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# Chapter ᚚ(P): The possible journeys

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ch-T27, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [Its glossary cards](ch27-cards.md) · [Its references](ch27-refs.md) · [Previous chapter](ch26.md) · [Next chapter](ch28.md)

*A footnote such as [r-27.3] is reference entry 27.3 and links to it. Section 27 of 28; chapters are named by their Ogham letter, never by number.*

ᚚ – [Peith](ch27-cards.md#g-og-peith): associated with the dwarf elder, Section 27

## Chapter ᚚ(P): The possible journeys

> “Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können. No one shall expel us from the paradise that Cantor has created for us.”

> — David Hilbert · Über das Unendliche, *Mathematische Annalen* 95, 1926 [r-27.21](ch27-refs.md#r-27.21)

Formula: ([|∅|](ch27-cards.md#g-empty)·[ℵ<sub>0</sub>](ch27-cards.md#g-aleph0)·[ℵ<sub>1</sub>](ch27-cards.md#g-aleph1)·[ℵ<sub>ω</sub>](ch27-cards.md#g-alephomega)·[𝔠](ch27-cards.md#g-cont)·[ℶ<sub>ω</sub>](ch27-cards.md#g-bethomega)) {brace: [cardinal arithmetic](ch27.md#ch-T27-cardinal-arithmetic)} + ([ω·0](ch27-cards.md#g-ordzero) + [∅](ch27-cards.md#g-vnzero)) {brace: [ordinal arithmetic](ch27.md#ch-T27-ordinal-arithmetic)} + ([|𝒫(∅)| − 1](ch27-cards.md#g-pset0)) {brace: [power sets](ch27.md#ch-T27-power-sets)} + ([|Hom<sub>Set</sub>(∅, X)|−1](ch27-cards.md#g-initialhom)) {brace: [category theory](ch27.md#ch-T27-category-theory)}

#### Type of number

Non-real object ·

Key concepts: Cardinals, ordinals, power sets, and categories. It includes symbols you may not have seen before, such as: [ℵ](ch27-cards.md#g-aleph0), [ℶ](ch27-cards.md#g-bethomega), [𝔠](ch27-cards.md#g-cont), and [∅](ch27-cards.md#g-empty). Zero here is the empty set itself: nothing times infinity is still nothing, and von Neumann builds 0 out of [∅](ch27-cards.md#g-empty). The braces are added, not divided, because cardinals do not divide.

The charts glow on the console, full of unfamiliar stars, while home turns slowly beneath us. We could count the destinations, arrange an order for visiting them, choose a collection to explore, or trace the connections between them. Those are different jobs, and mathematics lets us consider infinite collections far beyond any journey we could finish. While we plan, open your flight bag. Still empty? Good. Nothing is exactly what this chapter needs: we will examine the empty set through its size, its order, its subsets, and its maps into other sets.

You will notice there is no fraction this time. The four braces are simply added, and each comes to zero: 0 + (0 + 0) + 0 + 0 = 0. Leaving out division is deliberate. Infinite cardinals cannot be divided the way ordinary numbers can: [ℵ₀](ch27-cards.md#g-aleph0)·1, ℵ₀·2, and ℵ₀·ℵ₀ all equal ℵ₀, so knowing a product and one of its factors does not tell you the other. In a field, multiplying by a nonzero number can always be undone; in [cardinal arithmetic](ch27-cards.md#g-cardarith) it cannot. [r-27.7](ch27-refs.md#r-27.7)

Field notes

### [cardinal arithmetic](ch27.md#ch-T27-cardinal-arithmetic) {#ch-T27-cardinal-arithmetic}

Cantor’s definition of equal size is [a bijection](ch27-cards.md#g-bijection): match every member of one set with exactly one member of another, leaving none unmatched. The [natural numbers](ch01-cards.md#g-nats) and the even natural numbers have the same size by n ↦ 2n. [r-27.26](ch27-refs.md#r-27.26) [r-27.1](ch27-refs.md#r-27.1) Cardinal multiplication counts pairs: the size of A times the size of B is the size of their Cartesian product A × B, the set of every pair (a, b) with a from A and b from B. [r-27.7](ch27-refs.md#r-27.7)

#### The displayed product

[|∅|](ch27-cards.md#g-empty) = 0; the empty set’s size, the factor that silences all the infinities beside it.

The empty set has no members. To choose a pair (a, b), there must be an a to choose from the first set; if that set is empty, no amount of choice in the second set can supply one. A Cartesian product with an empty factor has no members, so its cardinality is 0. [r-27.7](ch27-refs.md#r-27.7)

[ℵ<sub>0</sub>](ch27-cards.md#g-aleph0) and [𝔠](ch27-cards.md#g-cont) = ℵ₀ and 2<sup>ℵ₀</sup>; the two infinities everyone meets first: the [counting numbers](ch01-cards.md#g-nats) and the [real line](ch01-cards.md#g-reals).

