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Set Theory
Andrew Marks
July 22, 2020
These notes cover introductory set theory. Starred sections below are op- tional. They discuss interesting mathematics connected to concepts covered in the course. A huge thanks to Spencer Unger for enlightening conversations, and the students in the class who asked excellent questions, and corrected countless typos in the midst of a global pandemic.
Contents
1 Introduction 3 1.1 Independence in modern set theory*................ 6
2 The axioms of ZFC 10 2.1 Classes and von Neumann-Bernays-G¨odel set theory*....... 12
3 Wellorderings 14
4 Ordinals 16
5 Transfinite induction and recursion 19 5.1 Goodstein’s theorem*........................ 23
6 The cumulative hierarchy 25
7 The Mostowski collapse 28
8 The axiom of choice 30 8.1 Fragments of the axiom of choice*................. 31
9 Cardinality in ZF 33 9.1 Cardinality in models of the axiom of determinacy....... 36 9.2 Resurrecting Tarski’s theory of cardinal algebras......... 36
10 Cofinality 38
11 Cardinal arithmetic in ZFC 41 11.1 The singular cardinals hypothesis*................. 44
- 12 Filters and ultrafilters
- 12.1 Measurable cardinals*
- 13 Ultraproducts
- 13.1 The Ax-Grothendieck theorem*
- 14 Clubs and stationary sets
- 15 Applications of Fodor: ∆-systems and Silver’s theorem
- 16 Trees
- 16.1 Compactness and incompactness in set theory*
- 17 Suslin trees and ♦
- 18 Models of set theory and absoluteness
- 19 The reflection theorem
- 20 G¨odel’s constructible universe L
- 20.1 G¨odel operations and fine structure*
- 21 Condensation in L and GCH
- 22 V = L implies ♦
- 23 L and large cardinals
- 23.1 Finding right universe of set theory*
- 24 The basics of forcing
- 25 Forcing = truth
- 26 The consistency of ¬CH
Exercise 1.6. Show that there is a bijection from P(N) to the real numbers R. [Hint: First, there is a bijection from R to (0, 1). Then show there is a bijection from (0, 1) to P(N) using binary expansions and Exercise 1.4.]
Using this bijection between P(N) and R, we can quickly prove that N has strictly smaller cardinality than P(N) (and hence R). Suppose f : N → P(N) is any function. Then f is not onto P(N). We prove this by constructing a subset of N that is not in ran(f ). Let D = {n ∈ N : n /∈ f (n)}. Then this set D diagonalizes against f. Since n ∈ D ↔ n /∈ f (n), D cannot equal f (n) for any n. Hence, D /∈ ran(f ) and f is not onto.
Figure 1: Cantor’s diagonal argument. In this figure we’re identifying subsets of N with infinite binary sequences by letting the where the nth bit of the infinite binary sequence be 1 if n is an element of the set.
This exact same argument generalizes to the following fact:
Exercise 1.7. Show that for every set X, there is no surjection f : X → P(X). [Hint: define D = {x ∈ X : x /∈ f (x)}. Then show D /∈ ran(f ).]
Thus, given any set X, its powerset P(X) has larger cardinality. Cantor had realized that as a consequence of this, there can be no universal set: a set containing all other sets. Every set would inject (via the identity function) into a universal set. But Exercise 1.7 shows that the powerset of the universal set could not inject into the universal set. Bertrand Russell traced through Cantor’s argument (letting f be the identity function and X be a supposed universal set), and isolated what is now known as Russell’s paradox. If D = {x : x /∈ x},
then is D ∈ D? If D /∈ D, then D ∈ D by definition, contradiction. But if D ∈ D, then D /∈ D by definition, contradiction.
Russell’s writings about this paradox caused a brief crisis in the foundations of set theory. Allowing ourselves to construct a set containing all mathemati- cal objects satisfying some given property leads to contradictions. What sets, then, should we be allowed to construct? Is the whole enterprise of set theory inconsistent? The resolution to Russell’s paradox that set theorists have adopted is the so called iterative conception of set theory^3. All sets are arranged into a cumulative hierarchy. We begin with a simple collection of sets, and then apply some basic operations to iteratively create more sets. This produces the hierarchy V of all sets. The precise set existence axioms we will use will be discussed in the next section. They are known as Zermelo-Frankel set theory or ZF. We use ZFC to denote ZF+ the axiom of choice. The first part of this class will be discussing these axioms of ZFC and axiomatic set theory.
