# Chapter ᚃ(F): The local park
*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ch-T5, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*
[Guide for AI readers](index.md) · [Its glossary cards](ch05-cards.md) · [Its references](ch05-refs.md) · [Previous chapter](ch04.md) · [Next chapter](ch06.md)
*A footnote such as [r-5.3] is reference entry 5.3 and links to it. Section 5 of 28; chapters are named by their Ogham letter, never by number.*
ᚃ – [Fearn](ch05-cards.md#g-og-fern): associated with the alder tree, Section 5
## Chapter ᚃ(F): The local park
> “Wir müssen wissen. Wir werden wissen. (We must know. We will know.)”
> — David Hilbert · closing words of his radio address at Königsberg, 8 September 1930; Gödel’s incompleteness theorem was announced at the same meeting [r-5.27](ch05-refs.md#r-5.27)
Formula: (([Legendre’s](ch05-cards.md#g-legendre) − 1))/(([EEB](ch05-cards.md#g-eb) + [ψ](ch05-cards.md#g-recipfib) + [Copeland–Erdős](ch05-cards.md#g-cope) + [ζ(3)](ch05-cards.md#g-apery) + [Mills’](ch05-cards.md#g-mills)) {brace: irrational, transcendence open} · ([γ](ch05-cards.md#g-gamma) + [δ](ch05-cards.md#g-feig) + |[αF](ch05-cards.md#g-feigA)| + [G](ch05-cards.md#g-catalan) + [K](ch05-cards.md#g-khin) + [A](ch05-cards.md#g-glaisher) + [B2](ch05-cards.md#g-brun) + [Gompertz](ch05-cards.md#g-gompertz) + [C2](ch05-cards.md#g-twinprime) + [Meissel–Mertens](ch05-cards.md#g-mm) + [Golomb–Dickman](ch05-cards.md#g-gd) + [Landau–Ramanujan](ch05-cards.md#g-lr) + [ζ(5)](ch05-cards.md#g-zeta5) + [γL](ch05-cards.md#g-levy) + [F](ch05-cards.md#g-fransen) + [β](ch05-cards.md#g-bernstein) + [1.1319…](ch05-cards.md#g-viswanath) + [cN](ch05-cards.md#g-niven) + [CArtin](ch05-cards.md#g-artin) + [KS](ch05-cards.md#g-sierpinski) + [MRB](ch05-cards.md#g-mrb) + [σ](ch05-cards.md#g-somos) + [ρKB](ch05-cards.md#g-keplerb) + [BN](ch05-cards.md#g-backhouse) + [ππ + ee](ch05-cards.md#g-pipi) + [(π + e) + π·e + πe](ch05-cards.md#g-pie)) {brace: not even known irrational} ) {brace: [unresolved constants](ch05.md#ch-T5-unresolved-constants)} · ([lim](ch05-cards.md#g-limop)n→[∞](ch05-cards.md#g-inf)(([TREE(3)](ch04-cards.md#g-tree3))/([Rayo’s](ch05-cards.md#g-rayo)))^(n))/(([Ω](ch05-cards.md#g-omega)·[BB(6)](ch05-cards.md#g-bb6))) {brace: [uncomputable and infeasible](ch05.md#ch-T5-uncomputable-and-infeasible)}
#### Type of number
Status open · Definable but uncomputable · Infinity or limit device ·
Key concepts: Unresolved constants, and uncomputable and infeasible numbers. It includes symbols you may not have seen before, such as: Chaitin’s [Ω](ch05-cards.md#g-omega), [BB(6)](ch05-cards.md#g-bb6), and [lim](ch05-cards.md#g-limop). Zero here comes from numbers we only half know: a constant that turned out to be exactly one, and a limit that flattens even TREE(3). A number can be perfectly definite and still unknown.
I see this park almost daily from my deck, but a familiar view can still contain blind spots. The pond has a depth nobody has measured, the old oak has more leaves than anyone will count, and the path I walk every evening has a length I have never checked: three ways of not quite knowing a thing you see daily. Here we explore numbers we *kinda* know. That sounds like an English major’s distinction, but it opens several good mathematical questions. We might know a number’s digits without knowing what kind of number it is. We might have a precise definition and no algorithm capable of calculating it to any requested accuracy. We might even be looking at an ordinary finite integer, as definite as 7, whose exact value we have yet to determine.
