By Isabelle Robinson, M.Sc.Apr 20 2018.jpg)
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To answer this question, we must first define what infinity is. It is now widely accepted that infinity is simply any number that isn’t finite. Therefore, infinity is not a number but rather a concept of describing a number or set of numbers, that does not end.
In 1874, the famous German mathematician Georg Cantor produced the first real proof of the existence of multiple infinities. Cantor built up the incomplete theories of Galileo Galilei in the 17th century, who attempted to draw a correspondence between all natural numbers and the squares of all these numbers. Galileo ultimately abandoned this mathematical experiment when it became obvious that many numbers were not squares, creating a paradox that is now aptly known as ‘Galileo’s Paradox.’ (DeHaan, 2011)
The basis of Cantor’s proof used a branch of seemingly useless mathematics called set theory. This concept has no direct application to real-world maths, however, is very helpful when dealing with complex numbers or theoretical concepts. (Herbert Enderton, 1999)
To put it simply, set theory is the idea that things can be placed into groups or sets. For example, the numbers 1,2,3,4 and 5 can be placed together into a set (1,2,3,4,5). However, the same could be done with anything. Another set could contain an umbrella, a cat, Tuesday, the letter X and the USA (Wilkins, 2011).
Although these two sets are wildly different regarding content, the number of terms in each set are equal. The two sets have the same cardinal number. A cardinal number indicates quantity. This is the key to Cantor’s theory and will play an important role as we delve further into infinity.
If we apply this theory of sets to all the natural numbers (any positive integer including zero) and group them into a set, then the set contains an infinite amount of numbers. It, therefore, stands to reason that it also has an infinite cardinality. This is the first and smallest infinity and is known as Aleph-null (א0).
It is at this point when everything becomes increasingly more difficult to comprehend. It would be simple to say that to make a number larger than Aleph-null, we could add one to it. However, this principle only works in a set of finite values.
To prove this, we can compare a set of infinite values (0,1,2,3…) and a set of one plus infinite values (1,0,1,2…). We shall call these set 1 and set 2 respectively. By using one-to-one correspondence of both sets (0 = 1, 1=0, 2=1, etc.), it becomes clear that the pattern can continue forever because both sets have an infinite number of terms in them. By this logic, both sets have a cardinality of Aleph-null. Cantor also compared an infinite set of rational numbers to an infinite set of natural numbers and found that these, counter-intuitively, also have the same number of terms. (Mastin, 2010)
So then, how can Aleph-null be the smallest infinity?
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To create a number larger than Aleph-null, we must use a set of numbers without a one-to-one correspondence. In layman’s terms, we must have an ‘uncountably infinite’ series of numbers. The most well-known of these sets includes all real numbers, including irrational numbers such as Pi.
Cantor created a method to get a number that is infinitely larger than Aleph-null. This method uses the notion of ordinal numbers. Ordinal numbers are different to cardinal numbers as they tell you the position of the numbers rather than the number of terms. It can, therefore, be assumed that the sequence (1,2,3,4…) is different to (2,3,4,1…).
Taking this principal, if we create an infinite amount of sequences with infinite amount of terms and write them as real numbers, so the sequence (1,2,3,4…) becomes 0.1234…, then we can create a real number that cannot be part of any of the original infinite number sequences.
Cantor imagined using a set of numbers with just two digits to show how a new number (Su) could be created (VSauce, 2016). This is done by associating each sequence with a term moving diagonally down the sequences.
S1 = 0.1000…
S2 = 0.0100…
S3 = 0.0010…
S4 = 0.0001…
Su = 0.1111…
Cantor then inverted this sequence so that the devised number does not match the corresponding terms of any of the original sequences. This means that the new number is not a duplicate of any other number and thus proved that an infinite set of numbers does not contain all possibilities.
By doing this, he proved that it is impossible to have a one-to-one correspondence of the real number set to a natural number set. This is known as the Continuum Hypothesis and has yet to be proved untrue by mathematicians to this day. The set of numbers that contain all the real numbers is known as Alpha-one and is infinitely larger than Alpha-null.
Can we go even further?
The answer to this is absolute, with the use of another simple mathematical principle, power sets. For this principle, a power set of any set (x) is defined as the set of all the subsets of x, including x itself and an empty set. The notation of a power set is P{X}. A simplified example of this is as follows;
Using the set(X) = {x,y,z}, we can see the subsets of this are {}, {x}, {y}, {z}, {x,y}, {x,z} {y,z} and {x,y,z}. Therefore, the power set of the P(X) = { {}, {x}, {y}, {z}, {x,y}, {x,z} {y,z}, {x,y,z} }.
It should be noted that the number of subsets in a power set is equal to 2N, where N is the number of terms of the original set. In the above example, there are three terms in the original set, and consequently, there are 23 (or 8) subsets in total in the power set.
By applying this logic to the Aleph-one, the new power set will have a larger cardinal number. This new set is known as Aleph-two. This process can be repeated an infinite amount of times, each time having a greater cardinality to the previous until we reach Aleph-Aleph-null.
Cantor himself believed in the ‘absolute infinite.’ This is the notion that encompasses all possible infinite. This idea is mathematically problematic. Russell’s paradox states that, according to set theory, “R” is the set of all infinite sets that is not a member of themselves. However, if ‘R’ is not a member of its set then by definition, it must contain itself. This is an obvious contradiction of the original statement. (Klement, 2002)
Controversially, Cantor supposed that absolute infinity was God and stated that "the Absolute can only be acknowledged and admitted, never known, not even approximately."
Sources and Further Reading
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