Figure 2: A picture of the set theoretic universe, known as V. At step α, we construct all sets of “rank” α. Vα denotes all sets of rank less than α.
Note that we will never define what a set is in these notes. We’re taking an axiomatic viewpoint. ZFC includes some true principles about sets, but not all of them. We caution that it is very false to say “a set is an element of a model of set theory”. First, this would be circular; a model is defined in model theory using sets. Second, there are strange models of set theory, which we wouldn’t want to use to define what sets are. It would be similarly false to say that a natural number is an element of a model of PA; there are nonstandard models of PA with infinite elements greater than any natural number^4.
(^3) Other alternatives to ZFC have been also explored such as Russell’s type theory, or Quine’s new foundations. They are rarely considered in modern set theory. (^4) Indeed, G¨odel’s incompleteness theorem says that it’s hopeless to try and axiomatize all true sentences about the natural numbers. It is similarly hopeless to try and axiomatize all true principles about sets.
verse restricted to height κ satisfies ZFC, and more generally will contain many “smaller” large cardinals. Many other interesting set theoretical statements end up being equivalent in consistency strength to large cardinal assumptions. For example, ZF + “all sets of real numbers are Lebesgue measurable”, is equiconsistent with an inaccessible cardinal, and ZFC + “there is a saturated ideal on ω 1 ” is equicon- sistent with a Woodin cardinal. One of the most important open problems in modern set theory is proving that the proper forcing axiom PFA is equiconsistent with a supercompact cardinal. Because of G¨odel’s incompleteness theorem, none of these large cardinals can be proved to exist from ZFC (and we cannot prove they are consistent without assuming the consistency of even “larger” cardinals). However, they are a vital part of the study of modern set theory, and they are viewed as the “natural” way to increase the consistency strength of the theory of ZFC. The consistency strength of all “natural” theories has been empirically found to be linearly ordered and indeed wellordered. This is important evidence that these theories are mathematically important^5. Large cardinals also create beautiful and intricate structure in the set theoretic universe which has important and concrete mathematical consequences (for example, in our understanding of the real numbers). They are freely used and investigated in modern set theory. We cannot prove they are consistent, but we deeply believe they are because of the beautiful and important mathematical structures they create. This, then, is one source of independence in set theory. Any statement that implies Con(ZFC) must be either false or independent of ZFC. However, there is a completely different method for proving independence from set theory. In 1938, G¨odel proved that inside any model of ZF set theory, there is an “inner model” L which consists of all the “constructible sets”. This is in a sense the “smallest” possible universe of set theory. It contains only the sets one must have by virtue of these sets being explicitly definable. G¨odel showed that this inner model known as L always satisfies both the axiom of choice and the continuum hypothesis. This was reassuring to mathematicians who were worried about the validity and acceptability of the axiom of choice. By G¨odel’s theorem if ZF is consistent and has a model, then ZF + the axiom of choice is also consistent. So using the axiom of choice cannot add new inconsistencies to set theory. A huge importance of inner models such G¨odel’s L is that they have an extremely detailed and canonical structure. Indeed, there is a whole study of “fine structure theory”, which analyzes these canonical models in great detail. Unfortunately, G¨odel’s L is deficient in that sufficiently large cardinals (e.g. measurable cardinals) cannot exist inside L. One aim of modern inner model theory is to construct inner models that are compatible with having all large cardinals, and understanding their structure. Complementing G¨odel’s constructible universe was Cohen’s 1963 invention (^5) The naturalness assumption is very important here; neither linearity or wellfoundedness are true if we consider all theories extending ZFC.