Some gaps may close with the next proof. Some are limits that mathematics has already proved. Others appear when a perfectly finite calculation asks more of the world than the world can supply. I have gathered the “kinda known” into two groups: unresolved constants, and [uncomputable](ch05-cards.md#g-uncomputable) and [infeasible](ch05-cards.md#g-infeasibility) quantities. What earns them a place in the formula is a simple fact: *well-defined* does not mean *computable*, yet we can sometimes settle the value of a whole expression without knowing the digits of everything inside it. [r-5.22](ch05-refs.md#r-5.22) [r-5.23](ch05-refs.md#r-5.23)
Field notes
### [unresolved constants](ch05.md#ch-T5-unresolved-constants) {#ch-T5-unresolved-constants}
Numbers whose digits we can compute (one of them only under a famous hypothesis, another only by extrapolation), but whose *kind* nobody has yet proved. The gap is in what they are, not in their digits.
#### Numerator
[Legendre’s](ch05-cards.md#g-legendre) − 1 = 0; a mystery that ended, placed as encouragement above a denominator full of open questions.
In 1808 Legendre, estimating [how many primes lie below a given size](ch18-cards.md#g-primecount), put a correction of about 1.08366 into his formula. In 1899 the right value of that same correction, in the long run, was proved to be exactly 1. The number never changed; our understanding of it did. [r-5.1](ch05-refs.md#r-5.1)
#### Denominator
Two sums, multiplied, with two different gaps in our knowledge. In the first we know the numbers are irrational but not whether they are transcendental. In the second we do not yet know whether they are fractions at all. Knowing a definition, or thousands of digits, need not settle what kind of number we have.
Irrational, transcendence open
Proved not to be fractions, but not yet shown to be algebraic, like [√2](ch04-cards.md#g-sqrt2), or transcendental, like [π](ch04-cards.md#g-pi).
[ζ(3)](ch05-cards.md#g-apery) ≈ 1.2021; Apéry’s constant, this sum’s flagship.
It is [the sum of the reciprocal cubes](ch05-cards.md#g-apery), 1 + 1/8 + 1/27 + …. Roger Apéry proved in 1978 that it is not a fraction; whether it is transcendental remains open. [r-5.5](ch05-refs.md#r-5.5)
[Mills’](ch05-cards.md#g-mills) ≈ 1.3064; chosen because two kinds of knowledge meet in one number.
It is the smallest θ > 1 whose powers θ3, θ9, θ27, … always round down to primes. Kota Saito proved it irrational with no conditions ([Saito’s paper](https://arxiv.org/abs/2404.19461)), but the familiar digits 1.3063778838… rely on [the Riemann Hypothesis](ch18-cards.md#g-rh) ([Caldwell and Cheng’s calculation](https://arxiv.org/abs/1010.4883)). [r-5.11](ch05-refs.md#r-5.11)
[EEB](ch05-cards.md#g-eb), [ψ](ch05-cards.md#g-recipfib), [Copeland–Erdős](ch05-cards.md#g-cope) ≈ 1.6067, 3.3599, 0.2357; the rest of this sum, each past the first threshold and stalled at the next.
The [Erdős–Borwein constant](ch05-cards.md#g-eb) adds one over each number one less than a power of two, 1 + 1/3 + 1/7 + 1/15 + …, and was proved irrational in 1948. The [reciprocal Fibonacci constant](ch05-cards.md#g-recipfib) does the same for 1, 1, 2, 3, 5, 8, …, giving 1 + 1 + 1/2 + 1/3 + 1/5 + 1/8 + …, and followed in 1989. The [Copeland–Erdős constant](ch05-cards.md#g-cope) is the primes written in a row, 0.2357111317…, and it is [normal in base 10](ch05-cards.md#g-normalnum): every block of digits turns up as often as chance says it should. A fraction’s digits eventually repeat, and a repeating pattern cannot give every block its fair share, so this one cannot be a fraction. [r-5.2](ch05-refs.md#r-5.2) [r-5.3](ch05-refs.md#r-5.3) [r-5.4](ch05-refs.md#r-5.4)
Not even known irrational
Here the question stalls a step sooner: nobody has even proved that these are not fractions. A million computed digits cannot rule out a repeating pattern that begins farther on.
[γ](ch05-cards.md#g-gamma) ≈ 0.5772; Euler’s constant, this sum’s flagship, which turns up everywhere.
Add up 1 + 1/2 + 1/3 + … + 1/n and the total grows without end, but so slowly that it keeps pace with ln n. Take ln n away and what is left settles, as n grows, on γ. [r-5.6](ch05-refs.md#r-5.6)
[δ](ch05-cards.md#g-feig) + |[αF](ch05-cards.md#g-feigA)| ≈ 4.6692 + 2.5029; Feigenbaum’s constants of chaos, which we will watch at work in Chapter ᚓ(E).
