(December 3, 2017 at 9:11 am)Kernel Sohcahtoa Wrote: In a countably infinite set, the elements in it can be arranged in an infinite list (ordered sequence of terms).  Thus, any countably infinite set, say S, has the same cardinality as the set of positive integers (let's call it P), which contains an infinite number of elements, and so, we can say that there is a 1-1 correspondence between S and P: there is a function, say g, that maps P to S such that, for every element (image) b in S, there exists an element (pre-image) a in P such that g(a)=b (onto function); and, for any two elements x and y in P with x does not equal y, it follows that g(x) does not equal g(y) (one to one function).  Hence, this one to one correspondence between sets is known as a bijection (a function that is both 1-1 and onto).  Regarding uncountable sets, no bijection exists, and so, the elements in an uncountable set cannot be arranged into an infinite list.

Interesting.

I don't know much about the countable/uncountable things... but am I correct that a countable infinite set is an infinite set of finite numbers or something like that?

I didn't understand the part about math. Lol.

My apologies for my post. I forgot that you were talking about countable sets.

(December 3, 2017 at 10:36 am)Kernel Sohcahtoa Wrote:

(December 3, 2017 at 10:10 am)Hammy Wrote: Interesting.

I don't know much about the countable/uncountable things... but am I correct that a countable infinite set is an infinite set of finite numbers or something like that?

I didn't understand the part about math. Lol.

Sets don't always contain numbers.  For example, the elements of a power set are sets. To clarify, the infinite set of all positive integers is an element in the power set of the set of all positive integers. Thus, in this instance, we have an infinite element in an infinite set.

Is a countable set more finite than an uncountable set?

What confuses me, is the uncountability of something kind of suggests infinity to me. If you have a truly infinite number of something . . . how can you count it?

I'm guessing they're just math terms. I suck at math.

(December 3, 2017 at 10:39 am)Hammy Wrote:

(December 3, 2017 at 10:36 am)Kernel Sohcahtoa Wrote: Sets don't always contain numbers.  For example, the elements of a power set are sets. To clarify, the infinite set of all positive integers is an element in the power set of the set of all positive integers. Thus, in this instance, we have an infinite element in an infinite set.

Is a countable set more finite than an uncountable set?

What confuses me, is the uncountability of something kind of suggests infinity to me. If you have a truly infinite number of something . . . how can you count it?

I'm guessing they're just math terms. I suck at math.

No no, you just haven't been told what countable means.technically, and its a subtle distinction.

Countable infinite means that you can devise a scheme how to count through the set that will sooner or later reach any arbitrary element of it although.you never finish all of them. You can never finish counting through it because its infinite!, but for any element you choose beforehand you can be sure that it will be reached sooner or later. The simplest example is the integers. 0,1,2,3,4... this counting scheme never ends but will eventually reach any number you want even if you can never finish all of them. This is what countable infinite means.

As opposed to, say, all the real numbers between 0 and 1. There is no way to count through them one by one that ensures that every single one will eventually be reached.

It's easy to say that infinity is just a symbol. But in fact it is very difficult to understand for every human being.
And the more you know about it, the less intuitive it seems....
We can imagine.... that there are different types of infinity. Some are bigger than others. The first one and at the same time the smallest infinity is a "countable" infinity. The point is that we can write it in the form of a list of numbers... example: 1,2,3,4,5,6 ... or 0, -1,1, -2,2, -3,3, .... What is interesting, the second string of numbers, also extending into infinity, is twice as large as the first. This is because in the second set we have all the numbers from the first string plus 0 and the negative values of all these numbers.
Well, now a curio.... There are also infinite-infinities, which can't be so nicely written down. The infinity of numbers between 0 and 1 is greater than the infinity of all natural numbers.
nah.., we don't even know how to start, because right after 0 we have to write a number: [0, (infinite number of zeros) 1] and right after it [0, (infinite number of zeros minus one) 1].. But it's not so easy, because if we subtract 1 from infinity, it still will be infinite!
So...now try to imagine the infinity of all possible points on the nimeric axis, since we can't even write a string of numbers from zero to one....





so ...∞ + ∞ = more than mathematics....maybe = philosophy?

That question is invalid. Are you a troll?

(December 3, 2017 at 12:19 pm)Shell B Wrote: That question is invalid. Are you a troll?

 What? No.

Okay, cool.

(December 3, 2017 at 11:02 am)Alex K Wrote: Countable infinite means that you can devise a scheme how to count through the set that will sooner or later reach any arbitrary element of it although.you never finish all of them. You can never finish counting through it because its infinite!

My bold. This is what confuses me. How is it countable if you can never finish counting them? I'm guessing it's a math term as I said, and a technical term as you say, that doesn't literally mean "countable" in the normal dictionary sense of the word.

Quote:, but for any element you choose beforehand you can be sure that it will be reached sooner or later. The simplest example is the integers. 0,1,2,3,4... this counting scheme never ends but will eventually reach any number you want even if you can never finish all of them. This is what countable infinite means.

I don't get it. But I don't know math terms like "element" and "integer" so that probably doesn't help.

Quote: As opposed to, say, all the real numbers between 0 and 1. There is no way to count through them one by one that ensures that every single one will eventually be reached.

This confuses me so much. I was starting to think maybe it means the real/natural numbers can be counted but not every number including ones with all decimals. But then you say it's opposed to all real numbers.

I don't get it.

I think unless I learn more math then you can keep trying to explain and I still won't get it hehe. Please forgive me if I'm being frustratingly dense.

(December 3, 2017 at 7:08 pm)Hammy Wrote:

(December 3, 2017 at 11:02 am)Alex K Wrote: Countable infinite means that you can devise a scheme how to count through the set that will sooner or later reach any arbitrary element of it although.you never finish all of them. You can never finish counting through it because its infinite!

My bold. This is what confuses me. How is it countable if you can never finish counting them? I'm guessing it's a math term as I said, and a technical term as you say, that doesn't literally mean "countable" in the normal dictionary sense of the word.

Quote:, but for any element you choose beforehand you can be sure that it will be reached sooner or later. The simplest example is the integers. 0,1,2,3,4... this counting scheme never ends but will eventually reach any number you want even if you can never finish all of them. This is what countable infinite means.

I don't get it. But I don't know math terms like "element" and "integer" so that probably doesn't help.

Quote: As opposed to, say, all the real numbers between 0 and 1. There is no way to count through them one by one that ensures that every single one will eventually be reached.

This confuses me so much. I was starting to think maybe it means the real/natural numbers can be counted but not every number including ones with all decimals. But then you say it's opposed to all real numbers.

I don't get it.

I think unless I learn more math then you can keep trying to explain and I still won't get it hehe. Please forgive me if I'm being frustratingly dense.

You're smart, Hammy. I'm sure math won't be so hard for you once you put your mind to it.

Anyway, a countable infinite refers to the sequence of numbers, whereby you can hypothetically (given an infinite amount of time) reach any number in the sequence.

With uncountable infinities, none of the numbers are reachable even if you are given an infinite amount of time.

For example, all the numbers between 0 and 1 comprise an uncountable infinity because you can't reach any of the numbers by counting.

Try and see if you can.

Try to reach 0.1 starting from 0 by counting all the numbers between them.