of the method of forcing. Given a countable model of ZFC, Cohen showed how one can add sets to the model to create a larger “outer model” of ZFC. Cohen used this technique to show that given any countable model of ZFC, one can add many real numbers to it in order to find an outer model where the continuum hypothesis is false. These two results combine to show that if there is a model of ZFC, then there is a model of both ZFC + CH and another model of ZFC + ¬CH. Thus, ZFC cannot prove that CH is either true or false, and CH is independent from ZFC. Philosophers of set theory still fiercely debate questions such as whether there could be new intuitively justified axioms for set theory that resolve the continuum hypothesis, or whether CH even has a definite truth value^6. This, then, is the second source of independence in set theory: we can prove a statement ϕ is independent from ZFC by constructing outer or inner models that satisfy both ϕ and its negation. An imperfect analogy is that starting with any field, we can study its subfields and field extensions. If we find two different fields, one of which has property ψ and the other does not, then we know the field axioms do not imply ψ. The invention of forcing led to a renaissance of independence results in set theory, many of which had stood open for many decades. For example, Suslin’s problem^7 , which had been open since 1920 was shown to be independent of ZFC by Solovay and Tennenbaum in 1971. Forcing is also intimately tied to inner model theory. The canonical structure given by inner models is often a necessary starting point for a good understanding of the outer models we force to create. Forcing and inner models also found applications in many different fields of mathematics. For example, Kaplansky’s conjecture in functional analysis, and Whitehead’s problem in group theory are independent of ZFC. There is a deep contrast between the type of independence that comes from having consistency strength, and the type that comes from forcing/inner mod- els. While very simple statements (e.g. Π^01 statements in arithmetic) can be independent of ZFC by virtue of having consistency strength, statements which are shown to be independent of ZFC by forcing and inner models must be very complicated by so-called absoluteness results. For example, we cannot use forc- ing to show any Σ^12 sentence is independent of ZFC by Shoenfield absoluteness. Indeed, assuming certain large cardinals exist, CH is in some sense the “sim- plest” statement that can be proved independent from ZFC by forcing.^8 Set theory remains a vibrant and active field of research, and many open problems remain. Indeed, even Cantor’s original goal of understanding basic cardinal arithmetic is still an unfinished puzzle; very simple-seeming questions about the possible behavior of cardinality in ZFC remain open. For example,
(^6) Classical large cardinals axioms cannot help resolve this question by a theorem of Levy and Solovay. (^7) Suslin’s problem asks the following: suppose (L, <L) is a dense complete linear order without endpoints in which every collection of pairwise disjoint open intervals is countable. Then must L be order-isomorphic to the real numbers? (^8) It is a theorem of Woodin that assuming the existence of large cardinals, all Σ 2 1 statements are absolute for set forcing, assuming CH (which is itself a Σ^21 statement).
2 The axioms of ZFC
The language of set theory L∈ consists of a single binary relation ∈ of set membership. We will begin by stating the axioms of ZFC. Throughout this section, we will introduce notation for certain sets, functions and relations which are defined in terms of the ∈ relation. For example, x ⊆ y will abbreviate ∀z(z ∈ x → z ∈ y). We will also use bounded quantifiers freely: (∃y ∈ x)φ is defined to mean ∃y(y ∈ x ∧ φ), and (∀y ∈ x)φ is defined to mean ∀y(y ∈ x → φ). We’ll also use the exists unique quantifier: ∃!yϕ(y) abbreviates ∃y(ϕ(y) ∧ (∀y′ϕ(y′) → y′^ = y)). The axiom of Extensionality: Every set is determined by its members.
∀x∀y[x = y ↔ ∀z(z ∈ x ↔ z ∈ y)]
This axiom essentially defines what it means to be a set. A set x is deter- mined precisely by what elements its contains. (A set has no order or other data). The axiom of Foundation: Every nonempty set contains a ∈-minimal element. ∀x[x 6 = ∅ → ∃y ∈ x∀z ∈ x(z /∈ y)]
Here x 6 = ∅ abbreviates ∃y(y ∈ x). The axiom of foundation says that the relation ∈ on every set has a minimal element: some y ∈ x with no predecessors under ∈ in x. The axiom of foundation also defines what it means to be a set, but in a more technical sense. We will prove shortly that the axiom of foundation is equivalent to the statement that every set is an element of the von Neumann universe V of sets; those that can be obtained from ∅ by iteratively applying the set existence axioms.^9 These first two axioms define what it means to be a set. All the other axioms of ZFC are set existence axioms which state that certain sets exist. The axiom of Pairing: Given two sets x and y, there is a set containing exactly these two sets.
