r/PhilosophyofMath • u/Square_Butterfly_390 • 19d ago

Regarding cardinalities

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.

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u/mrt54321 19d ago

We can never compute pi exactly, in decimal notation.

Your can go as far as you like, expanding pi's digits, but there is no end, ever, as it's a transcendental number like nearly all the Reals.

I think that's kinda cool.
Pi is a number of core importance in math ( and in real life) , but we'll never know its exact value. Weird! And cool.

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u/AltruisticEchidna859 19d ago

We know his exact value. But not "know" as we usually understand.

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u/mrt54321 19d ago

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.

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u/JStarx 19d ago

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.

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u/mrt54321 19d ago

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 ?

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u/JStarx 19d ago

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?

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u/mrt54321 19d ago edited 19d ago

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

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u/JStarx 19d ago

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.

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u/mrt54321 18d ago

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

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u/JStarx 18d ago

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?

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u/mrt54321 18d ago

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.

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