2956: Number Line Branch

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Number Line Branch

Title text: Attention all passengers: This is an express sequence to infinity. If your stop is not a power of two, please disembark now.

Explanation

This comic likens the number line to a line of a railroad or subway system. These often have branches where different trains continue on to a different destination, with different stops along the way, or travel on parallel lines to allow faster trains to bypass slower ones. In the number line, one branch (presumably the original) contains ordinary numbers, while the newly opened branch consists of some completely different numbers, denoted with various symbols as an analogue to those we use as digits. The branches seem to split at π. The new branch maintains the same scale-separations as the 'normal' one (as far as it goes) but, due to a longer initial curve away from the junction, the new-branch digits are also consistently slightly offset from the horizontal positions of the respective old-branch ones.

The sequence ending with a bold mark at Δ (whereas the original number line fades out) suggests that it is the end of this branching sequence. Mathematicians, apparently, could only afford to construct 5 additional numbers, or their research hasn't yet found other numbers. The branch may have been intended to run much further, but been scaled back due to budget overruns and cutbacks.

The title text makes a parallel between a train stopping at a station and a numerical sequence "stopping" at a number – that is, taking it as a value. It's a spoof of announcements that are typically made on trains, so that passengers can confirm that they're on a train that goes to their desired station; an "express train" typically makes fewer stops so it can serve the most popular stops and reach its final destination sooner. In this case, the express train only stops at powers of 2; presumably the "local" stops at every integer. Powers of 2 are 2, 4, 8, 16, 32, and so on, such that the interval between stops grows exponentially larger.

Mathematically, an express train like this would get to its scheduled stops much faster, but it would not actually have any fewer stops overall. Mathematicians that study infinities generally regard all "countably" infinite sets as being the same "size." Infinity is not a fixed value, rather it's the concept of "does not end," so it's paradoxical to try to take a train to a destination that is, by definition, not a single destination. By way of analogue, it's akin to promising to stop hitting your little brother only after you've done so forever.

A fictional number was previously shown in 899: Number Line ("gird"), and fictional numerals were shown in 2206: Mavis Beacon. And similar treatment of mathematics as public infrastructure was seen in 2735: Coordinate Plane Closure.

Transcript

This transcript is incomplete. Please help editing it! Thanks.

[A diagram reminiscent of both the number line, and a transit system diagram. The line starts at the left and goes right through points labeled 0, 1, 2, and 3, at which point there is a bifurcation into two branches. The top branch continues: 4, 5, 6, 7, 8, 9, 10, …. The bottom branch is shorter, labeled with five curious glyphs: a square, a near-vertical with two short horizontals jutting out from it, a phi-like/lollipop symbol, a spiral and a delta/triangle symbol.]

[Caption below the panel:]

Good news!

After thousands of years, mathematicians have finally opened a second branch on the number line to reduce congestion.

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Discussion

Is it significant that the branch point is close to the value of π? Barmar (talk) 20:28, 8 July 2024 (UTC)

I was thinking the same thing, but decided it was probably nothing worth mentioning - probably just an arbitrary starting point. *Possibly* referencing the strange appearance of π but I doubt it. Anything can be significant if you believe hard enough, anyway.--162.158.158.60 20:30, 8 July 2024 (UTC)

Keep in mind π isn't special. Most real numbers are like π. The rational numbers, and the roots of polynomials with rational coefficients (algebraic completion), are the aberration. --172.71.160.71 07:25, 10 July 2024 (UTC)

How does adding a new branch to a railway line reduce congestion? Isn't this more like a highway? 141.101.105.47 23:30, 8 July 2024 (UTC)

Read about the 2nd avenue subway. 172.70.111.168 02:22, 9 July 2024 (UTC)

What is "thrembo"? Your favorite aura doggo (talk) 04:12, 9 July 2024 (UTC)

I think the whole section can be removed. The point is to explain the comic, not to describe what is seen (unless it's relevant for the explanation, which, so far, seems not be the case). "Various symbols" should cover it. Elektrizikekswerk (talk) 06:35, 9 July 2024 (UTC)