∀x∀y∃w[x ∈ w ∧ y ∈ w ∧ ∀z(z ∈ w → z = x ∨ z = y)]
We let {x, y} denote this set w whose only two elements are x and y, and we’ll use {x} to denote {x, x}. Assuming the pairing axiom, the axiom of foundation implies that there is no set x such that x ∈ x. To see this, assume for a contradiction there is such a set x, and consider {x}. Then the only element of {x} is x. However, since x ∈ x, x is not -minimal. Since {x, y} = {y, x} we will also define an ordered pair, where the order of the two elements matters. We define (a, b) = {{a}, {a, b}}. It is straightforward
(^9) Precisely, we’ll show that ZFC − Foundation proves that Foundation ↔ ∀x(x ∈ V ). Here V is defined as the class of sets that are in Vα for some ordinal α, where V 0 = ∅, Vα+1 = P(Vα) and Vλ = ⋃ α<λ Vα.
to show that for all sets a, b, c, d, we have (a, b) = (c, d) if and only if a = c and b = d.
Exercise 2.1. Show that for all sets a, b, c, d, we have (a, b) = (c, d) if and only if a = c and b = d.
The axiom of Union: Given any set of sets x, there is a set containing exactly all the element of these sets, denoted
x. Precisely, letting y =
x denote ∀z[z ∈ y ↔ ∃w ∈ x(z ∈ w)], the axiom of union states
∀x∃y[y =
x]
Writing z = x ∪ y for ∀w(w ∈ z ↔ w ∈ x ∨ w ∈ y), pairing and union prove that for all sets x and y, x ∪ y exists, since x ∪ y =
{x, y}. The axiom of Nullset: There is a set with no elements. We let x = ∅ abbreviate ¬∃y(y ∈ x). Nullset states:
∃x[x = ∅]
The axiom of Infinity: There exists an infinite set.
∃x[∅ ∈ x ∧ ∀y(y ∈ x → y ∪ {y} ∈ x)]
We say a set x is inductive if ∅ ∈ x and y ∈ x implies y ∪ {y} ∈ x, so the above axiom says that an inductive set exists. As we’ll see later, if y is an ordinal, then y ∪ {y} is the ordinal successor of y. Note that the infinite set x whose existence is guaranteed by the axiom of infinity must have the following set as a subset: {∅, {∅}, {∅, {∅}},.. .}. We will eventually call this set ω: the set containing the ordinals { 0 , 1 , 2 ,.. .}. The axiom of Powerset: For every set x, there is a set containing all the subsets of this set. We let y = P(x) abbreviate ∀z(z ∈ y ↔ z ⊆ x)
∀x∃y[y = P(x)]
The axiom schema of Separation: If x is a set, then every subset of x that’s definable (from parameters) exists. Formally, for every formula ϕ in the language of set theory, the following is an axiom
∀x, w 1 ,... , wn∃y∀z[z ∈ y ↔ z ∈ x ∧ ϕ(x, z, w 1 ,... , wn)]
We will use {z ∈ x : ϕ(z, w 1 ,... , wn)} to abbreviate the set whose existence is given by this axiom. More generally, we will use {z : ϕ(z, w 1 ,... , wn)} to denote the collection of all sets z satisfying the formula ϕ(z, w 1 ,... , wn). In general, this will not be a set (e.g. {z : z /∈ z}). We will instead call such a collection a class, and by a class in these notes, we mean all sets z satisfying some formula ϕ(z, w 1 ,... , wn) where w 1 ,... , wn are fixed set parameters. In the case where such a collection is a set then y = {z : ϕ(z, w 1 ,... wn)} abbreviates ∀z(z ∈ y ↔ ϕ(z, w 1 ,... , wn)).
There are alternate ways of axiomatizing set theory where we explicitly give classes formal existence instead of just associating to each formula the class it defines. Having classes as formally defined objects is often convenient. For example, many large cardinal axioms say there is a proper class inner model M of ZFC and an class function j : V → M which is an elementary embedding. One such way to axiomatize set theory and directly talk about classes is von Neumann-Bernays-G¨odel set theory. In this axiomatization, all the objects of study are classes. We define a set to be a class which is an element of some other class. A proper class is a class which is not a set. By convention, uppercase letters in NBG denote classes, while lowercase letters denote only sets. So for example, ∃xϕ abbreviates ∃x∃Y (x ∈ Y ∧ ϕ). The axioms of von Neumann-Bernays-G¨odel set theory, abbreviated NBG include the axiom of ex- tensionality, and all the remaining axioms of ZF, where all quantifiers in these other axioms range just over sets^11. There is one final axiom schema, the class comprehension axiom scheme: for every formula ϕ, the axiom:
∀X 1 ,... , Xn∃Y [∀x(x ∈ Y ↔ ϕ(x, X 1 ,... , Xn)]
saying that ϕ defines a class. NBG is conservative over ZF; it proves exactly the same formulas about sets. This is easily proved by showing every model of ZF can be extended to a model of NBG by adding all definable proper classes to our universe. Similarly, if we remove all the proper classes from a model of NBG, we obtain a model of ZF. Hence, if we want to discuss classes in this formal way, we can work with NBG without changing any of the facts we’ll prove about sets. NBG also has other advantages, for example it is finitely axiomatizable, while ZF is not. Choice in NBG is generally taken to be the axiom that there is a global choice class; a class function F so that for every nonempty set F (x) ∈ x.
