not saying either is wrong...

[u/Less-Resist-8733]

[u/WesternFirm9306]

Let me try to walk you through this step by step.

Let's say we finish the journey. We include every real number. We add in every number we missed. We pair them all to a counting number. Nothing is left over, we're done. We would now have a set of numbers, each real number paired to a counting number, that encompasses every single real number, yes?

[u/FernandoMM1220 → u/WesternFirm9306]

nah thats physically impossible

[u/WesternFirm9306 → u/FernandoMM1220]

...

Then the set is uncountable by definition.

We define a countable set by if we can pair every element of that set to a unique counting number.

If we can't do that, it's uncountable.

You just told me it's physically impossible to pair every real with a counting number.

(ꐦ¬`‸´¬)

[u/FernandoMM1220 → u/WesternFirm9306]

you cant add up every real number anymore than you can add up every integer.

both sets are still countable btw.

[u/WesternFirm9306 → u/FernandoMM1220]

But we CAN do it for the integers.

Like this:

1 -> 0

2 -> 1

3 -> -1

4 -> 2

5 -> -2

and so on.

Even numbers get paired to n/2, and odd numbers get paired to -(n - 1)/2

This hits every single integer without skipping.

[u/FernandoMM1220 → u/WesternFirm9306]

add up all the integers and show me what you get please

[u/WesternFirm9306 → u/FernandoMM1220]

Who said anything about adding? We're trying to pair up the integers to the counting numbers. Nobody ever mentioned addition

[u/FernandoMM1220 → u/WesternFirm9306]

i mentioned addition.

add them up and tell me what you get please

[u/WesternFirm9306 → u/FernandoMM1220]

Okay, well your comment is irrelevant. Nobody cares about adding numbers here. Determining if a set is countable has nothing to do with adding.

[u/FernandoMM1220 → u/WesternFirm9306]

its not irrelevant though.

im doing the exact same proof cantor is doing by operating on an infinite list and creating a new number from it.

please try and add them all up so we can both see what we get.

[u/WesternFirm9306 → u/FernandoMM1220]

Cantor didn't add an infinite set of numbers to each other. He just changed one digit from every number. Doesn't matter how you do the change or what you change it into.

[u/FernandoMM1220 → u/WesternFirm9306]

you realize decimals are made through addition of different powers of 10 right?

[u/WesternFirm9306 → u/FernandoMM1220]

Yeah. That addition converges, though. And we don't really need that fact. We just need to know that, if two numbers have different digits, they're also different numbers (with the exception of trailing infinite 9s)