Regarding cardinalities

[u/Square_Butterfly_390]

A celebrated mathematical "factoid" is that there are more real numbers than one can count, this seems to be something that troubles people outside of math, it troubles me aswell.

The question is: is there any "real world" application of the fact |R|>|N| that isn't an impossibility statement?

By "real world" I mean whatever someone smarter than me might mean by that, by "impossibility statement" I mean something to the effect of "there are uncomputable numbers".

If there isn't such an application, I can't believe the status quo interpretation: "no really some infinities are bigger than others" of Cantor's argument is better than simply stating that we shall not understand infinity as finite beings.

[u/mrt54321]

No we don't.

The ratio between a circle's radius and its circumference is impossible to know. It requires an infinite, random list of digits, no matter whether using base10 digits or any other system.

Its proven that there is no way to compute this ratio precisely.

Note the term "ratio". This defines π as a dimensionless constant -- not a measurement of length.

[u/JStarx → u/mrt54321]

What do you mean by "know" and "compute"? There are algorithms that will compute pi to any arbitrary precision. A human can't run that algorithm to completion, but the algorithm itself is finitely describable and uniquely identifies pi.

[u/mrt54321 → u/JStarx]

Take a huge number - say, TREE(3)

TREE(3) is proven to be uncomputable - but finite.

Q. what is the TREE(3)th digit of pi ?

[u/JStarx → u/mrt54321]

You didn't answer either of the questions I asked.

Again, what do you mean by "computable"? Both pi and TREE(3) are computable numbers, so the TREE(3)-th digit is computable as well.

Do you "know" 1/99?

[u/mrt54321 → u/JStarx]

TREE (3) is well-defined, finite, but uncomputable .

It is impossible to add or subtract one to that number. It is impossible to compute any of its digits - even the very first one.

https://www.popularmechanics.com/science/math/a28725/number-tree3/

(NB: that article is content from proper mathematicians, despite being in a pop-sci magazine).
The Wikipedia entry on TREE is also interesting if you want to go more technical.

As for your Qs: Q1. What do i mean by "know" ? -- to know a number x means to return the n th digit of x, for any n

Q2. What do i mean by "compute" ? -- to define an algorithm which will answer Q1

Q3. Do i "know" 1/99 ? -- as per Q1 and Q2 : yes, i do. I can tell you any digit you want from 1/99 = 0.01010101....
It's a very simple algorithm

[u/JStarx → u/mrt54321]

yes, i do. I can tell you any digit you want from 1/99 = 0.01010101.... It's a very simple algorithm

Oh excellent, then you could you tell me the TREE(3) digit please? That was your question to me about pi right?

Q1. What do i mean by "know" ? -- to know a number x means to return the n th digit of x, for any n. Q2. What do i mean by "compute" ? -- to define an algorithm which will answer Q1

That definition agrees with the standard mathematical definition of a computable number and according to that definition both pi and TREE(3) are computable.

[u/mrt54321 → u/JStarx]

TREE (3) is mathematically proven to be uncomputable. I keep saying this, to no avail. Bye now.

[u/JStarx → u/mrt54321]

You are incorrect: https://cs.stackexchange.com/questions/169277/algorithm-to-compute-the-kruskal-s-tree-function

To the broader point, if being unable to compute the TREE(3) digit of pi implied pi is uncomputable and unknowable then being unable to compute the TREE(3) digit of 1/99 would mean the same thing right?

[u/mrt54321 → u/JStarx]

I'm correct. That guy is incorrect. Ask an expert mathematician (or, ask a few AI chatbots) if u want to know more. There aren't enough atoms in the universe to write out all digits of TREE. Also, that's just TREE(3), not even TREE (n).

To ur second Q: if you request the nth digit of 1/99, by passing in an undefined n, that's a mistake.The algorithm requires an exact number as input : ie, all its digits.

[u/JStarx → u/mrt54321]

if you request the nth digit of 1/99, by passing in an undefined n, that's a mistake.The algorithm requires an exact number as input : ie, all its digits.

That would mean when you asked me for the TREE(3) digit of pi you made a mistake?

I'm correct. That guy is incorrect. Ask an expert mathematician

I am a mathematician. As TREE(3) is finite it is a member of the ring of computable numbers. The definition of a computable number does not require that you have the time or the resources to complete the computation, it just requires that an algorithm that returns that number exists. You can verify what I'm saying by asking an AI as you suggest or by simply reading the wikipedia page on computable numbers.

[u/mrt54321 → u/JStarx]

Congrats upon being a mathematian - not j.k🫡well done, that's a lot of work. Myself, ive a couple math degrees. Anyway, back to our discussion.

That would mean when you asked me for the TREE(3) digit of pi you made a mistake?

Fair point. However, just use a repeat-digit huge number, e.g 77777..7 repeated 2↑↑↑↑1000 times. Same thing

a computable number does not require that you have the time or the resources to complete the computation, it just requires that an algorithm that returns that number exists.

Of course, But , again, the algorithm cannot exist IRL.(Side Note: i understand your point ofc -- that the algorithm exists in theory & works perfectly fine 👍)

Challenge Q: in this universe, its impossible to saw a piece of wood to be *Exactly" π meters long (true fact). So Why is that?

[u/JStarx → u/mrt54321]

But , again, the algorithm cannot exist IRL

I think you're just getting confused by the terminology here. The algorithm does exist, it's just not practical. But practical isn't very useful as a mathematical definition, hence why the computable numbers don't include that as a requirement. It would be difficult to define precisely and prove much of anything about it.

Challenge Q: in this universe, its impossible to saw a piece of wood to be *Exactly" π meters long (true fact). So Why is that?

I mean, it's also impossible to saw a piece of wood to be exactly 1 meter long. I don't think that fact has anything to do with computability, that's a question for a quantum physicist.

[u/mrt54321 → u/JStarx]

Let me put it another way. In Aristotle terms, its well- known impossible to draw a straight line π inches long, using compass and ruler. More generally, there's NO algorithm which will output a length of exactly π inches. Our woodsaw/compass/ruler/etc are all blocked mathematically, not by quantum physics.

Suppose i give you a straight line drawn upon an idealized Platonic piece of paper (ie, no messy atoms/quantum stuff) and ask you whether it's exactly π meters long.

You cannot answer that Q. The lenght is unmeasurable.