用户: Scavenger/Set Theory (and conventions)

We use [1] as our standard reference.

1Ordinals

Definition 1.1. A set $T$ is transitive if $T\subseteq P(T)$, where $P()$ is the power set.

Definition 1.2. A set $T$ is an ordinal if it is transitive and well ordered by $\in$.

Proposition 1.3. Every well ordered set is isomorphic (as a well ordered set) to a unique ordinal (whose well ordering is provided by $\in$).

Proposition 1.4. The class of ordinals form a well ordered class, with order provided by $\in$ (of course this is not strictly a set theoretic statement since classes do not exist in ZFC, but we will leave it to the reader (who cares) to figure out the rigorous set theoretic statement). An element of an ordinal is an ordinal.

Thus in the case where $\alpha$ and $\beta$ are ordinals we will also write $\alpha <\beta$ for $\alpha \in \beta$.

Remark 1.5. Proposition 1.4 is the substantive form of transfinite induction and recursion.

Convention 1.6. Let $n$ be a natural number, we will sometimes use $n$ to denote the unique ordinal with $n$ elements. (In fact, the natural numbers can (and should) be constructed from the finite ordinals.)

Definition 1.7. Let $\alpha$ be an ordinal, we define the successor of $\alpha$, denoted $\alpha +1$, to be the minimal ordinal $>\alpha$.

Definition 1.8. An ordinal is called a limit ordinal if it is not the successor of another ordinal.

2Cardinals

Definition 2.1. An ordinal $\alpha$ is a cardinal if for every $\beta \in \alpha$, there exists no bijection from $\alpha$ to $\beta$.

Proposition 2.2. Every set is in bijection with a unique cardinal.

Definition 2.3. Let $X$ be a set, we denote the unique cardinal in bijection with $X$ by $\mathrm{card}(X)$. We call $\mathrm{card}(X)$ the cardinality of $X$.

Proposition 2.4. Let $X$ and $Y$ be sets. There exists an injection from $X$ to $Y$ if and only if $\mathrm{card}(X)\le \mathrm{card}(Y)$. If there exists a surjection from $X$ to $Y$, then $\mathrm{card}(X)\ge \mathrm{card}(Y)$.

Proposition 2.5. The supremum of a set of cardinals is a cardinal.

Arithmetic of Cardinals

Definition 2.6. Let $\kappa$ and $\lambda$ be cardinals. We define

$\kappa +\lambda =\mathrm{card}(\kappa \coprod \lambda )$

$\kappa \cdot \lambda =\mathrm{card}(\kappa \times \lambda )$

${\kappa }^{\lambda }=\mathrm{card}({\kappa }^{\lambda })$

Proposition 2.7. Let $\kappa ,\lambda$ and $\mu$ be cardinals, then:

(i) $+$ and $\cdot$ are associative, commutative and distributive.

(ii) $(\kappa \cdot \lambda {)}^{\mu }={\kappa }^{\mu }\cdot {\lambda }^{\mu }$.

(iii) ${\kappa }^{\lambda +\mu }={\kappa }^{\lambda }\cdot {\kappa }^{\mu }$.

(iv) $({\kappa }^{\lambda }{)}^{\mu }={\kappa }^{\lambda \cdot \mu }$.

(v) If $\kappa \le \lambda$, then ${\kappa }^{\mu }\le {\lambda }^{\mu }$.

(vi) If $0<\lambda \le \mu$, then ${\kappa }^{\lambda }\le {\kappa }^{\mu }$.

(vii) ${\kappa }^0=1$, $1^{\kappa }=1$, $0^{\kappa }=0$ if $\kappa >0$.

Theorem 2.8 (Cantor). Let $\kappa$ be a cardinal, then $\kappa <2^{\kappa }$.

Proof. See [1] Theorem 3.1.

$\square$

Proposition 2.9. Let $\kappa$ and $\lambda$ be cardinals such that at least one of them is infinite, then $\kappa +\lambda =\kappa \cdot \lambda =\max (\kappa ,\lambda )$.

Proof. See [1] Theorem 3.5.

$\square$

3Cofinality

Definition 3.1. Let $\alpha$ be an ordinal. A subset $A\subseteq \alpha$ is called a cofinal subset if for every $\beta \in \alpha$, there exists a $\gamma \in A$ such that $\beta \le \gamma$.

Definition 3.2. Let $\alpha$ be an ordinal. We define $\mathrm{cf}(\alpha )$, called the cofinality of $\alpha$, to be the minimal ordinal $\beta$ such that there exists a cofinal subset $A\subseteq \alpha$ which is isomorphic as a well ordered set (the ordering of $A$ comes from that of $\alpha$) to $\beta$.

Proposition 3.3. Let $\alpha$ be an ordinal, then $\mathrm{cf}(\alpha )$ is a cardinal, and $\mathrm{cf}(\mathrm{cf}(\alpha ))=\mathrm{cf}(\alpha )$.

Definition 3.4. An infinite cardinal $\alpha$ is called regular if $\mathrm{cf}(\alpha )=\alpha$.

Definition 3.5. An cardinal $\kappa$ is called inaccessible (or strongly inaccessible) if:

(i) $\kappa$ is uncountable.

(ii) $\kappa$ is regular.

(iii) Let $\lambda$ be a cardinal such that $\lambda <\kappa$, then $2^{\lambda }<\kappa$.

Proposition 3.6. Let $\kappa$ be an inaccessible cardinal, let $\lambda ,\mu <\kappa$ be cardinals. Then ${\lambda }^{\mu }<\kappa$.

Proof. We may assume $\lambda$ and $\mu$ are infinite, then ${\lambda }^{\mu }\le (2^{\lambda }{)}^{\mu }=2^{\lambda \cdot \mu }=2^{\max (\lambda ,\mu )}<\kappa$.

