Countably Infinite Unions and the Real Numbers: Can They Really Be Uncountable?

[Zedertie Dessen]

Hello experts,

Full disclosure: I am a total layman at math, nothing in my training aside from high school courses and one college calculus class. I'm sure a week doesn't pass without someone posting a question about or challenge to Cantor. I am not here to challenge anything but rather to learn where the error is in my thinking. I hope I don't get mistaken for a crank who's trying to disprove Cantor.

Like many people, I suppose, I'm having difficulty wrapping my head around Cantor's proof that the real numbers are uncountable and that there are different sizes of infinity. I can follow along with the diagonal proof for the most part, but another way of visualizing the problem keeps making me think that I don't understand how the real numbers diverge from other numbers in countability.

Here is what I mean. I think it is true that any given real number is made up of a countable set of digits, following this general pattern or mask:
...NNN.NNN...
where N is any digit 0-9. To the left of the decimal point, one could imagine an endless string of 0s leading up to the actual number (if it is non-zero). Such as ...000003.14159...

The set of those digits is countably infinite, as I understand it, because it can be put into a 1:1 correspondence with the natural numbers. If that is wrong, I'd be grateful for an explanation why.

Now - and this is where I get confused - I believe I have read that the union of two countably infinite sets is also countable. I got that from some Google searches, so I have no idea if that is really true. Whether it is or not affects whether I can even get to the next step of my thinking.

If it is true that the union of countably infinite sets is also countable, then it seems to me that one could simply do another union with the next countably infinite set of digits for another real number, and so on. Even if in principle there are real numbers that cannot be listed as has been proved by the diagonal argument, the real numbers are out there, each one consisting of a countably infinite set of digits, so I don't understand how their "total" could comprise an uncountably infinite set.

How could such an ongoing sequence of unions produce an uncountably infinite set? Is there even a proof or explanation that a layman like me could comprehend?

Thanks for any help.

 

[fresh_42]

You can write down each real number as a sequence of digits, but you can't write down all of them.

 

[Dale]

If it is true that the union of countably infinite sets is also countable, then it seems to me that one could simply do another union with the next countably infinite set of digits for another real number, and so on.

The union of a countable number of countably infinite sets is countably infinite. An uncountable number of countable sets is uncountable.

The number of digits in each real number is countable, so a countable number of real numbers has a countable number of digits. But there are an uncountable number of real numbers, so they do not have a countable number of digits.

 

[Zedertie Dessen]

Thanks for the responses! I was still a little confused, so I searched YouTube for a devil's advocate sort of video. I found the one below. (More commentary by me below the video.)

Cantor's Diagonal Argument Applied to Integers - Fatal Flaw


The video claims to find a contradiction in Cantor's Diagonal Argument that seemed pretty persuasive - basically trying to show that the diagonal argument can also "prove" that integers are uncountable. But the comment to the video explains why the video is not correct:

When you try to bring this argument to the integers, you run into a big problem. Yes, every integer can be represented as a string of digits in base ten, but not every string of digits in base ten represents an integer. In base ten, each integer has only finitely many nonzero digits. There are infinitely many integers, and there is no global upper bound on the number of digits applied to the entire set of integers, but each individual integer has only finitely many nonzero digits.

I think the bolded part of what I quoted pertains to why one could not simply keep UNIONing the sets of the digits of real numbers and then say the overall collection of real numbers has a countable number of elements. I think that statement, combined with Dale's comment ("But there are an uncountable number of real numbers, so they do not have a countable number of digits.") explains the issue clearly enough for me at the moment. Makes my head hurt but also makes sense lol.

Thanks again!

 

[Stephen Tashi]

If it is true that the union of countably infinite sets is also countable, then it seems to me that one could simply do another union with the next countably infinite set of digits for another real number, and so on.

The final cardinality of "all real numbers" would depend on how many times you need to take another union. A countable union of countably infinite sets would be countably infinite. But you must consider that you might need to take an uncountable union of countably infinite sets.