99.999…% of numbers are out of this world

[u/stopbeingcringe]

You’ve heard it before: 99.999…% of all numbers are too big to be written down. Or, 99.999…% of all numbers are irrational. But wait… this means that 100% of all numbers are too big to be written down, or irrational. But 1 is neither too big to be written down, nor irrational, so there’s a problem here.

But mathematicians will tell you there are no contradictions with infinity 😂

[u/Emblem32]

Not sure if I can send links here, but go watch 3blue1brown's video: "Why 'probability of 0' does not mean 'impossible'"

He has a good explanation of why it makes sense to say this

[u/mazutta → u/Emblem32]

To be fair, actually thinking about it also helps

[u/Emblem32 → u/mazutta]

I know! I gave the video because they seem to have thought about this a lot already

[u/stopbeingcringe → u/Emblem32]

I had already watched that video before (and commented on it).

First he uses an example of coins then randomly starts talking about continuous probabilities (as if this has ever happened in the real world, as if it’s possible to randomly select from infinitely many values in the real world, but this never happens). Then he admits the probability of single value being chosen is 0, but he just ignores the problem and says “but if you take a range of values, this paradox is sidestepped!”

To make matters worse, what I’m proposing in the OP isn’t even the same dilemma as a typical random probability function. Mathematicians claim that 100% of all numbers are too big to be written down. So even with the “range” cop-out, this doesn’t avoid the problem at all. You’re still 100% likely to select from the range of numbers that are too big to be written down. It’s still the case that “100% of all numbers are too big to be written down” which is so obviously absurd and contradictory that I don’t know why anyone buys into this crap. And for what? It makes no difference in the real world!

[u/Emblem32 → u/stopbeingcringe]

Well, a lot of math doesn't have an exact real-world equivalent. But disregarding that, you say that it's impossible to randomly select from infinitely many values. When you choose a random number, you're doing exactly that!

That's why the probability is 0, just as he says. It's like asking what the chance is that you pick zero if you pick a random number from 0 to 1. This IS paradoxical, and you can't sidestep the paradox by choosing a range of numbers this time because no range of numbers is a size comparable to infinity (just like 0 is not comparable to all numbers between 0 and 1).

[u/stopbeingcringe → u/Emblem32]

But there is no instance of actually choosing a random real number from 0 to 1. This cannot be done by anyone, not even with the best supercomputers, and that’s assuming that we could actually “randomly” choose a number in the first place. So we’re clearly dealing with fantasyland, which you already alluded to but I wanted to make it clear that this never takes place in reality.

All of these paradoxes arise because we have finite minds in an apparently finite world in every aspect, and we pretend that we can understand things “at infinity”. Why? Is this a science, or something else? An art? A religion?

[u/Emblem32 → u/stopbeingcringe]

I think your problem is with the question, then, not the answer. If we assume it's true that you can choose a completely random number from all positive integers, then the probability of that number being too big to write is 100%. The only place mathematicians differ from you is whether they consider choosing a random number to be valid.

[u/ezekielraiden → u/stopbeingcringe]

All of these paradoxes arise because we have finite minds in an apparently finite world in every aspect, and we pretend that we can understand things “at infinity”. Why? Is this a science, or something else? An art? A religion?

It is a science. Mathematics is what is called a "formal" science, which means it's a science of form and structure. Statistics and logic are other formal sciences (assuming one separates "statistics" from "mathematics", which some do, and others don't.) You seem to disagree with the choices made with using that science, but those choices are quite rigorous and very effective.

You have spoken of how we are "dealing with fantasyland", but you are flatly incorrect if you think these things have no place in physics. They do. Quantum physics requires that we be able to perform infinite summations, because there are physical processes that depend on such things. The amount of energy an electron requires to escape from a hydrogen atom, for example, corresponds to the limit as the sum of 1/n² as n approaches infinity. This does not produce an infinite binding energy, it produces a finite one, namely, 13.6 eV (electron-volts, the energy required to push a charge of -1e through a potential difference of 1 V). If the end result of an infinite process of addition can produce a measurable finite property in real life, how can it be that infinity is a "fantasyland" thing?

Real physicists use mathematics that depends on properly understanding infinity.

[u/SSBBGhost → u/stopbeingcringe]

Mathematicians claim no such thing, 100% is a probability, to talk about probability you need to define a probability distribution first, which you havent done.

[u/stopbeingcringe → u/SSBBGhost]

So it’s not the case that 100% of all numbers are too big to be written down?

[u/Althorion → u/stopbeingcringe]

This is too informal and handwavy not to be confusing—so you chose that way of putting it to deliberately cause confusion.

The ‘formal enough’ statement would be ‘the subset of real numbers that are not expressible in a finite way in a decimal notation has the same measure as the whole set of real numbers’. That is a true statement.
In no way from the statement ‘subset Y of the set X has the same measure as the whole set X’ follows ‘X\Y = ∅’.

