequence (aα​)α∈κ​ where κ is an ordinal. Then, we can define a function f where f(α) = aα​ for each α ∈ κ. Here, f is a function whose domain is the ordinal κ and whose codomain is the set A, and this function is indeed a set of ordered pairs {(α, aα​)}.

Well-Ordering Principle

Every well-ordered set is uniquely order-isomorphic to a unique ordinal Example: The natural numbers are order-isomorphic to the ordinal ω

Transfinite Counting and Ordinal Numbers

Extends counting into the infinite, with ω representing the first infinite ordinal Example: ω, ω+1, ω+2, etc.

Any linearly ordered set is well-ordered (T/F?)

False

If (S, <) is a well-ordering, then (S, <⁻¹) is a well ordering (T/F?)

False

If f is surjective, then f[A\B] = f[A] \ f[B] for any A, B ⊆ dom (f) (T/F?)

False

If f: A → B is injective, and g: B → C is a bijection, then g o f: A → C is a bijection (T/F?)

False

The union of any two functions is a function (T/F?)

False

If A is equipotent to a proper subset of B, then B is not equipotent to a proper subset of A (T/F?)

False Example: A = ℕ \ {0}, B = ℕ \ {1}

What is the theorem regarding transfinite sequences?

For a set A and ordinal Ω, there exists a unique function for all transfinite sequences of elements of A of length less than Ω. The unique function is f : Ω→A such that: f(α) = g(f \ α) for all α < Ω Example: Sequences of sets, where each set includes all members of the previous set.

Lemma about Increasing Functions on Well-Ordered Sets

For a well-ordered set W and increasing function f: W → W, then f(x) ≥ x for all x ∈ W. An increasing function on a well-ordered set never maps an element to something smaller than itself. Example: The identity function on ℕ.

Recursion Theorem

For any operation G and element a, there exists a unique infinite sequence aₙ​ such that: (a) a₀​ = a and (b) aₙ₊₁ ​= G(aₙ​) for all n ∈ N Example: If G(a) = a + 1 and a = 0, the resulting sequence is the natural numbers

Axiom of Replacement

For any set A and property P that assigns a unique y for each x ∈ A, there exists a set B containing all such y Example: If A is a set of natural numbers, and P(x,y) is the property "y is the square of x", then B would be the set of squares of elements in A Example 2: If A is the set of natural numbers and F is a function defined by a property P, then F[A] denotes the image of A under F, which is a set by the Axiom of Replacement F(x) could be "2 times x", and F[A] would be the set of even natural numbers

The Axiom of Choice

For any set of nonempty sets, there exists a choice function that selects an element from each set Example: Given a collection of nonempty sets such as {{1, 2}, {3, 4}, {5, 6}}, the Axiom of Choice guarantees that there is a function f that could, for instance, pick 1 from the first set, 3 from the second, and 5 from the third.

Corollary to Cantor's Theorem

For any system of sets S, there exists a set Y such that the cardinality of Y is greater than the cardinality of every set X ∈ S. Example: If S is a collection of sets, then P(∪S), the power set of the union of all sets in S, has a greater cardinality than any set in S.

Theorem about Comparison of Well-Ordered Sets

For any two well-ordered sets W1​ andW2​, exactly one of the following holds: (a) W1​ and W2 are isomorphic or (b) W1​ is isomorphic to an initial segment of W2 or (c) W2 is isomorphic to an initial segment of W1​ Example: ℕ is isomorphic to an initial segment of ℕ ∪ {ω}

Union of Finite Sets

If X and Y are finite, then X ∪ Y is finite, and if X and Y are disjoint, ∣X ∪ Y ∣ = ∣X ∣ + ∣Y ∣. Example: For X = {a, b} and Y = {c, d}, X ∪ Y = {a, b, c, d } is finite with ∣X ∪ Y ∣ = 4

Cardinality and Subsets

If X is a finite set and Y ⊂ X, then Y is finite and ∣Y ∣ ≤ ∣X ∣ Example: If X = {1, 2, 3, 4, 5} and Y = {2, 4}, then Y is finite with ∣Y ∣ = 2 and ∣Y ∣ < ∣X ∣