(^11) For example, the axiom of infinity becomes ∃x∃Z(x ∈ Z ∧ x 6 = ∅ ∧ ∀y ∈ x(y ∪ {y} ∈ x).
3 Wellorderings
A strict partial order is a pair (P, <P ) where P is a set and <P is a bi- nary relation on P that is irreflexive and transitive, so (∀a ∈ P )a ≮P a), and (∀a, b, c ∈ P )(a <P b ∧ b <P c → a <P c). We say that (P, <P ) is linear if for all a, b ∈ P such that a 6 = b, either a <P b or b <P a. We write a ≤P b to mean a <P b or a = b. A strict partial order (P, <P ) is wellfounded if every nonempty subset X ⊆ P contains an element that is <P -minimal inside X. That is, (∀X ⊆ P )(∃a ∈ X)(∀b ∈ X)(b ≮P a). A linear wellfounded strict partial order is called a wellordering. Suppose (P, <P ) and (Q, <Q) are strict partial orders. Then we say a function f : P → Q is order-preserving if for all a, b ∈ P a <P b implies f (a) <Q f (b). We say that a bijection f from P to Q is an isomorphism from (P, <P ) to (Q, <Q) if for all a, b ∈ P , a <P b ↔ f (a) <Q f (b). Hence, both f and f −^1 are order preserving.
Lemma 3.1. If (P, <P ) is a wellfounded strict partial order and f : P → P is an order preserving function from (<P , P ) to (<P , P ), then f (a) ≮P a for all a ∈ P.
Proof. Let X = {a ∈ P : f (a) <P a} be the set of points which are moved “downward”. Assume for a contradiction that X is nonempty. Then by defini- tion of wellfoundedness, X must have a <P -minimal element a. By definition of X, f (a) <P a. Since a is minimal in X, and f (a) <P a, we must have f (a) ∈/ X. Now since f is order preserving, and f (a) <P a, we must have f (f (a)) <P f (a). But then f (a) ∈ X by definition of X. Contradiction!
Lemma 3.2. If (P, <P ) is a wellordering and f : P → P is an isomorphism from (P, <P ) to (P, <P ), then f is the identity.
Proof. By the previous lemma, f (a) ≥P a for all a ∈ P. Since f −^1 is also an isomorphism from (P, <P ) to (P, <P ), for all b ∈ P , f −^1 (b) ≥P b, so letting b = f (a), we see a ≥P f (a) for all a ∈ P. Hence, f is the identity.
Corollary 3.3. If (P, <P ) and (Q, <Q) are isomorphic wellorderings, then there is a unique isomorphism between them.
If (P, <P ) is a wellordering and x ∈ P , then the initial segment of (P, <P ) below x, noted (P, <P ) x is the wellordering (Q, <P Q × Q), where Q = {a ∈ P : a <P x}. An initial segment of (P, <P ) is an ordering (P, <P ) x for some x ∈ P.
Lemma 3.4. No wellordering (P, <P ) is isomorphic to an initial segment of itself.
Proof. Suppose x ∈ P , and (P, <P ) is isomorphic to (P, <P ) x for some x ∈ P via the function f , which is therefore an order preserving function from P to itself. Then f (x) <P x contradicting Lemma 3.1.
4 Ordinals
We call a set x transitive if ∀a ∈ x∀b(b ∈ a → b ∈ x). Careful: the ∈ relation on a transitive set need not be transitive.
Definition 4.1. An ordinal is a transitive set x so that the ∈ relation on x is a wellordering. We let ORD denote the class of all ordinals. If α, β are ordinals, we define α < β iff α ∈ β. We will use lowercase Greek letters α, β, γ, λ,... for ordinals.
