Hey Park said: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?
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.Hey Park said: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?
Hey Park said:How can one infinity be larger than another?
Hey Park said: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?Hey Park said: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?
Infinity is actually a concept, not a numberHey Park said: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?
That is incorrect. The two infinities have the same size.Quds Akbar said: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 said:Infinity is actually a concept, not a number
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 said:All ideas are concepts. But there are infinite numbers just as there are finite numbers.
phinds said: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.
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.lavinia said: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?
My question was asking what was trying to be said not whether the arithmetic is different.HallsofIvy said:Since you specifically referred to the ordinals, yes, arithmetic on the ordinals is different from arithmetic on cardinal numbers.
"Two Types of Infinity" is a concept in mathematics and philosophy that explores the idea that there are different sizes of infinity. This idea was first introduced by mathematician Georg Cantor in the late 19th century.
The two types of infinity are countable infinity and uncountable infinity. Countable infinity refers to sets that can be put into one-to-one correspondence with the natural numbers (1, 2, 3, ...), while uncountable infinity refers to sets that cannot be put into one-to-one correspondence with the natural numbers.
The concept of "Two Types of Infinity" is important in mathematics because it helps us understand the different sizes and properties of infinite sets. It has applications in fields such as calculus, topology, and set theory.
An example of countable infinity is the set of all natural numbers (1, 2, 3, ...). An example of uncountable infinity is the set of all real numbers between 0 and 1, as there is no way to count or list all of these numbers.
The concept of "Two Types of Infinity" has sparked debate and controversy among mathematicians and philosophers. Some argue that there is only one type of infinity, while others believe in the existence of multiple types. Additionally, the idea challenges our understanding of infinity and the concept of "counting" in mathematics.