10.2 Cardinal numbers
Definition 10.2.1.
The type of cardinal numbersis the 0-truncation of the type \\setof sets:
Thus, a cardinal number, or cardinal, is an inhabitant of .Technically, of course, there is a separate type associated to each universe .
As usual for truncations, if is a set, then denotes its image under the canonical projection ; we call the cardinality of .By definition, is a set.It also inherits the structure of a semiring from \\set.
Definition 10.2.2.
The operation of cardinal addition
is defined by induction on truncation:
Proof.
Since is a set, to define for all , by induction it suffices to assume that is for some .Now we want to define , i.e. we want to define for all .However, since is a set, by induction it suffices to assume that is for some .But now we can define to be .∎
Definition 10.2.3.
Similarly, the operation of cardinal multiplication
is defined by induction on truncation:
Lemma 10.2.4.
is a commutative semiring, i.e. for we have the following.
where and .
Proof.
We prove the commutativity of multiplication, ; the others are exactly analogous.Since is a set, the type is a mere proposition, and in particular a set.Thus, by induction it suffices to assume and are of the form and respectively, for some .Now and , so it suffices to show .Finally, by univalence, it suffices to give an equivalence .But this is easy: take and its obvious inverse.∎
Definition 10.2.5.
The operation of cardinal exponentiation is also defined by induction on truncation:
Lemma 10.2.6.
For we have
Proof.
Exactly like \\autorefcard:semiring.∎
Definition 10.2.7.
The relation of cardinal inequality
is defined by induction on truncation:
where is the type of injections from to .In other words, means that there merely exists an injection from to .
Lemma 10.2.8.
Cardinal inequality is a preorder, i.e. for we have
Proof.
As before, by induction on truncation.For instance, since is a mere proposition, by induction on 0-truncation we may assume , , and are , , and respectively.Now since is a mere proposition, by induction on -truncation we may assume given injections and .But then is an injection from to , so holds.Reflexivity is even easier.∎
We may likewise show that cardinal inequality is compatible with the semiring operations.
Lemma 10.2.9.
Consider the following statements:
- 1.
There is an injection .
- 2.
There is a surjection .
Then, assuming excluded middle:
- •
Given , we have 12.
- •
Therefore, if is merely inhabited, we have 1 merely 2.
- •
Assuming the axiom of choice
, we have 2 merely 1.
Proof.
If is an injection, define at as follows.Since is injective, the fiber of at is a mere proposition.Therefore, by excluded middle, either there is an with , or not.In the first case, define ; otherwise set .Then for any , we have , so is surjective.
The second statement follows from this by induction on truncation.For the third, if is surjective, then by the axiom of choice, there merely exists a function with for all .But then must be injective.∎
Theorem 10.2.10 (Schroeder–Bernstein).
Assuming excluded middle, for sets and we have
Proof.
The usual “back-and-forth” argument applies without significant changes.Note that it actually constructs an isomorphism (assuming excluded middle so that we can decide whether a given element belongs to a cycle, an infinite chain, a chain beginning in , or a chain beginning in ).∎
Corollary 10.2.11.
Assuming excluded middle, cardinal inequality is a partial order, i.e. for we have
Proof.
Since is a mere proposition, by induction on truncation we may assume and are and , respectively, and that we have injections and .But then the Schroeder–Bernstein theorem gives an isomorphism , hence an equality .∎
Finally, we can reproduce Cantor’s theorem, showing that for every cardinal there is a greater one.
Theorem 10.2.12 (Cantor).
For , there is no surjection .
Proof.
Suppose is any function, and define by .If , then but , a contradiction.Thus, is not surjective.∎
Corollary 10.2.13.
Assuming excluded middle, for any , there is a cardinal such that and .
Proof.
Let .Now we want to show a mere proposition, so by induction we may assume is , so that .Using excluded middle, we have a function defined by
And if , then , so ; hence is injective.Thus, .
On the other hand, if , then we would have an injection .By \\autorefthm:injsurj, since we have and excluded middle, there would then be a surjection , contradicting Cantor’s theorem.∎