[raattgift]

I'll try to physicalize countability.

Start counting the naturals: 1, 2, 3, ...

At future timelike infinity you'll reach infinity.

Now for the reals. Your goal is to step from 0 to 1, by way of 0.1, 0.01, and so forth.

0 0.0........

At future timelike infinity you still haven't stopped adding in zeroes to the right of the decimal point.

In the first case, at any finite time before \breve{i}^+ you will have counted out some finite natural number. In the second case, you will not yet have counted out your first nonzero real.

This survives across changes of positional counting systems, and almost certainly survives arbitrary choices of non-lossy notation, as long as you start with a finite representation of 0.

[stymaar]

> I'll try to physicalize countability.

> Start counting the naturals: 1, 2, 3, ...

> At future timelike infinity you'll reach infinity.

> Now for the reals. Your goal is to step from 0 to 1, by way of 0.1, 0.01, and so forth.

> 0 0.0........

> At future timelike infinity you still haven't stopped adding in zeroes to the right of the decimal point.

> In the first case, at any finite time before \breve{i}^+ you will have counted out some finite natural number. In the second case, you will not yet have counted out your first nonzero real.

The same reasoning applies for rationals, yet they can still be counted.

The definition of «can be counted» means there is a bijection between your set and the naturals. Such kind of bijection can easily be created for the rationals[1] and Cantor's diagonal argument[2] shows that you can't create such bijection for reals.

There is nothing really intuitive about this concept, but fortunately the proofs are pretty straigtforward which give a kind of «intuition» around this.

[1] https://en.wikipedia.org/wiki/Pairing_function#/media/File:D...

[2] https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

[hzhou321]

> but fortunately the proofs are pretty straigtforward which give a kind of «intuition» around this.

It is only straightforward once you accepted infinity and all other definitions/description based on infinity.

If you, like me, cannot accept infinity, then all the proofs/descriptions that contain infinity become apparent non-sensical.

>> Start counting the naturals: 1, 2, 3, ... >> At future timelike infinity you'll reach infinity.

This highlights the flaw. You reach infinity with infinity. Nothing is really being said about infinity. But somehow if you accepted the understanding infinity here, the rest of thesis such as cantor's diagonal argument may seem to be natural, except you forget that you really didn't know what infinity is.

All property of infinity cannot be finitely described. So anything about infinity is built on top of infinity. Turtle all the way down (or up), and we don't really know what it is.

[Dr_Segfault]

The mathematical concept of infinity generally derives from the axiom of infinity in Zermelo-Fraenkel set theory, which asserts the existence of the inductive set (which is the basis for the construction of the natural numbers). Specifically, it states that there exists a set N such that the empty set is in N, and for all x in N, x union {x} is also in N. This is an axiom, so it cannot be proven to be right or wrong. You can either accept this axiom, or attempt to create your own system of mathematics that does not require the axiom of infinity.

[hzhou321]

Infinite defined as not finite is meaningless. For example, infinity is not really a number in a conventional sense. What it is is not clear other than it is not finite.

A real number with infinite precision is equally meaningless.

(Above are not directly related to your comments. I simply like to summarize my thought).

Now to your comment. You can start counting from 0 by 2s, you'll never hit 1, but that doesn't show 1 is not countable. It only shows that 1 is not countable in this particular counting scheme. Yes, you can devise a counting scheme that never hits some numbers, doesn't really contribute to either proof or insight.

I assume you are familiar with the counting scheme of rational numbers, and in that scheme, it can hit any number within any (finite) precisions. Just as infinity, a number with infinite precision is unclear. You certainly can define it, but the definition will have infinite built-in, and it is not clear what meaning does such definition adds.

[raattgift]

Counting integers from 0 by 2s and never hitting 1 (or 3 or 5 ...) still results in a finite number of natural numbers counted in finite time, skipping a finite number of natural numbers at each step. How many reals do you have to skip between 0.0 and 2.0 and between 2.0 and 4.0 ?

Returning to my previous attempt, you could think of instead a successor function; for any finite natural number the immediately adjacent natural number can be found in finite time. For any real number, the immediately adjacent real number cannot be found in finite time because the step from one real number to the next is infinitesimally small.

All of these examples are "de-generalizations" of the mapping argument. Counting integers from 0 by 2s maps bijectively onto the natural numbers. The naturals map injectively and surjectively onto the reals; you exhaust all the naturals counting between 0.0 and 1.0, or 1.0 and 2.0, or even between 0.01 and 0.011.

"A real number with infinite precision is equally meaningless": uhm, integration of infinitesimals (dS, dV, ...) ?

[hzhou321]

> How many reals do you have to skip between 0.0 and 2.0 and between 2.0 and 4.0 ?

Here you sneaked the concept of reals in. Remember reals are defined on top of infinity. You can't have reals if we are still debating what infinity is. There are infinite amount of numbers between 2.0 and 4.0, in the same sense there are infinite amount of numbers in the natural set.

Your successor function defines any finite natural number, it does not define infinity. In the rational counting scheme, we can reach any number within any finite precision. A real number that is defined on the base of infinity precision requires infinity time to reach with the same counting scheme -- the same way infinity requires infinity time to reach by 1, 2, 3, ... So if you allow infinity time, the same way you allowed infinity in your definition of real, then all real numbers can be reached (including infinity time) by counting -- not that provide any meaning.

Calculus is based on taking limit -- that is assuming a finite precision, albeit arbitrary. Infinitesimals are still finite, not infinite. Otherwise, you cannot divide them.