$\square$

4Grothendieck Universes

Definition 4.1. A set $\text{U}$ is a universe if it satisfies the following conditions:

(i) If $x\in \text{U}$ and $y\in x$, then $y\in U$.

(ii) If $x,y\in \text{U}$, then $\{x,y\}\in \text{U}$.

(iii) If $x\in \text{U}$, then $P(x)\in \text{U}$, where $P()$ means the power set.

(iv) If $I\in \text{U}$ and $\{x_i{\}}_{i\in I}$ is a family of sets with each $x_i\in \text{U}$, then ${\cup }_{i\in I}{x}_i\in \text{U}$.

(v) $\omega \in \text{U}$, where $\omega$ is the smallest infinite ordinal.

We will mostly use the symbol $\text{U}$ (or some variants of it) to refer to a universe.

We can now formulate the Axiom of Universes: For every set $x$, there exists a universe $\text{U}$ such that $x\in \text{U}$.

The main inituition about a universe $\text{U}$ is that sets inside $\text{U}$ form a smaller model of ZFC, inside ZFC itself. This allows us to talk about notions like “small sets”, “big sets” and even bigger sets. This intuition leads to the following convention:

Convention 4.2. A definition without a prefix $\text{U}$- is meant to be the general definition suitable for all sets, without refering to some specific universe. The same definition with a prefix $\text{U}$- is meant to be the version of the definition inside the model of ZFC in the universe $\text{U}$. The same definition with the adjective $\text{U}$-small means something that is isomorphic to a $\text{U}-$ version.

As a simple example, $\text{U}$-sets means sets in $\text{U}$, and $\text{U}$-small sets means sets that are in bijection with a set in $\text{U}$.

Most of the time, a definition is the “filtered colimit” version of its $\text{U}$- versions. In good situations (which is the major case), a definition is independant of a choice of universe, meaning that it coincides with its $\text{U}$- version when restricted to $\text{U}$-sets.

Convention 4.3. When constructing (set theoretically) small mathematical objects (for example the natural numbers and other constructions derived from this), we always suitably arrange the definition such that the resulting object lies in every universe.

5The Hierarchy of Sets

Definition 5.1. We define, by transfinite recursion, a set $V_{\alpha }$ for every ordinal $\alpha$,by letting $V_0=\varnothing$, $V_{\alpha +1}=P(V_{\alpha })$ ($P()$ means power set), and for a limit ordinal $\alpha$, $V_{\alpha }={\cup }_{\beta <\alpha }{V}_{\beta }$.

Proposition 5.2. Let $\alpha$ and $\beta$ be ordinals, then:

(i) $V_{\alpha }$ is transitive.

(ii) If $\alpha <\beta$, then $V_{\alpha }\subseteq V_{\beta }$.

(iii) $\alpha \subseteq V_{\alpha }$.

Proposition 5.3. Every set is an element of $V_{\alpha }$ for some ordinal $\alpha$.

Definition 5.4. The rank of a set $X$ is defined to be the minimal ordinal $\alpha$ such that $X\in V_{\alpha +1}$.

Proposition 5.5. A set is a universe if and only if it is equal to $V_{\kappa }$ for some inaccessible cardinal $\kappa$.

Proof. See [2], Exp. I, Appendice.

$\square$

For this reason, we will often abuse notation to identify an inaccessible cardinal $\kappa$ with the corresponding universe $V_{\kappa }$ (though they are never equal). For example, when we say $\kappa$-sets we mean $V_{\kappa }$-sets, i.e. those sets of rank $<\kappa$.

Proposition 5.6. Let $\kappa$ be an inaccessible cardinal, then a set $X$ is isomorphic to an element of $V_{\kappa }$ if and only if $\mathrm{card}(X)<\kappa$.

Proof. To prove $\mathrm{card}(X)<\kappa$ if $X\in V_{\kappa }$, one prove by transfinite induction that $\mathrm{card}(V_{\alpha })<\kappa$ for every ordinal $\alpha <\kappa$. To prove the converse, notice that $\alpha \in V_{\alpha +1}$ if $\alpha$ is an ordinal.

$\square$

6$\alpha$-inaccessible cardinals

Definition 6.1. We define, by induction, the notion of $\alpha$-inaccessible cardinals for every ordinal $\alpha$:

(i) $0$-inaccessible cardinals are the same as the inaccessible cardinals.

(ii) Let $\alpha >0$ be an ordinal. Then a cardinal $\kappa$ is $\alpha$-inaccessible if it is inaccessible, and for every ordinal $\beta <\alpha$, the set of $\beta$-inaccessible cardinals less than $\kappa$ is unbounded in $\kappa$.

Convention 6.2. We work in the ZFC system with an additional axiom:

For every pair of ordinals $\alpha ,\kappa$, there exists an $\alpha$-inaccessible cardinal ${\kappa }^{\prime }>\kappa$.

We refuse to manipulate classes (though in very few situations we will need to consider classes), so all our mathematical objects will be sets required to satisfy certain properties. Instead we will use ($\alpha$-)inaccessible cardinals (or universes) to distinguish between small and large sets.

References

[1]	Thomas Jech, “Set Theory”, Springer Monographs in mathematics, Springer, 2002.
[2]	Alexander Grothendieck, “Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos.” Lecture Notes in Mathematics, Vol. 269. Séminar de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4), Dirigé par M.Artin, A.Grothendieck, et J.L.Verdier. Avec la collaboration de N.Bourbaki, P.Deligne et B.Saint-Donat. Berlin: Springer-Verlag, 1972, pp. xix+525.