Theorem of Finiteness and Infinite Sets

If X is infinite, then ∣X ∣ > n for all n ∈ N Example: The set of natural numbers N is infinite because for any natural number n, there are more than n natural numbers

Lemma for Well-Founded Sets and Sequences

If a set is well-founded, then it is impossible to find an infinite descending sequence of elements X1​ ∈ X2​ ∈ X3​ ∈ ... within it Example: In the set of natural numbers, you cannot find an infinite sequence such that a1 ​> a2 ​> a3 ​> ..., where each ai​ is a natural number

Uniqueness of Isomorphism

If two well-ordered sets are isomorphic, their isomorphism is unique Example: The isomorphism from ℕ to itself that maps each n to n is unique

Lemma about subsets and membership for ordinal numbers

If α and β are ordinal numbers such that α ⊂ β, then α ∈ β. Example: If β = {∅, {∅}, {∅, {∅}}} and α = {∅, {∅}}, then α ∈ β because {∅, {∅}} is one of the elements of β.

Lemma about membership for ordinal numbers

If α is an ordinal number, then α ∉ α.

State the Transfinite Recursion Theorem

It allows the definition of functions over ordinals. Given a function G, it uniquely defines a function F over all ordinals such that F(α) = G(F \ α). Example: Defining a sequence where each term is determined by applying G to the previous term.

What is the parametric version of the Transfinite Recursion Theorem?

It extends the theorem to operations that also depend on an additional parameter set z.

What is the Transfinite Induction Principle?

It's a method of proof used in set theory to show that a property P holds for all ordinal numbers. It states that: (a) if P(β) holds for all β < α, then P(α) must also hold. Example: If P is "being countable," and each β < α is countable, then α is also countable.

Lemma for Cardinal Multiplication

Lemma 1.4 verifies the multiplication of cardinals is well-defined. If |A| = |A'| and |B| = |B'|, then |A × B| = |A' × B'|. This means the result of cardinal multiplication, like addition, does not depend on the specific sets chosen.

Let A = ℕ, and let b ∉ ℕ. Define the binary relation ≺ on ℕ υ {b} by putting: x ≺ y iff either x, y ∈ ℕ and x ∈ y or x ∈ ℕ and y = b Does every non-empty subset of ℕ υ {b} have a greatest element?

No

Let A = ℕ, and let b ∉ ℕ. Define the binary relation ≺ on ℕ υ {b} by putting: x ≺ y iff either x, y ∈ ℕ and x ∈ y or x ∈ ℕ and y = b Is ℕ υ {b} with the ordering ≺ isomorphic to ℕ with the usual ordering? If so, write down an isomorphism; if not, say why not.

No because b is an upper bound (if it is on the right-hand side of the relation, it is greater than) and ≺ would not be well-ordered

If α is an ordinal number, what can be said about S(α)?

S(α), the successor of α, is also an ordinal number. Example: If α = 2 (represented by {∅, {∅}}), then S(α) = α ∪ {α} = {∅, {∅}, {∅, {∅}}} is also an ordinal.

Definition of a Well-Ordered Set

Set W is well-ordered by < if (a) it is linearly ordered and (b) every nonempty subset has a least element Example: Any segment of natural numbers, e.g., {1,2,3,...}

Now assume all the axioms we have learned thus far. Does the collection of all infinite sets form a set? Why or why not? (proof)

Show Russell's Paradox and the set R is infinite by: ℕ ℕ \ {0} ℕ \ {1} ℕ \ {2} ... Each of these sets don't contain themselves as members, and they go into infinity

Let A = ℕ, and let b ∉ ℕ. Define the binary relation ≺ on ℕ υ {b} by putting: x ≺ y iff either x, y ∈ ℕ and x ∈ y or x ∈ ℕ and y = b Is ≺ a strict or partial order?

Strict order

Initial Segments of Well-Ordered Sets

Subset S of W is an initial segment if S ⊂ W (i.e. S ≠ W) if there exists a ∈ W, such that S = {x ∈ W ∣ x < a} Example: In ℕ, for a = 5, the initial segment is {0, 1, 2, 3, 4}

Uniqueness of Cardinal Numbers of Finite Sets

The cardinal numbers of finite sets are unique. For any natural number n, there is no one-to-one mapping of n onto a proper subset of n. Example: For the natural number 3, there is no one-to-one correspondence between the set {1, 2, 3} and any proper subset like {1, 2}

The Continuum Hypothesis

The hypothesis states that there is no set with a size between that of the integers and the real numbers Example: It's equivalent to the statement that the cardinality of the continuum, the set of real numbers, is the second smallest infinity, ℵ1​.

