“Mathematics is the science of the infinite.” - Hermann Weyl
Can you imagine something infinite? Rarely do people think about this concept deeply, because discussions about the infinite cause many paradoxes. For example, we know that the set of natural numbers - all positive numbers that we use for counting - is infinite, just like the set of integers - both positive and negative whole numbers. So are these two infinities equal, if technically, there are twice as many positive and negative numbers as just positive ones? In this article we will try to understand these paradoxes behind the concept of infinity.
What is infinity in math?
First of all, it is important to understand that infinity is NOT a number, but more like a concept or an idea of something endless. Infinity can be both potential or actual. When something is potentially infinite, then it can be theoretically endless, such as any straight line according to Euclid’s second postulate, but an actual existence of such an independent object as an infinite straight line does not follow from it. Vice versa, we consider actually infinite objects as something real, completed, and we can operate with them. For instance, the actually infinite set of natural numbers are operated by modern scientists.
Are all infinite sets equal?
Now we will try to answer the question from the introduction: is an infinite set of natural numbers equal to an infinite set of all integers? In order to find the answer, we will use cardinality, meaning how many items are in one set. Two sets have the same cordiality if they contain the same amount of items. But, we can’t count the number of all integers and naturals, so we are going to connect the elements in two sets, and each element in one set will be uniquely matched with one element in the other set.
In this case, we can connect numbers in this way:
It is easy to see a pattern: all even natural numbers are connected with positive integers, and all odd naturals are connected to negative integers. When we achieved one-to-one correspondence, we can claim that these two sets are equal, and both infinities are equal as well.
Let’s do the same thing with naturals and rational numbers (fractions). Now it is harder to match them, since there are an infinite number of fractions between each two integers.
We will never reach a fraction with 2 as its numerator. In this case, we need to find some other approach. Let's place all fractions into the table, so fractions in the same row will have a common numerator, and fractions in the same column will have a common denominator.
And now we are going to move diagonally, skipping fractions with values we have already encountered like 2/4 and 3/6. Thus, we have a continuous infinite series of fractions that can be compared with natural numbers:
Now, we can assert that sets of naturals, integers and rationals have the same cardinality, so the amount of numbers in the sets are equally infinite!
Of course, the set of rational numbers isn't the last set. We still have irrational numbers, so the method we have investigated is not applicable. The point is that there are more real numbers than rational numbers, even though both sets are infinite.
We now introduce Cantor's diagonal argument, which proves that the set of real numbers in the interval [0,1) is infinite. Imagine a list of all the real numbers between 0 and 1, including decimals. The diagonal of this list can be taken by the digits along the diagonal, and constructing a new number by changing each digit.
We can see that this new number (0.0256604283...) differs from every number in the original list by at least one digit. For example, the first number in the list starts with 0.0, but the next number begins with 0.0 and has a different digit (2) in the second decimal place. The second number in the list starts with 0.26, but the new number starts with 0.02 and has different digits (5) in the third and fourth decimal places.
This contradiction shows that you cannot list all real numbers, indicating that the set of real numbers is uncountably infinite, unlike natural numbers.
Cardinalities of the infinities
Up to this point, we have talked about the same and different cardinalities of infinities without naming them. ℵ₀(aleph-zero) represents the cardinality of the set of natural numbers, integers, rationals and many other sets, marking it as the smallest form of infinity in set theory. This is a key concept because it serves as a stepping stone to understanding larger infinities.
In contrast, real numbers are infinite, with a cardinality of 2^ℵ0. The set of real numbers includes all rational numbers plus all irrational numbers, and its greater cardinality indicates a "larger" type of infinity.
Conclusion
Of course, this article does not answer all the questions about the concept of infinity, but addresses the main ideas. People always strive to go beyond the limits of reality, and the concept of infinity is an example - mathematicians can only build theories and propose axioms about this concept that cannot be seen or counted.
To infinity and beyond!
Works Cited
Vsauce (2016). How To Count Past Infinity. [online] www.youtube.com. Available at: https://www.youtube.com/watch?v=SrU9YDoXE88&t=1s&ab_channel=Vsauce [Accessed 4 May 2024].
Wikipedia Contributors (2019). Infinity. [online] Wikipedia. Available at: https://en.wikipedia.org/wiki/Infinity.
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