Notes on set theory

Much of this (up to and including cofinality) is taken from Set Theory by Thomas Jech [Jec03], but you can probably find it in any reasonable set theory book. The rest is from Hovey's book Model Categories [Hov99].

Well-ordered sets

A linearly ordered set (P, < ) is well-ordered if every nonempty subset of P has a least element. A map f of ordered sets is increasing if x ≥ y ⇒ f (x) ≥ f (y).

Lemma 1.1 Suppose that (W, < ) is well-ordered. If f : W → W is an increasing function, then f (x) ≥ x for all x ∈ W.

Corollary 1.2 The only automorphism of a well-ordered set is the identity.

Corollary 1.3 If W₁ and W₂ are isomorphic well-ordered sets, there is a unique isomorphism between them.

Given a well-ordered set W and an element u ∈ W, the initial segment given by u is {x ∈ W : x < u}.

Lemma 1.4 No well-ordered set is isomorphic to an initial segment of itself.

Theorem 1.5 For any well-ordered sets W₁ and W₂, exactly one of the following holds:

W₁ is isomorphic to W₂.
W₁ is isomorphic to an initial segment of W₂.
W₂ is isomorphic to an initial segment of W₁.

If W₁ and W₂ are isomorphic well-ordered sets, say that they have the same order type.

Ordinals

A set T is transitive if every element of T is a subset of T. (That is, T is a subset of its power set.) A set is an ordinal number (or an ordinal) if it is transitive and well-ordered by ∈. For ordinals α and β, define α < β if α ∈ β.

0 = ∅ is an ordinal.
If α is an ordinal and β ∈ α, then β is an ordinal.
If α and β are distinct ordinals and α ⊂ β, then α ∈ β.
If α and β are ordinals, then either α ⊂ β or β ⊂ α.

The class of ordinals, Ord, has these properties:

Ord is linearly ordered by <.
For each α ∈ Ord, α = {β : β < α}.
For each α, α ∪ {α} is an ordinal, and α ∪ {α} = inf{β : β > α}.

So define α + 1 to be α ∪ {α}; this is called the successor to α. If α = β + 1 for some β, then α is called a successor ordinal. If α is not a successor ordinal, then α = sup{β : β < α}, and α is called a limit ordinal.

Theorem 2.1 Every well-ordered set is isomorphic to a unique ordinal number.

Let β be a limit ordinal. A β-sequence is a function whose domain is β:

⟨a_ξ : ξ < β⟩.

If ⟨ α_ξ : ξ < β⟩ is a nondecreasing sequence of ordinals (so ξ < η implies α_ξ < α_η), then the limit of the sequence is

lim_{ξ → β} α_ξ = sup {α_ξ : ξ < β}.

Examples of ordinals:

0	= {}
1	= {0}
2	= {0, 1}
3	= {0, 1, 2}
...
ω	= {0, 1, 2, ... }
ω + 1	= {0, 1, 2, ... ;ω}
...

Cardinals

An ordinal α is a cardinal number if |α| ≠ |β| for all β < α.

If W is well-ordered, there is an ordinal α such that |W| = |α|, so let |W| denote the least ordinal α such that |W| = |α|. (By the axiom of choice, every set can be well-ordered, so we can extend this notation to any set W.)

Note: all infinite cardinals are limit ordinals.

Cofinality and κ-filtered ordinals

Let α > 0 be a limit ordinal. An increasing β-sequence ⟨ α_ξ : ξ < β⟩, for β a limit ordinal, is cofinal in α if lim_{ξ → β} α_ξ = α. Similarly, A ⊂ α is cofinal in α if sup A = α.

For α an infinite limit ordinal, let cf α be the least limit ordinal β such that there is an increasing β-sequence ⟨ α_ξ : ξ < β⟩ with lim_{ξ → β} α_ξ = α; this is called the cofinality of α. Note that cf α ≤ α always.

From [Hov99, Section 2.1]: given a cardinal κ, an ordinal α is κ-filtered if it is a limit ordinal and, if A ⊆α and |A| ≤ κ, then sup A < α. This holds if and only if the cofinality of α is greater than κ.

For example, if κ is finite, then a κ-filtered ordinal is just an infinite limit ordinal. If κ is infinite, the smallest κ-filtered ordinal is the first cardinal κ₁ larger than κ.

Smallness

(This material is taken from [Hov99, Section 2.1].)

Suppose that C is a category with all small colimits, D is a collection of morphisms of C, A is an object of C and κ is a cardinal. We say that A is κ-small relative to D if, for all κ-filtered ordinals λ and all λ-sequences

X₀ → X₁ → ... → X_β → ...

such that each map X_β → X_β+1 is in D, the map of sets

colim_{β < λ} C(A, X_β) → C(A, colim_{β < λ} X_β)

is an isomorphism. We say that A is small relative to D if it is κ-small for some κ. We say that A is small if it is small relative to the whole category C. We say that A is finite relative to D if it is κ-small relative to D for some finite cardinal κ, and similarly, A is finite if it is finite relative to C. Finite means that maps from A commute with colimits of arbitrary λ-sequences, for any limit ordinal λ.

Examples:

In the category of sets, every set is small. The finite sets (in the usual sense) are precisely the sets that are finite according to the above definition.
In the category of R-modules, every module is small. Every finitely presented R-module is finite.
In the category of topological spaces, the set {0, 1} with the indiscrete topology is not small.

Bibliography

Hov99: M. Hovey, Model categories, American Mathematical Society, Providence, RI, 1999.
Jec03: Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003, The third millennium edition, revised and expanded.

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Notes on set theory

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