Two Types of Infinity

[Hey Park]

I am having trouble grasping the idea of countable and uncountable infinity. How can one infinity be larger than another? Also, are there any other types of infinity that exist?

 

[lavinia]

I am having trouble grasping the idea of countable and uncountable infinity. How can one infinity be larger than another? Also, are there any other types of infinity that exist?

What exactly is the trouble that you are having?

 

[Hey Park]

Infinity goes on forever, it's more of a concept than an actual number (I think, I could be wrong) so how can one infinity be greater than another?

 

[lavinia]

Infinity goes on forever, it's more of a concept than an actual number (I think, I could be wrong) so how can one infinity be greater than another?

In mathematics there are things that are infinite. There are the counting numbers, the real numbers, the complex numbers to name three, These exist completely are are not going on forever. So it makes sense to say that there are infinitely many counting numbers or complex numbers. One can also ask if an infinite set has the same size as the integers, If not it is called uncountable.

 

[phinds]

Google "aleph number".

 

[PeterDonis]

How can one infinity be larger than another?

To expand a bit on what lavinia said, suppose you ask whether there are as many natural numbers as there are real numbers. Georg Cantor proved in the late 1800s that the answer is no: there are more real numbers than natural numbers. He did this using an argument called the "diagonal argument" (which you can look up), which basically showed that, no matter how you try to set up a one-to-one correspondence between the natural numbers and the real numbers, you will always end up leaving some real numbers out.

The key point of all this is that the relative "size" of sets is determined by trying to put them into one-to-one correspondence with each other. If you can do it, both sets are the same size; if you can't, the set that ends up having elements left out is the "larger" of the two. So the real numbers are larger than the natural numbers, even though both sets are infinite.

Also, are there any other types of infinity that exist?

If you mean, other than "countable" and "uncountable", mathematically speaking, no, because "uncountable" by definition includes all sets that are not countable.

The more interesting question is, are there uncountable sets of different sizes? (By "size" I mean "cardinality", which is the technically correct term in set theory, but "size" is easier to type. ;) ) The answer to that is yes. We know this because of the following fact: given any set, we can construct its "power set", which is the set of all subsets of the given set. And we can show (using a version of Cantor's diagonal argument) that the power set of any given set must be larger than the given set. So given any set, we can always find another set that is larger.

 

[Quds Akbar]

I am having trouble grasping the idea of countable and uncountable infinity. How can one infinity be larger than another? Also, are there any other types of infinity that exist?

Imagine all the infinite decimals between 0 and 1, they would be 0.1, 0.11, 0.111, or 0.12,0.123,0.1234, and so on. So there is an infinite number of decimals between 0 and 1, right? Now take all the infinite decimals between 1 and 2, they would be larger than all the infinite number of decimals between 0 and 1, does it make sense now because its a somewhat hard subject?

 

[Quds Akbar]

Infinity goes on forever, it's more of a concept than an actual number (I think, I could be wrong) so how can one infinity be greater than another?

Infinity is actually a concept, not a number

 

[lavinia]

Imagine all the infinite decimals between 0 and 1, they would be 0.1, 0.11, 0.111, or 0.12,0.123,0.1234, and so on. So there is an infinite number of decimals between 0 and 1, right? Now take all the infinite decimals between 1 and 2, they would be larger than all the infinite number of decimals between 0 and 1, does it make sense now because its a somewhat hard subject?

That is incorrect. The two infinities have the same size.

 

[lavinia]

Infinity is actually a concept, not a number

All ideas are concepts. But there are infinite numbers just as there are finite numbers.

 

[phinds]

All ideas are concepts. But there are infinite numbers just as there are finite numbers.

That's a point of view that requires some care, since if you say infinity is a number, some people will try to USE it as a number and that leads to problems.

 

[lavinia]

That's a point of view that requires some care, since if you say infinity is a number, some people will try to USE it as a number and that leads to problems.

Npt sure what you mean. Can you elaborate? Are you saying that the arithmetic of the infinite ordinals is different than the arithmetic of the integers?

 

[phinds]

Npt sure what you mean. Can you elaborate? Are you saying that the arithmetic of the infinite ordinals is different than the arithmetic of the integers?

I'm saying that if you perform arithmetic operation on the infinity symbol, as though it represented a number, you can prove that any number = any other number.

 

[HallsofIvy]

Since you specifically referred to the ordinals, yes, arithmetic on the ordinals is different from arithmetic on cardinal numbers.

 

[lavinia]

Since you specifically referred to the ordinals, yes, arithmetic on the ordinals is different from arithmetic on cardinal numbers.

My question was asking what was trying to be said not whether the arithmetic is different.