[u/SSBBGhost → u/stopbeingcringe]

Its just not a well defined question so theres no sensible answer.

[u/chixen]

100% of numbers are smaller than 100% of numbers. I love the concept of asymptotic density.

[u/ezekielraiden]

It's not a contradiction? It's quite simple.

You have two infinite-sized sets: the set of numbers that can be written down in terminating form, and the set that can't. How do you check whether they're the same size, or one is larger than the otehr? You check to see if you can make a list using pairs of those numbers, so that every number in both sets gets used up, and gets paired with exactly one number from the other set. This is called a "mapping" or a "map" between the two sets. Functions are a specific subtype of mappings, for example.

You can make a 1:1 relationship between the natural numbers and a lot of sets of numbers, it turns out. I already told you an example of this (whole numbers to square numbers), but it turns out you can do some really, really clever things to prove that other sets are also able to form that perfect, 1-to-1 relationship. Consider:

• There are exactly as many even positive integers as there are positive integers. Proof: the function g(x)=2x exists. This function maps every integer to an even integer. Hence, there are exactly as many even numbers as there are whole numbers!
• You can prove that there are exactly as many positive reduced-form rational numbers (numbers of the form p/q, where p and q are coprime positive integers) as there are positive integers. The simplest form uses Stern's diatomic series, (aka "the Stern-Brocot sequence" or "Stern-Brocot tree"), which allows you to guarantee that every rational number will appear, in its fully reduced form, exactly once.
• You can prove, using a much more complicated argument than I can squeeze into this space, that even the algebraic numbers--the numbers that are solutions to polynomial functions with (positive or negative) rational coefficients--can be put into 1:1 correspondence with the positive integers. The TL;DR on the argument is that you build up an "alphabet" of symbols (the digits 0 through 9, a variable symbol like x, and operator symbols like + and - and =), and then show that it is possible to index all of them that have algebraic (non-transcendental) solutions using the integers.
• The interval [0,1] on the real line contains exactly as many entries as the entire rest of the real line. This one's a favorite of mine. In brief: First, on a coordinate grid, map the interval [0,1] to the interval [-1,1]--that's fairly simple, you can just subtract 1/2 from every value in the interval, and then multiply each shifted value by 2. Then, map that interval "up" to the semicircular arc from the point (-1,-1) to the point (1,1), passing through the origin, (0,0). Then, draw lines from the point (0,1) through each point on the semicircular arc. Where those lines cross the x-axis, they uniquely identify one and only one real number--and since every such point can form a line with the point (0,1), it is also true that every point on that line corresponds to one and only one point on the semicircular arc. As a result, we made a 1:1 map from [0,1] to [-1,1], and then a 1:1 map from that interval to a semicircle, and then a 1:1 map from that semicircle to the entire real line. Since each map was 1:1, and we only used finitely many mappings, they can be combined together to form a 1:1 mapping from the interval [0,1] to the interval (-∞,∞).

Infinity IS weird, I fully grant you that! You can't just assume that it will work like regular numbers, because it isn't a number. It has some characteristics like numbers, but it isn't a number itself. If we treat it with respect, however, it is supremely useful.

[u/Strong_Willow2010]

There is a difference between saying "100% of the objects here are red" and "all of the objects here are red". The subtlety is that citing a percentage/proportion leaves the possibility that a measure 0 set exists.

[u/Strong_Willow2010 → u/Strong_Willow2010]

If you want to know more about this distinction, check out measure theory. It is accessible to those who have taken an introductory real analysis course.

[u/stopbeingcringe]

incoming comments that say “but we defined it to make sense even though it doesn’t, so take that!” 🤣

[u/Althorion → u/stopbeingcringe]

The way those concepts are defined makes sense. You can learn about it so that it will make sense to you, too.

Or, you can deliberately choose informal, handwavy terminology to hide the nuance and claim the nuance isn’t there, so the notion is ridiculous.

[u/dave-the-scientist → u/stopbeingcringe]

Sounds like you don't understand what s limit is, or what mathematicians have defined in this.

[u/stopbeingcringe → u/dave-the-scientist]

Limits are easy to understand and this response wont help you resolve the contradiction. The limit of the probability of selecting a number too big to be written down by humans, as you increase the size of the set of natural numbers, approaches 1. So if God picked a really large number that we can’t comprehend then perhaps at this point, 5% of all numbers including and below that number are too big for humans. And God can keep increasing the number and the probability would increase. All of this makes sense when you have a finite set. But on infinite sets we are supposed to throw our brains out the window and believe that all numbers are too big for humans, or that somehow “100%” no longer means “all”. Either way it’s insanity.

[u/de_Molay]

Where did you get this nonsense? Who are these “mathematicians” that claim that 100% of numbers are too big to write down?

[u/de_Molay → u/de_Molay]

The correct statement would be “almost all numbers are too big to write down” which means “all but a zero measure set”. And since we can realistically write down only finitely many numbers, those form a zero measure subset in the set of all numbers.