[ℵ₀](ch27-cards.md#g-aleph0) is the smallest infinite cardinal, the size of any infinite set that can be listed. The [real numbers](ch01-cards.md#g-reals) cannot be listed; their size is [the continuum](ch27-cards.md#g-cont), [𝔠](ch27-cards.md#g-cont) = 2<sup>ℵ₀</sup>. It is also the number of ways to pick a subset of the counting numbers: each number is either in or out, two choices made ℵ₀ times over. [r-27.3](ch27-refs.md#r-27.3) [r-27.7](ch27-refs.md#r-27.7)

[ℵ<sub>1</sub>](ch27-cards.md#g-aleph1) = the next cardinal after [ℵ₀](ch27-cards.md#g-aleph0); chosen for the most famous question about infinity.

Whether 𝔠 equals [ℵ₁](ch27-cards.md#g-aleph1) is [the continuum hypothesis](ch27-cards.md#g-ch). If the standard [axioms](ch08-cards.md#g-axioms) of set theory, [ZFC](ch27-cards.md#g-zfc), are free of contradiction, they can neither prove it nor refute it. We do not need an answer to evaluate our product. [r-27.2](ch27-refs.md#r-27.2) [r-27.4](ch27-refs.md#r-27.4)

[ℵ<sub>ω</sub>](ch27-cards.md#g-alephomega) and [ℶ<sub>ω</sub>](ch27-cards.md#g-bethomega) = two limit cardinals; two ladders of infinity that climb by different rules.

Both ladders start at [ℵ₀](ch27-cards.md#g-aleph0) and have a rung for every counting number; above all those rungs sits a limit, the first size beyond every one of them. On the aleph ladder each rung is simply the next size up, ℵ₀, ℵ₁, ℵ₂, and so on, and their limit is [ℵ<sub>ω</sub>](ch27-cards.md#g-alephomega). On the beth ladder, from ℶ₀ = [ℵ₀](ch27-cards.md#g-aleph0), each rung is the size of [the power set](ch27-cards.md#g-powerset) of the rung below, the set of all its subsets, and their limit is [ℶ<sub>ω</sub>](ch27-cards.md#g-bethomega). [r-27.5](ch27-refs.md#r-27.5) [r-27.6](ch27-refs.md#r-27.6)

The first term is 0: there are no tuples with a coordinate chosen from [∅](ch27-cards.md#g-empty), so the whole product is cardinal zero.

**Fine print.** This is a legitimate operation on sizes of sets, unlike 0·[∞](ch05-cards.md#g-inf) on the extended real line, which is usually left undefined. Zero absorbs any cardinal in the same way; the formula simply picks five infinite ones to silence.

### [ordinal arithmetic](ch27.md#ch-T27-ordinal-arithmetic) {#ch-T27-ordinal-arithmetic}

An [ordinal](ch27-cards.md#g-ordarith) describes a [well-order](ch27-cards.md#g-wellorder): an ordering in which every nonempty subset has a least member. The [first infinite ordinal](ch27-cards.md#g-ordomega) ω has the order type 0, 1, 2, 3, … . Add one point after the whole sequence and we have ω + 1, a different order type with a final element. Put one point before it and the result is still ω, so 1 + ω = ω. The two orders have the same cardinality; size does not record where the extra point went. [r-27.27](ch27-refs.md#r-27.27)

#### The displayed sum

[ω·0](ch27-cards.md#g-ordzero) = 0; infinity multiplied by nothing, and in [ordinal arithmetic](ch27-cards.md#g-ordarith) that is allowed.

Ordinal multiplication arranges copies in succession. In [ω·0](ch27-cards.md#g-ordzero) there are zero copies of ω to arrange, so the result is 0. [r-27.8](ch27-refs.md#r-27.8) [r-27.9](ch27-refs.md#r-27.9)

[∅](ch27-cards.md#g-vnzero) = 0; von Neumann’s zero, where the empty set *is* the number.

Under von Neumann’s construction, each [ordinal](ch27-cards.md#g-ordarith) is the set of its predecessors: [0 = ∅](ch27-cards.md#g-vnzero), 1 = {∅}, and 2 = {∅,{∅}}. So the separate ∅ in the brace is itself [ordinal](ch27-cards.md#g-vnzero) zero. [r-27.9](ch27-refs.md#r-27.9)

The second term is (0 + 0): two [ordinal](ch27-cards.md#g-vnzero) zeros, each read as the ordinary integer 0.