[δ](ch05-cards.md#g-feig) measures how fast the period-doublings crowd together on the road to chaos, and [αF](ch05-cards.md#g-feigA) how much each new fork shrinks. [r-5.7](ch05-refs.md#r-5.7)
[ζ(5)](ch05-cards.md#g-zeta5) ≈ 1.0369; chosen for a theorem that only knows “at least one.”
At least one of [ζ(5)](ch05-cards.md#g-zeta5), ζ(7), ζ(9), and ζ(11) is irrational (Zudilin, 2001), but the theorem does not say which. [r-5.13](ch05-refs.md#r-5.13) [r-5.31](ch05-refs.md#r-5.31)
[ππ + ee](ch05-cards.md#g-pipi) + [(π + e) + π·e + πe](ch05-cards.md#g-pie) ≈ 36.46 + 15.15 + 5.86 + 8.54 + 22.46; built from the two most famous transcendentals of the previous chapter.
At least one of [π + e](ch05-cards.md#g-pie) and π·e must be transcendental, yet neither has been proved even irrational: we know one of them is, but not which, and both may be. [Schanuel’s conjecture](ch05-cards.md#g-schanuel) would settle all five of these at once, but it is unproved. [r-5.12](ch05-refs.md#r-5.12) [r-5.24](ch05-refs.md#r-5.24)
[G](ch05-cards.md#g-catalan), [K](ch05-cards.md#g-khin), [A](ch05-cards.md#g-glaisher), [B2](ch05-cards.md#g-brun), [Gompertz](ch05-cards.md#g-gompertz), [C2](ch05-cards.md#g-twinprime), [Meissel–Mertens](ch05-cards.md#g-mm), [Golomb–Dickman](ch05-cards.md#g-gd), [Landau–Ramanujan](ch05-cards.md#g-lr), [γL](ch05-cards.md#g-levy), [F](ch05-cards.md#g-fransen), [β](ch05-cards.md#g-bernstein), [1.1319…](ch05-cards.md#g-viswanath), [cN](ch05-cards.md#g-niven), [CArtin](ch05-cards.md#g-artin), [KS](ch05-cards.md#g-sierpinski), [MRB](ch05-cards.md#g-mrb), [σ](ch05-cards.md#g-somos), [ρKB](ch05-cards.md#g-keplerb), [BN](ch05-cards.md#g-backhouse) each finite and positive; the rest of the gallery, each with its own story (hover or tap for it).
They come from very different places: [Khinchin’s constant](ch05-cards.md#g-khin) from [continued fractions](ch04-cards.md#g-contfrac), [Brun’s](ch05-cards.md#g-brun) from the twin primes, the [Landau–Ramanujan constant](ch05-cards.md#g-lr) from sums of two squares. What they share here is that none is yet known to be irrational. [r-5.9](ch05-refs.md#r-5.9)
The first fraction is 0/(a positive product): every entry in both sums is finite and positive, so each sum is positive, and so is their product. An unsettled classification does not make a value indefinite. [r-5.23](ch05-refs.md#r-5.23)
**Fine print.** Decimal values are rounded; the formula uses the exact constants. The bars around αF are there because it is conventionally negative; the formula uses its magnitude. No conjecture’s consequences are treated as established here. [Brun’s constant](ch05-cards.md#g-brun) is written B₂ by long convention; it is not [the Bernoulli number B₂](ch04-cards.md#g-bern2) = ⅙ of the previous chapter.
### [uncomputable](ch05-cards.md#g-uncomputable) and [infeasible](ch05-cards.md#g-infeasibility) {#ch-T5-uncomputable-and-infeasible}
Numbers with exact definitions whose digits are out of reach, each for a different reason: too big to write, forbidden to any algorithm, or simply not yet found.
#### Numerator
[lim](ch05-cards.md#g-limop)n→[∞](ch05-cards.md#g-inf) ([TREE(3)](ch04-cards.md#g-tree3) / [Rayo’s](ch05-cards.md#g-rayo))n = 0; two giants too big to write out, chosen because we can still send their ratio to zero.
