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单词 102CardinalNumbers
释义

10.2 Cardinal numbers


Definition 10.2.1.

The type of cardinal numbersMathworldPlanetmathis the 0-truncation of the type \\setof sets:

𝖢𝖺𝗋𝖽:\\set0

Thus, a cardinal number, or cardinal, is an inhabitant of Card\\set0.Technically, of course, there is a separate type CardU associated to each universePlanetmathPlanetmath U.

As usual for truncations, if A is a set, then |A|0 denotes its image under the canonical projection \\set\\set0𝖢𝖺𝗋𝖽; we call |A|0 the cardinality of A.By definition, 𝖢𝖺𝗋𝖽 is a set.It also inherits the structureMathworldPlanetmath of a semiring from \\set.

Definition 10.2.2.

The operationMathworldPlanetmath of cardinal additionMathworldPlanetmath

(+):𝖢𝖺𝗋𝖽𝖢𝖺𝗋𝖽𝖢𝖺𝗋𝖽

is defined by inductionMathworldPlanetmath on truncation:

|A|0+|B|0:|A+B|0.
Proof.

Since 𝖢𝖺𝗋𝖽𝖢𝖺𝗋𝖽 is a set, to define (α+):𝖢𝖺𝗋𝖽𝖢𝖺𝗋𝖽 for all α:𝖢𝖺𝗋𝖽, by induction it suffices to assume that α is |A|0 for some A:\\set.Now we want to define (|A|0+):𝖢𝖺𝗋𝖽𝖢𝖺𝗋𝖽, i.e. we want to define |A|0+β:𝖢𝖺𝗋𝖽 for all β:𝖢𝖺𝗋𝖽.However, since 𝖢𝖺𝗋𝖽 is a set, by induction it suffices to assume that β is |B|0 for some B:\\set.But now we can define |A|0+|B|0 to be |A+B|0.∎

Definition 10.2.3.

Similarly, the operation of cardinal multiplication

():𝖢𝖺𝗋𝖽𝖢𝖺𝗋𝖽𝖢𝖺𝗋𝖽

is defined by induction on truncation:

|A|0|B|0:|A×B|0
Lemma 10.2.4.

𝖢𝖺𝗋𝖽 is a commutativePlanetmathPlanetmathPlanetmathPlanetmath semiring, i.e. for α,β,γ:Card we have the following.

(α+β)+γ=α+(β+γ)
α+0=α
α+β=β+α
(αβ)γ=α(βγ)
α1=α
αβ=βα
α(β+γ)=αβ+αγ

where 0:|0|0 and 1:|1|0.

Proof.

We prove the commutativity of multiplication, αβ=βα; the others are exactly analogous.Since 𝖢𝖺𝗋𝖽 is a set, the type αβ=βα is a mere proposition, and in particular a set.Thus, by induction it suffices to assume α and β are of the form |A|0 and |B|0 respectively, for some A,B:\\set.Now |A|0|B|0|A×B|0 and |B|0×|A|0|B×A|0, so it suffices to show A×B=B×A.Finally, by univalence, it suffices to give an equivalence A×BB×A.But this is easy: take (a,b)(b,a) and its obvious inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.∎

Definition 10.2.5.

The operation of cardinal exponentiation is also defined by induction on truncation:

|A|0|B|0:|BA|0.
Lemma 10.2.6.

For α,β,γ:Card we have

α0=1
1α=1
α1=α
αβ+γ=αβαγ
αβγ=(αβ)γ
(αβ)γ=αγβγ
Proof.

Exactly like \\autorefcard:semiring.∎

Definition 10.2.7.

The relationMathworldPlanetmathPlanetmath of cardinal inequality

():𝖢𝖺𝗋𝖽𝖢𝖺𝗋𝖽𝖯𝗋𝗈𝗉

is defined by induction on truncation:

|A|0|B|0:𝗂𝗇𝗃(A,B)

where inj(A,B) is the type of injections from A to B.In other words, |A|0|B|0 means that there merely exists an injection from A to B.

Lemma 10.2.8.

Cardinal inequality is a preorderMathworldPlanetmath, i.e. for α,β:Card we have

αα
(αβ)(βγ)(αγ)
Proof.

As before, by induction on truncation.For instance, since (αβ)(βγ)(αγ) is a mere proposition, by induction on 0-truncation we may assume α, β, and γ are |A|0, |B|0, and |C|0 respectively.Now since |A|0|C|0 is a mere proposition, by induction on (-1)-truncation we may assume given injections f:AB and g:BC.But then gf is an injection from A to C, so |A|0|C|0 holds.ReflexivityMathworldPlanetmath 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. 1.

    There is an injection AB.

  2. 2.

    There is a surjection BA.

Then, assuming excluded middle:

  • Given a0:A, we have 12.

  • Therefore, if A is merely inhabited, we have 1 merely 2.

  • Assuming the axiom of choiceMathworldPlanetmath, we have 2 merely 1.

Proof.

If f:AB is an injection, define g:BA at b:B as follows.Since f is injective, the fiber of f at b is a mere proposition.Therefore, by excluded middle, either there is an a:A with f(a)=b, or not.In the first case, define g(b):a; otherwise set g(b):a0.Then for any a:A, we have a=g(f(a)), so g is surjective.

The second statement follows from this by induction on truncation.For the third, if g:BA is surjective, then by the axiom of choice, there merely exists a function f:AB with g(f(a))=a for all a.But then f must be injective.∎

Theorem 10.2.10 (Schroeder–Bernstein).

Assuming excluded middle, for sets A and B we have

𝗂𝗇𝗃(A,B)𝗂𝗇𝗃(B,A)(AB)
Proof.

The usual “back-and-forth” argument applies without significant changes.Note that it actually constructs an isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath AB (assuming excluded middle so that we can decide whether a given element belongs to a cycle, an infinite chain, a chain beginning in A, or a chain beginning in B).∎

Corollary 10.2.11.

Assuming excluded middle, cardinal inequality is a partial orderMathworldPlanetmath, i.e. for α,β:Card we have

(αβ)(βα)(α=β).
Proof.

Since α=β is a mere proposition, by induction on truncation we may assume α and β are |A|0 and |B|0, respectively, and that we have injections f:AB and g:BA.But then the Schroeder–Bernstein theorem gives an isomorphism AB, hence an equality |A|0=|B|0.∎

Finally, we can reproduce Cantor’s theorem, showing that for every cardinal there is a greater one.

Theorem 10.2.12 (Cantor).

For A:\\set, there is no surjection A(A2).

Proof.

Suppose f:A(A𝟐) is any function, and define g:A𝟐 by g(a):¬f(a)(a).If g=f(a0), then g(a0)=f(a0)(a0) but g(a0)=¬f(a0)(a0), a contradictionMathworldPlanetmathPlanetmath.Thus, f is not surjective.∎

Corollary 10.2.13.

Assuming excluded middle, for any α:Card, there is a cardinal β such that αβ and αβ.

Proof.

Let β=2α.Now we want to show a mere proposition, so by induction we may assume α is |A|0, so that β|A𝟐|0.Using excluded middle, we have a function f:A(A𝟐) defined by

f(a)(a):{1𝟐a=a0𝟐aa.

And if f(a)=f(a), then f(a)(a)=f(a)(a)=1𝟐, so a=a; hence f is injective.Thus, α|A|0|A𝟐|02α.

On the other hand, if 2αα, then we would have an injection (A𝟐)A.By \\autorefthm:injsurj, since we have (λx. 0𝟐):A𝟐 and excluded middle, there would then be a surjection A(A𝟐), contradicting Cantor’s theorem.∎

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