A minor technical detail in this section is that we will not use the axiom of foundation. This is because we’ll want to use the ordinals later to prove that the axiom of foundation is equivalent to ∀x(x ∈ V ). Note for example that if α is an ordinal, α /∈ α just by the definition of ordinal: since ∈ is a strict partial order on α it is irreflexive. Our first goal is to prove that the order < on the ordinals is a wellordering.
Lemma 4.2. If α 6 = β are ordinals, and α ( β, then α ∈ β.
Proof. Let γ be the ∈-least element of the set β \ α. Since α is transitive, it follows that α is the initial segment of β given by γ. Thus, α = {ξ ∈ β : ξ < γ} = {ξ ∈ β : ξ ∈ γ} = γ, so α ∈ β.
Lemma 4.3. If α is an ordinal and β ∈ α, then β is an ordinal, and β is an initial segment of α under <.
Proof. Every element of β is an element of α by transitivity of α. So β is transitive since α is transitive. Next, β = {γ : γ ∈ β} = {γ ∈ α : γ ∈ β} = {γ ∈ α : γ < β} since every element of β is an element of α by transitivity. Finally, ∈ is a wellordering of β, since β is an initial segment of α.
It follows from this lemma that each ordinal is equal to the set of ordinals that are less than it.
α = {β : β ∈ α} = {β ∈ ORD : β ∈ α} = {β ∈ ORD : β < α}.
Now we’re ready to prove the trichotomy property for the ordering < on ORD.
Lemma 4.4. If α, β are ordinals and α 6 = β, then either α ⊆ β or β ⊆ α.
Proof. Clearly γ = α ∩ β is an ordinal. Suppose γ is not equal to α or β. Then γ ∈ α and γ ∈ β by Lemma 4.2. So γ ∈ γ, which contradicts the definition of an ordinal.
Applying Lemma 4.2 gives the following corollary:
Corollary 4.5. If α 6 = β are ordinals, then α < β or β < α. So < is a linear ordering of the class of ordinals.
Next, we show that < is a wellordering of ORD, which we’ve already shown is linear.
Lemma 4.6. < is a wellfounded ordering of the class of ordinals.
Proof. Suppose A ⊆ ORD is a nonempty class of ordinals, and α ∈ A. If α is not the least element of A, then A ∩ α is a nonempty subset of α. Hence, it has a least element.
Definition 4.7. If A is a nonempty class of ordinals, we let inf(A) denote its least element. If A is a nonempty set of ordinals, we let sup(A) denote the least ordinal that is greater than or equal to every element of A. It easy to check that sup(A) =
A.
Next, we show every wellorder is isomorphic to a unique ordinal. First we give an exercise:
Exercise 4.8. If α is a set of ordinals, then α is an ordinal iff α is a transitive set.
Note that a set α of ordinals being transitive is equivalent to being closed downwards under <; i.e. α is transitive if β ∈ α, and γ < β, implies γ ∈ α.
Lemma 4.9. If (P, <P ) is a wellordering, then (P, <P ) is isomorphic to a unique ordinal.
Proof. Every wellordering is isomorphic to at most one ordinal by Lemma 3.4. Consider F = {(x, α) : α is isomorphic to the initial segment (P, <P ) x}. It is clear that F is an order preserving map, that dom(F ) is closed downwards, and ran(F ) is a transitive set of ordinals. Finally, if dom(F ) 6 = P , then letting x be the least element of P not in dom(F ), we see that F is an isomorphism from (P, <P ) x to a set of ordinals which is closed downwards, and thus must be an ordinal. Contradiction. Hence, F is an isomorphism from P to an ordinal.
Definition 4.10. If P is a wellordering, we let ot((P, <P )) denote the unique ordinal isomorphic to P ; the ordertype of P.
There are two types of ordinals: successor ordinals and limit ordinals:
Definition 4.11. If α is an ordinal, we define α + 1 to be α ∪ {α}. We say α is a successor ordinal if there is an ordinal β so that α = β + 1. If α is not a successor ordinal, we call α a limit ordinal.
We have the following easy lemma:
Lemma 4.12. If α is an ordinal, α + 1 is an ordinal.
Proof. First, α + 1 is transitive. Suppose a ∈ α + 1 and b ∈ a. We want to show b ∈ α + 1. Case 1: if a ∈ α, then b ∈ α since α is transitive, hence b ∈ α + 1. Case 2: if a = α, then since b ∈ α, b ∈ α ∪ {α} = α + 1. The verification that α + 1 is linearly ordered by is similarly simple; since every element of α + 1 is either α, or an element of α.
