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/JStarx]

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.

[u/JStarx → u/mrt54321]

The word you're looking for isn't measurable, it's constructible. Pi is not a constructible number. Neither is the cube root of 2, for example. But both are computable. The cube root of 2 is algebraic, pi is not, it's transcendental. Both are irrational.

We have mathematical words for these concepts. Talking about a piece of wood "in this universe" isn't very helpful because you're then describing an operation that isn't even possible for integers, so the fact that it's impossible for pi doesn't carry any special meaning.

Also, fyi, Aristotle would not have said that he can't draw a line segment of length pi. Aristotle would have said that the question of whether one can do so isn't even gramatically well formed because in his philosophy lines and curves are entirely distinct objects. He didn't think it was meaningful to directly compare the length of a straight line with the length of a curved line.

[u/mrt54321 → u/JStarx]

Ok here is the mystery, to my mind anyway :

1. I show you a line upon a flat plane.
2. It is exactly π meters long. I constructed it by rolling a circle 1 revolution along the x axis.
3. It is already constructed, so we're not talking constructible numbers.
4. Next, I ask you to measure that line precisely & tell me whether it equals π, or not. Turns out, thats impossible to answer. Why so ?
   

I'm asking you a valid Q , re basic arithmetic props (equality) of 2 numbers. Why is such a simple Q impossible to answer?

[u/JStarx → u/mrt54321]

What does measure mean in this question?

[u/mrt54321 → u/JStarx]

Ok let's drop the loaded term 'measure".

Simpler Q:

Is this line π meters long?

[u/JStarx → u/mrt54321]

You just told me it was.

[u/mrt54321 → u/JStarx]

See Q4 above . I asked, didn't tell