Relationship between natural numbers and ordinal numbers

The natural numbers are exactly the finite ordinal numbers. Every ordinal that is not a natural number is an infinite set, and every natural number is a well-ordered set under the well-ordering ε. Example: The set of natural numbers, denoted by ω, is an ordinal number.

Order-Type of a Well-Ordered Set

The order-type of a well-ordered set W is the unique ordinal isomorphic to W

Transitive Closure

The smallest transitive set containing set X TC(X) includes X and all elements of elements of X Example: For X = {a, {a, b}}, TC(X) = {a, b, {a, b}}, since it includes all elements and elements of elements of X

What is the second version of the Transfinite Induction Principle?

This version separates the successor and limit ordinals. Let P(x) be a property. Assume that: (a) P(0) holds (b)P(α) implies P(α+1) and (c) if P(β) for all β<α where α is a limit ordinal, then P(α).

Any finite linearly ordered set is well-ordered (T/F?)

True

For any distinct natural numbers m and n, either m ⊂ n or n ⊂ m (T/F?)

True

If f is injective, then f[A\B] = f[A] \ f[B] for any A, B ⊆ dom (f) (T/F?)

True

The intersection of any set of functions from A to B is a function whose domain is a subset of A (T/F?)

True

Let A = ℕ, and let b ∉ ℕ. Define the binary relation ≺ on ℕ υ {b} by putting: x ≺ y iff either x, y ∈ ℕ and x ∈ y or x ∈ ℕ and y = b Are any two elements in ℕ υ {b} comparable?

Yes

Are all natural numbers ordinal numbers?

Yes, every natural number is an ordinal number because it is transitive and well-ordered by ∈.

Is the power set of a finite set also finite?

Yes, if X is finite, then the power set P(X) is finite Example: For X = {1, 2}, P(X ) = {∅, {1}, {2}, {1, 2}} is finite with ∣P(X ) ∣ = 4

Let A = ℕ, and let b ∉ ℕ. Define the binary relation ≺ on ℕ υ {b} by putting: x ≺ y iff either x, y ∈ ℕ and x ∈ y or x ∈ ℕ and y = b Is ℕ υ {b} with the ordering ≺ isomorphic to a proper subset of itself (with the ordering ≺ restricted to that subset)? If so, write down such an isomorphism; if not, say why not.

Yes; an example is ℕ \ {1}

Cantor's Theorem Restated

|P(X)| = 2^|X| κ < 2^κ for every cardinal number κ

Cantor's Theorem

|X| < |P(X)| For any set X, the cardinality of the power set P(X) is strictly greater than the cardinality of X. Example: For a set X with 3 elements, there are 2³ = 8 subsets of X, including the empty set and X itself. Therefore, the power set P(X) has more elements than X itself.

Definition of Cardinal Addition

κ + λ = |A ∪ B|. Where κ and λ are cardinal numbers and there are two disjoint sets A and B with cardinalities |A| = κ and |B| = λ. Example: If set A has 3 elements and set B has 2 elements, and they are disjoint, A ∪ B has 5 elements, so 3 + 2 = 5. This extends to infinite sets where κ and λ are cardinals of infinite sets.

Definition of Cardinal Multiplication

κ · λ = |A × B| The product of two cardinal numbers κ and λ is the cardinality of the Cartesian product of sets A and B (|A| = κ and |B| = λ). Example: If set A has 3 elements and set B has 2 elements, A × B has 6 ordered pairs, so 3 · 2 = 6. This concept extends to infinite sets as well.

Definition of Cardinal Exponentiation

κ^λ = |Aᴮ| κ^λ is defined as the cardinality of the set of all functions from B to A, where |A| = κ and |B| = λ. Example: For sets A and B where |A| = 2 and |B| = 3, the number of functions from B to A is 2³ = 8, since each element of B has 2 choices in A, and there are 3 such decisions to make.