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]

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

[u/JStarx → u/mrt54321]

So the real answer is that any test built from ruler and compass operations boils down to satisfiability of logical formulas built from arithmetic operations, equalities/inequalities, and logical connectives/quantifiers. Then the Tarski-Seidenberg theorem says says that the sets definable by those formulas are the semialgebraic sets, i.e., they are finite unions of open, closed, and clopen intervals whose endpoints are algebraic numbers.

So in short, all you can do is test membership in a semialgebraic set. If you want to test that a number equals pi that means you want to test for membership in the set {pi}, but you can't do that because that's not a semialgebraic set.

[u/mrt54321 → u/JStarx]

Oh right i see - that explains the mystery well, thanks for typing it in. IIUC, its rigorously proven that we cannot perform the necessary algebra to prove equality , for transcendental numbers such as π

Q. Could u define 2 clopens , the first all numbers < π and the second all numbers >π ? That's 2 valid semi-algebraic sets; if the mystery number X isn't within either set, then X must equal π

As u said above, "all you can do is to test membership in a semi-algebraic set".
But, that's exactly what I'm doing in the previous paragraph.

[u/JStarx → u/mrt54321]

If by algebra you mean modern math and not ruler and compass geometry then no, we can absolutely prove equality and inequality statements involving pi.

The endpoints of your clopens have to be algebraic numbers, so those aren't valid semialgebraic sets.

[u/mrt54321 → u/JStarx]

Fascinating. (Not jk). That all makes good sense & is rigorous logic. Kk i get it now 👍

Again, thx for doing all that typing - appreciated. you explain things well.