Both are finite integers with short descriptions and decimal expansions we could never lay out. That is **[infeasibility](ch05-cards.md#g-infeasibility)**: too big to print, but not beyond reasoning. [r-5.15](ch05-refs.md#r-5.15) [r-5.16](ch05-refs.md#r-5.16) [Rayo’s number](ch05-cards.md#g-rayo), from a 2007 big-number duel at MIT, exceeds every integer that set theory can name within a [googol](ch04-cards.md#g-googol) symbols. [TREE(3)](ch04-cards.md#g-tree3) has such a name, so it is smaller ([Rayo’s account and definition](https://web.mit.edu/arayo/www/bignums.html)). Their ratio therefore lies strictly between 0 and 1, and any such ratio shrinks toward zero under ever-higher powers: a half, a quarter, an eighth. [r-5.17](ch05-refs.md#r-5.17)
#### Denominator
[Ω](ch05-cards.md#g-omega) has no algorithm that can supply all its digits; [BB(6)](ch05-cards.md#g-bb6) is one finite integer whose exact value has not yet been found.
[Ω](ch05-cards.md#g-omega) between 0 and 1; Chaitin’s halting probability, the famous [uncomputable](ch05-cards.md#g-uncomputable) number.
Fix an optimal universal computer whose programs delimit themselves, and feed it random bits; Ω is the chance that it eventually halts. It is **[uncomputable](ch05-cards.md#g-uncomputable)**: an algorithm giving Ω to any requested accuracy would solve [the halting problem](ch09-cards.md#g-halting) ([Lance Fortnow’s explanation](https://blog.computationalcomplexity.org/2002/10/chaitins-omega-halting-probability.html)). Some initial bits can still be established, and Ω is known to be transcendental. [r-5.18](ch05-refs.md#r-5.18) ([Calude and colleagues on computing initial bits](https://www.cs.auckland.ac.nz/~mjd/misc/167omega.pdf).)
[BB(6)](ch05-cards.md#g-bb6) a positive integer; chosen because it is one fixed number that nobody has found.
[BB(n)](ch09-cards.md#g-busybeaver) is the longest run of any halting n-state, two-symbol machine started on a blank tape. No algorithm computes [BB(n)](ch09-cards.md#g-busybeaver) for every n ([Scott Aaronson, *The Busy Beaver Frontier*](https://www.scottaaronson.com/papers/bb.pdf)), but [BB(6)](ch05-cards.md#g-bb6) is a single value. In the way stand six-state machines that behave like the [Collatz problem](ch05-cards.md#g-collatz), the unsolved puzzle of halving even numbers and sending each odd n to 3n + 1: to learn whether they ever stop, we would first have to solve a puzzle of the same kind. So far we have only lower bounds ([the BB(6) research record](https://wiki.bbchallenge.org/wiki/BB%286%29)). [r-5.19](ch05-refs.md#r-5.19) [r-5.20](ch05-refs.md#r-5.20) [r-5.29](ch05-refs.md#r-5.29)
The second and final fraction is 0/(a positive product nobody can write out): neither factor is zero, so Ω · [BB(6)](ch05-cards.md#g-bb6) > 0 without our knowing either value. [r-5.22](ch05-refs.md#r-5.22)
**Fine print.** Ω depends on the choice of optimal universal computer whose programs delimit themselves; once that is fixed, it is a definite real number. Every single integer, [BB(6)](ch05-cards.md#g-bb6) included, is technically computable, since some program simply prints it; we just do not know which program. In the limit, [∞](ch05-cards.md#g-inf) names where n is heading; it is not used as an exponent.
What this chapter adds to the formula: 0 / (a positive product) · 0 / (Ω · [BB(6)](ch05-cards.md#g-bb6)) = 0 · 0 = 0
“Kinda know” turns out to cover a great deal of mathematics, and the missing knowledge has not stopped us from evaluating the chapter. We have spent these opening chapters getting acquainted with individual numbers. Ahead, the interesting question is what happens when they have to work together.
Can I suggest a bicycle helmet for this next part? There are bikes for rent by the park exit, and we can cover a lot more ground on two wheels. Come along—the city streets are just ahead, with a few tempting places to stop.
Watch and read
- [Uncomputable functions and numbers (Busy Beaver, Rayo, Chaitin Ω)](https://www.youtube.com/watch?v=2_P8_kXNeUc) Numberphile’s big-numbers series on exactly the three objects in the right-hand column: the Busy Beaver function, Rayo’s number, and Chaitin’s Ω.
- [The mystery of 0.577 - Numberphile](https://www.youtube.com/watch?v=4k1jegU4Wb4) Tony Padilla on Euler’s constant γ, the flagship of the not-even-known-irrational brace, and on why a number that appears everywhere can stay unclassified.
- [Busy beaver](https://en.wikipedia.org/wiki/Busy_beaver) Wikipedia
- [Euler’s constant](https://en.wikipedia.org/wiki/Euler%27s_constant) Wikipedia