The symbols seem well chosen TBH, I can totally see how they substitute for 4, 5, 6, 7, 8 162.158.146.32 14:42, 9 July 2024 (UTC)

Except that, according to the title text, they should be 4, 8, 16, 32, 64. 172.71.242.159 15:59, 9 July 2024 (UTC)

I think that the express train travels on the regular number line, so I think the second branch parallels the regular number line. Thus, 4, 5, 6, 7, 8. Sarah the Pie(yes, the food) (talk) 02:11, 10 July 2024 (UTC)

Mathematicians been there, done that: https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic Although a nonstandard model of the integers can't branch by Peano axioms. 172.71.160.71 15:08, 9 July 2024 (UTC)

Hold my beer! [Prompts Claude 3.5 Sonnet to create a non-standard model of arithmetic.]

"Here is a non-standard model of arithmetic consistent with the Peano postulates:

Axioms:

∃0, ω : 0 ≠ ω

∀x : S(x) = x + 1 if x ∈ ℕ; S(ω) = ω

Theorems:

T1. ∀n ∈ ℕ : n < ω

T2. ∀n ∈ ℕ : n + ω = ω

T3. ω + ω = ω

Lemma:

L1. ∀n ∈ ℕ : S(n) ≠ ω

"Induction" Principle:

For any property P, if P(0) ∧ P(ω) ∧ (∀x : P(x) → P(S(x))), then ∀x : P(x)

Conclusions:

C1. This model "satisfies" Peano axioms while introducing a non-standard element.

C2. Arithmetic with ω leads to paradoxical results.

C3. Use of this model may violate conservation of sanity."

Oh, no! .... Tentacled one sleeps. Ph'nglui mglw'nafh Cthulhu. Accept this new soul. 172.70.210.103 03:24, 10 July 2024 (UTC)

That is just the [later edit: positive] integers plus infinity, it should place you in the thrall of Asmodeus, not Cthulhu. 108.162.245.42

The Russell-Riemann-Cantor diagonalization applies to integers with infinity, mapping each integer to a tentacle on an Eldritch manifold. 172.70.210.53 03:59, 10 July 2024 (UTC)

Personally I would have branched off between 9 and 10, and had single character symbols for 10, 11, 12, 13, 14, and 15 so that you could do base 16 without having to use letters. Randell just lacks vision. Andyd273 (talk) 15:12, 9 July 2024 (UTC)

Letters are single character symbols! I think he should extend the number line with all the letters, getting to 36 (z) before needing any new symbolsPotatoGod (talk) 21:46, 9 July 2024 (UTC)

In my eyes they all seem like geometry or geometry-related symbols. A square, sideways pi, phi (the golden ratio), a spiral, and a triangle. That should probably be noted somewhere.--Rerere284 (talk) 18:23, 10 July 2024 (UTC)

I was thinking that the first symbol looks like a square too, so what is this stuff about a Brahmi letter instead of a square? How do we get that complicated with a square? Ianrbibtitlht (talk) 18:57, 10 July 2024 (UTC)

assuming particular mappings of the depicted symbols to phonemes, and saying "Putting these 5 phonemes together gives a word that sounds a bit like 'bisect,'" is absolutely a stretch and should be removed. --162.158.159.208 22:20, 10 July 2024

I'm rather surprised that these mathematicians have decided to subdivide each integer by 8, instead of by 10, like good metricists.172.70.86.132 08:13, 11 July 2024 (UTC)

It looks to me that the split happens just beyond 3, probably at around 3.14156... Xplora1a (talk) 13:09, 11 July 2024 (UTC)

The new branch comes up between 3 and 4, which may reference The Secret Number, a sci-fi novel in which a mathematician find a number between 3 and 4. 799571388 (talk) 07:41, 16 July 2024 (UTC)

It's called bleem btw (Anonymous) 14:51, 24 Sep 2024 (UTC) ~actually 172.70.43.93 (talk) 14:52, 24 September 2024 (please sign your comments with ~~~~)

Not even a mention of the line with two origins. Who are you people?162.158.154.206 07:52, 21 July 2024 (UTC)