5 Transfinite induction and recursion
We can now formulate the principles of transfinite induction and recursion. Transfinite induction is a proof technique we use to prove statements about all ordinals, analogously to how we use ordinary induction to prove statements about all natural numbers. Transfinite recursion lets us recursively define func- tions on the ordinals, similarly to how ordinary recursion lets us recursively define functions on ω.
Theorem 5.1 (Transfinite induction). Suppose C is a class of ordinals such that
- 0 ∈ C,
- For all α, α ∈ C → α + 1 ∈ C,
- If λ is a nonzero limit ordinal (∀α < λ)(α ∈ C) → λ ∈ C.
Then C = ORD.
Proof. Suppose λ is the least ordinal such that λ /∈ C. Then apply one of the three conditions above.
If X and Y are classes (perhaps proper classes), then a class function F from X to Y is a class that is a subclass of X × Y = {(x, y) : x ∈ X ∧ y ∈ Y }, such that for every x ∈ X, there is a unique y ∈ Y such that (x, y) ∈ F.
Theorem 5.2 (Transfinite recursion). Let G be a class function (on V ). Then there is a unique class function F such that for all α ∈ ORD,
F (α) = G(F α) (*)
Note that F α = F {β : β < α}.
Proof. First we prove uniqueness. Suppose f, f ′^ are two class functions on ordinals, or ORD satisfying () for all α ∈ dom(f ) ∩ dom(f ′). Then we claim f = f ′^ on dom(f ) ∩ dom(f ′). Suppose not, and let α be the least ordinal such that f (α) 6 = f ′(α). Then have a contradiction, since by choice of α, f ′^ α = f α. So since () is true on dom(f ) and dom(f ′),
f (α) = G(f α) = G(f ′^ α) = f ′(α).
Now we define F. Let F be the set of pairs (β, y) such that there exists an f such that dom(f ) ∈ ORD, and f (α) = G(f α) for all α ∈ dom(f ), β ∈ dom(f ), and f (β) = y. F is a function by the uniqueness we’ve proved above. We claim dom(F ) = ORD, if not, let β be the least element not in dom(F ). Then by definition, there is some function f with dom(f ) = β such that () is true on dom(f ). Now let f ′^ = f ∪ {(β, G(f ))}. Then f ′^ also satisfies (), and has domain β + 1. Contradiction!
We give some examples of transfinite induction and recursion. We will start with some operations on ordinals.
Definition 5.3. Define ordinal addition by recursion as follows:
- α + 0 = α.
- α + (β + 1) = (α + β) + 1
- α + λ = sup{α + β : β < λ}.
Technically for each fixed α, we’re defining the function β 7 → α + β by recursion. There is a different way to conceive of ordinal addition rather than this recursive definition of “iterated successor”. The ordertype of α + β is the same as the ordertype of the order of α followed by β.
Lemma 5.4. For all ordinals α, β, the ordertype of α + β is the same as the ordertype of the set α t β = { 0 } × α ∪ { 1 } × β equipped with the ordering ≺ where (m, γ) ≺ (n, δ) if n < m, or n = m and γ ≺ δ.
Proof. We prove this for each α by transfinite induction on β. Our base case is that α is isomorphic to α t ∅, and if α + β is isomorphic to α t β, then if we add one more point to each order, α + (β + 1) is isomorphic to α t β + 1. Finally, given any two isomorphic wellorders, there is a unique isomorphism between them. So if α + β is isomorphic to α t β for each β < λ via the isomorphism fβ , then
β<λ fβ^ is an isomorphism between^ α^ +^ λ^ and^ α^ t^ λ.
Similarly, we can definition ordinal multiplication and exponentiation:
Definition 5.5. Define ordinal multiplication by recursion:
- α · 0 = 0.
- α · (β + 1) = (α · β) + α
- α · λ = sup{α · β : β < λ} for limit λ.
Exercise 5.6. Show that α · β is isomorphic to “β many copies of α”. That is, α · β is isomorphic to the lexicographic order on α × β, where (γ, δ) <lex (λ, ξ) iff γ < λ or γ = λ and δ < ξ.
Caution: neither + nor times are commutative. For example, 1 + ω = ω 6 = ω + 1 and 2 · ω 6 = ω · 2.