**Fine print.** This ω is the [ordinal](ch27-cards.md#g-ordarith), not the surreal. The surreal [ω − 1](ch26-cards.md#g-omm1) from the previous chapter does not give an [ordinal](ch27-cards.md#g-ordarith) immediately before ω; no such predecessor exists in the [ordinal](ch27-cards.md#g-ordarith) order. The ordinals do sit among the surreals, but the two arithmetics add differently: in [ordinal arithmetic](ch27-cards.md#g-ordarith) 1 + ω = ω, while surreal addition gives 1 + ω = ω + 1, one more than ω. [r-27.10](ch27-refs.md#r-27.10)

### [power sets](ch27.md#ch-T27-power-sets) {#ch-T27-power-sets}

[The power set](ch27-cards.md#g-powerset) 𝒫(X) contains every subset of X. For every set X, [Cantor’s theorem](ch27-cards.md#g-cantorthm) says its [power set](ch27-cards.md#g-powerset) has strictly greater cardinality, and the proof is a [diagonal argument](ch27-cards.md#g-diagonal). Repeating the operation produces the beth sequence above, and shows that there is no largest cardinal. [r-27.3](ch27-refs.md#r-27.3)

#### The displayed difference

[|𝒫(∅)| − 1](ch27-cards.md#g-pset0) = 0; the first step of Cantor’s ladder, taken from nothing.

For [∅](ch27-cards.md#g-empty) the only choice is to choose no elements, so 𝒫(∅) = {∅}. This set is not empty: it has one member, and that member happens to be empty. That fits the rule met above: each member is in or out, so a set of n members has 2ⁿ subsets, and 2⁰ = 1. [r-27.7](ch27-refs.md#r-27.7) Hence [|𝒫(∅)| − 1](ch27-cards.md#g-pset0) = 1 − 1 = 0. Tuck your empty flight bag inside mine: my bag now holds one object. A bag holding an empty bag is not the same collection as a bag holding nothing. Even the empty set begins Cantor’s process by stepping from zero members to one. [r-27.12](ch27-refs.md#r-27.12)

The third term is 0: 1 − 1 = 0.

### [category theory](ch27.md#ch-T27-category-theory) {#ch-T27-category-theory}

Category theory studies objects together with their allowed maps and how those maps compose. It compares structures through the relationships between them, instead of requiring every structure to have the same kind of elements. [r-27.24](ch27-refs.md#r-27.24) Follow one leg of a journey with another, A to B and then B to C, and you have a route from A to C: that chaining is [composition](ch10-cards.md#g-maps), and every category must allow it. A route out need not come with a route back. In the [category](ch27-cards.md#g-category) of sets, the maps are ordinary functions. [r-27.24](ch27-refs.md#r-27.24)

#### The displayed difference

[|Hom<sub>Set</sub>(∅, X)|−1](ch27-cards.md#g-initialhom) = 0; [the one arrow out of the empty set](ch27-cards.md#g-emptyfn), which makes it *initial*.

Hom<sub>Set</sub>(∅, X) is the collection of functions from [∅](ch27-cards.md#g-empty) to X. A function must give an output to every input; with no inputs, there are no assignments to make and nothing to distinguish one such function from another. There is exactly one, [the empty function](ch27-cards.md#g-emptyfn), whose list of input–output pairs is empty; so the count is 1, and 1 − 1 = 0. Follow that unique map ∅ → X by any map X → Y and the result must be the unique map ∅ → Y: [composition](ch10-cards.md#g-maps), the operation categories are built on, is already visible here. An object with exactly one arrow to every object is called *initial*, and ∅ is initial in the [category](ch27-cards.md#g-category) of sets. [r-27.25](ch27-refs.md#r-27.25) [r-27.14](ch27-refs.md#r-27.14) In the other direction there is no function from a nonempty X to ∅: an input would need an output where none exists.

The fourth and final term is 0: 1 − 1 = 0. The four terms are added: 0 + (0 + 0) + 0 + 0 = 0.

**Fine print.** This does not claim that every [category](ch27-cards.md#g-category) has [an initial object](ch27-cards.md#g-initialobj), or that every [initial object](ch27-cards.md#g-initialobj) is an empty set.

What this chapter adds to the formula: 0 + (0 + 0) + 0 + 0 = 0

By size the empty set gave 0; as an ordinal it was 0; by subsets it gave one choice; by maps it gave [one arrow to every set](ch27-cards.md#g-emptyfn). Four viewpoints recognized the same small object in four distinct ways.

Take your bag back; both are still empty, and that is fine. We have learned to tell apart how many, in what order, which selections, and which connections. Did you hear the overhead announcement that there is a problem with the engine? There are always problems—oh well, looks like we are heading home. Speaking of problems: some of the hardest problems of all were waiting beside the road we already traveled.

Watch and read

- [Infinity is bigger than you think - Numberphile](https://www.youtube.com/watch?v=elvOZm0d4H0) Numberphile with James Grime: Cantor’s diagonal argument in ten minutes

- [How To Count Past Infinity](https://www.youtube.com/watch?v=SrU9YDoXE88) Vsauce, 2016: ordinals, cardinals, ω, and the alephs, for a general audience

- [Cardinal number](https://en.wikipedia.org/wiki/Cardinal_number) Wikipedia

- [Ordinal number](https://en.wikipedia.org/wiki/Ordinal_number) Wikipedia