And that's not even the half of it. See this Youtube video, "All the Numbers," from an old thread:
https://www.youtube.com/watch?v=5TkIe60y2GI&list=PLt5AfwLFPxWKuRpivZd_ivR2EvEzKrDUu&index=4

Not all numbers are expressible as fractions, only the rational numbers are. All infinities made up of rational numbers are equivalent - they are countable. Numbers not expressible as fractions, e.g. pi, make up a larger infinity, they are not countable.

Not all numbers are expressible as fractions, only the rational numbers are. All infinities made up of rational numbers are equivalent - they are countable. Numbers not expressible as fractions, e.g. pi, make up a larger infinity, they are not countable. — T Clark

A few days ago, I saw a YouTube video where a topological proof of the uncountability of the reals is offered instead of Cantor's more familiar diagonal proof. So, I asked ChatGPT o3 about it.

Was there a question?

Sorry, way over my head.

Sorry, way over my head. — T Clark

I asked o3 for an informal and intuitive explanation of the proof, which doesn't appeal to concepts from topology. It fulfilled this task quite elegantly.

On edit: I spotted an inconsequential non sequitur on o3's proof. I pointed out this fact to the model and it found it on its own.

Aleph Null + Aleph Null = Aleph Null

Numbers not expressible as fractions, e.g. pi, make up a larger infinity, they are not countable. — T Clark

Is this the same as saying that the infinity of all integers is larger than the infinity of all even integers? Or, is it the same as saying that that is you have two sets, one composed of all numbers and the other composed of all numbers except the number 3, the first set is larger than the second?

In my first question, both sets are countable.

In my second question, neither are countable because both contain irrational numbers.

As to my first question, if you subtract the total number of even integers from the total number of integers, what is your sum?

As to the second question, if you do the same, is the answer 1?

I think the answer is that if you subtract two countable infinities from each other, the answer is 0, meaning they are the same.

I think the answer is that if you remove one element from an otherwise infinite set, there is no difference in number even with the removed element. That would be because the removed element represents 1/infinity, which would be equivalent to zero.

This then means that the removal of any number of finite elements from the complete set of all numbers (rational and irrational) would result in no difference in size of infinity.

However, the removal of an infinite set of countable numbers from an infinite set of all numbers would result in a reduction in the infinity of all numbers.

I think.

All of my answers are based on my non-mathematicians understanding.

Is this the same as saying that the infinity of all integers is larger than the infinity of all even integers? — Hanover

No

you have two sets, one composed of all numbers and the other composed of all numbers except the number 3, the first set is larger than the second? — Hanover

No

As to my first question, if you subtract the total number of even integers from the total number of integers, what is your sum? — Hanover

The difference between the two sets is the same size as the sets themselves, which are equal.

As to my first question, if you subtract the total number of even integers from the total number of integers, what is your sum? — Hanover

You get a set of numbers which is the same size as the initial two sets.

As to the second question, if you do the same, is the answer 1? — Hanover

I’m not sure

There is a nice mathematical way to cash our the intuition the original poster is gesturing towards. See The continuum as a final coalgebra shows that the real numbers (a.k.a. the continuum) can be constructed from infinite steaming interactions over infinite sequences of natural numbers. I like this definition because it gives a more operational sense of how to think of the reals as being generated out of the naturals.

I’m not sure — T Clark

Yeah, well, I liked my answer which is that the removal of a single number from an infinite set does not make the set any smaller because a single number is infinitely small when compared to an infinite set which is mathematicall equivalent to zero.

I left the "y" off mathematically by accident. Consider it being there.

There is a nice mathematical way to cash our the intuition the original poster is gesturing towards. See The continuum as a final coalgebra shows that the real numbers (a.k.a. the continuum) can be constructed from infinite steaming interactions over infinite sequences of natural numbers. — FirecrystalScribe

Category theory. Beyond my grasp and interest. Nevertheless, what is an "infinite steaming interaction"?

Is this the same as saying that the infinity of all integers is larger than the infinity of all even integers? Or, is it the same as saying that that is you have two sets, one composed of all numbers and the other composed of all numbers except the number 3, the first set is larger than the second?

In my first question, both sets are countable.

In my second question, neither are countable because both contain irrational numbers — Hanover

Regarding question 2, consider two horizontal non-negative real lines, one above the other. The line on top is missing the number 3. Do they have the same cardinality? That means a one-to-one correspondence between the two lines. Start at the bottom line at 0 <-> 0 on the top line. Let x be on bottom line and y on top line. Then x=y until the "hole" where y=3. You have a 3 on the bottom, but not on the top. So you introduce the correspondence 3 <-> 4, then it's as usual, x=y, until you reach 4 on top, which has been taken. So, stipulate 4 <-> 5. Since this pattern persists indefinitely the one-to-one correspondence holds forever.

Not sure i followed that, but isn't it also correct that the length of the "hole" is zero?

While you're right in saying that there are an infinite number of infinities, the size of infinity of the fractions is a smaller infinity than the size of infinity of the reals. Moreover, this "every pair of numbers has a number strictly between them" property is a slightly different one than infinite size. That property is called density or denseness.

↪jgill
Not sure i followed that, but isn't it also correct that the length of the "hole" is zero? — Hanover

The "length" of any number on R is zero. Numbers are positions on the real line, designated as points, none of which have any "length".

(To see if you comprehend what I say, Counselor, you might show that the interval [0,1] is in 1:1 correspondence with the interval [0,1). But don't worry about it.)

what is an "infinite steaming interaction"?

In computer science an infinite stream is a data structure where you can pull off the front part of an infinite series. For example, you can have the list of all the Fibonacci numbers as an infinite stream. Of course, you can't actually store infinite values in memory, but effectively it means that as you ask for values initial segment of the stream those values are computed on the fly and you receive a new stream which is shifted.

So e.g., if you start with the infinite stream [1, 1, 2, 3, 5, 8, 13, 21,... ]

Then you take the first three, you deconstruct your original stream to get a finite list [1,1,2] and a new infinite stream [3, 5, 8, 13, 21,... ] which is a shifted version of the original.

Now applying this to real numbers. We can represent a real number in the open unit interval as an infinite stream of natural numbers (so we don't have to worry about where the decimal point lies). If we take the head (first value) and tail (a new stream shifted over by 1), the new stream is also a real number. So this is what I meant by "an infinite streaming interaction". This operation arises naturally as the final coalgebra of the ordinal product functor in the category of partially ordered sets. And the carrier set of this coalgebra is order isomorphic to the reals (i.e., it is equivalent as a partially ordered set).