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VS Battles Wiki Forum

Agnaa
Agnaa
No.
GreatIskandar14045
GreatIskandar14045
How exactly?
Agnaa
Agnaa
Being on the real number line does not make something uncountable. 2 is not uncountable.
GreatIskandar14045
GreatIskandar14045
I don't really understand why, so you're saying it's still just a subset of all set of R?
Agnaa
Agnaa
It's not a subset of R. Like that wikipedia page says, infinity isn't a part of R, the real numbers have to be extended to include infinity as a number.

Frankly it's hard to answer because I don't understand why you think that adding infinity would make that infinity uncountable.
GreatIskandar14045
GreatIskandar14045
I just think that is the extension of real number line which includes the infinite decimal expansion.
Agnaa
Agnaa
And the real numbers are an extension of the natural numbers, which includes infinite countable numbers. That does not make decimals countably infinite in size.

Or to put it another way, let's say that we create a number system where decimals can only go up to one place. So counting from 0, you'd go 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and then 1.

Even though 1 comes after 10 different numbers, it does not mean it has a size of 10. Its size is only 1.

Similarly, extending the real numbers with an infinity does not make that number uncountably infinite, even though it's after uncountably infinltely many numbers. It makes as much sense to say that 1 is uncountably infinite because there's uncountably infinitely many decimals between 1 and 0.
GreatIskandar14045
GreatIskandar14045
I probably misinterpreted what you're saying, but I don't really say that decimals (such as irrational numbers) are infinite-sized. This is what I meant for the following description from wikipedia which you probably already known:

"a real number is a value of a continuous quantity that can represent a distance along a line (or alternatively, a quantity that can be represented as an infinite decimal expansion)"

What I think is, or I thought, an already represented the entire line hence is uncountable. I never know if it's just extend the real numbers to infinity.
GreatIskandar14045
GreatIskandar14045
Wait, I probably did a blunder. Does that a real number line refers to just infinite decimal or all set of real numbers?
Agnaa
Agnaa
I don't really understand what you're saying anymore.

but I don't really say that decimals (such as irrational numbers) are infinite-sized


Yeah, that's exactly the point I'm getting at. The size of a number isn't how many numbers came before it.

What I think is, or I thought, an ∞ already represented the entire line hence is uncountable. I never know if it's just extend the real numbers to infinity.


I have no clue what most of this means so I can't even respond to it.

Wait, I probably did a blunder. Does that a real number line refers to just infinite decimal or all set of real numbers?


What?
GreatIskandar14045
GreatIskandar14045
Alright, please listen. These are what I'm getting and what I used to think:

-The number line of R aka real line is all the set of R -> Checked it, correct.
-The infinity represent the real line, hence it is equal to all the set of R and is uncountable -> You said it's incorrect, since infinity just extend the real numbers rather than appear as a notation for the cardinality of the set of all reals. As I think of it now, I think it does make sense.

For that last question, I'm asking if whether a real number (I have typo there as mistake) refers to infinite decimal expansion or all the set of R. It's kind of dumb question and I already have it answered, you can ignore it.

So for the last question, is infinity () has the same size as aleph-0?
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