Numbers Larger Than Infinity!

Written By: Vishrut Kinikar

Infinity, in the minds of the common man, has been thought of as the “final number” in a seemingly unending set of whole numbers. The common-sense notion of the everyday man also, without any logical or mathematical premise, assumes the integers and the fractions between them have a fixed value and there are no numbers quantitatively greater that the aforementioned set. However, the principle of reality that science has unearthed over millennia is that nothing is ever what it seems to be on the surface, and that our common assumptions of the world around are almost never scientifically accurate. This has been summed up in the following passage from the book “Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimension” written by Michio Kaku:

“If all our common-sense notions about the universe were correct, then science would have solved the secrets of the universe thousands of years ago. The purpose of science is to peel back the layer of the appearance of the objects to reveal their underlying nature. In fact, if appearance and essence were the same thing, there would be no need for science.”

Given this principle in mind, we can now explore the numbers greater than infinity.

This study of the infinite forms a topic of set theory, the abecedarian concepts of which were known in the West since the era of Aristotle, but was refined and improved upon by German mathematician Georg Cantor.

The 1845-born mathematician had spent the most substantial part of his mathematical career in Halle, Germany, where he laid the foundation for the formulation and acceptance of set theory into the mathematical academia in the West. During his time at Halle, Cantor had published his paper “Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen” meaning “On a Property of the Collection of All Real Algebraic Numbers“. This research paper laid in front of the German mathematical academia at Halle that, without an iota of doubt, there exist infinities greater than the amount of whole numbers (which tends to be the common-sense notion for infinity). He propounded that certain infinite sets contain more elements than other infinite sets. In fact, Cantor proved that the amount of real numbers is greater than the amount of natural numbers. Real numbers are those numbers for x which are satisfy the equation y = x² for y≥0. Real numbers can also be thought of in a less abstruse and more simplistic way, as every single number (including decimals) which exists. Natural numbers are those numbers used for counting, such as 1, 2, 3, 4, 5, etc.

The cardinality (amount of elements) in the set of natural numbers is an infinite quantity known as aleph-null and is denoted by ℵ₀. If there are an infinite number of objects, it means that there are aleph-null objects. Aleph-null is the cardinality of a countable set. A countable set is a set in which each element in the set can be paired up with each element in the set of natural numbers. For example, suppose that there is a set of four apples. It can be proven that there are four apples by pairing one apple with the number 1, the next apple with the number 2, the next apple with the number 3, and the final apple with the number 4. Contrariwise, an uncountable set is a set in which the elements do not have the aforesaid one-to-one correspondence with the set of natural numbers. The set of all real numbers is an example of an uncountable set. Between 0 and 1, there exists an infinite number of numbers. However a new decimal can be generated indefinitely which causes this set of numbers between 0 and 1 to resist a one-to-one correspondence with natural numbers. This resistance to a one-to-one correspondence with the natural numbers is also the case with all the real numbers. This concept is proven using Cantor’s diagonal argument.

The diagonal argument established the existence of uncountably infinite sets and even subsequently laid the groundwork of its exploration in mathematics. The concept of uncountable set was also used by Cantor as a substantiation of his famous theorem, which states that the power set of a given set is always greater than the given set itself. However, before exploring the diagonal argument, it is imperative that properties of aleph-null and omega are properly understood. Aleph-null (alternatively known as aleph-naught) is the cardinality of the natural numbers and is the smallest infinity. Suppose that there exists an aleph-null number of books. It can be proven that the set of books has an aleph-null number of books by pairing up each book with a natural number and seeing the one-to-one correspondence. Importantly, if one more book is added to the aleph-null number of books, the cardinality does not change. The cardinality does not became ℵ₀ + 1 as one may suppose. The cardinality still remains ℵ₀. This is due to the fact that infinite amounts are distinct from finite amounts. After adding one more book to the aforesaid endless collection of books, it can be proven that this new collection still has ℵ₀ books by pairing each book with a natural number. Incredibly, even if another aleph-null books are added to the collection, the cardinality still remains ℵ₀. This, as elucidated above, can be proven by pairing each book with a natural number. From the above information, the important property about aleph-null that no matter how many elements are added to a set of aleph-null elements, the cardinality remains aleph-null can be deduced. This property is also reminiscent of even and odd numbers. One may think that that there are half as many even numbers as natural numbers, but the truth is that the cardinality of the set of natural numbers is equal to the cardinality of the set of even numbers. This, as mentioned above, can be proven by assigning each element of the set of even numbers to every element in the set of natural numbers (i.e. 0 to 1, 2 to 2, 4 to 3, 6 to 5, etc.) The case is the same with the set of odd numbers.

It has been established that the cardinality of an aleph-null-sized set cannot be changed regardless of the cardinality of any new elements which may be added. However, one may ask what labels (or ordinal number) has to be assigned to these new elements which are added to the aleph-null-sized set. To answer this question, it will be helpful to return to the example of the set of aleph-null books. It will also be helpful to understand the concept of ordinal numbers beforehand.

An ordinal number is a number which does not signify a quantity. An ordinal number simply exists to act as a label which makes sure that elements in a set are properly ordered. For example, if there are four apples, each number (namely 1, 2, 3, and 4) serves as a number which gives the apples a numerical order. In other words, each number (1, 2, 3, and 4) is a label which gives each apple an order. The ordinal number a set has is called the set’s order type. For finite sets, cardinality and order type are the same. The cardinality and order type of the aforementioned set of four apples is 4.

The order type of a set of aleph-null objects is an ordinal number called “omega”. This ordinal number is denoted using the symbol ω, which is the lowercase Greek letter omega. Going back to the example of an aleph-null amount of books will help readers better comprehend the nature and properties of omega. If there is a collection of aleph-null books, and one more book is added, the cardinality, as described previously, still remains aleph-null. On the other hand, the order type of this infinite set changes. The order type is now ω+1. If one more book is added, the order type is ω+2. If yet another book is added, the order type is ω+3. If another aleph-null number of books was added to the set, the order type becomes ω+ω, or 2(ω). All of the ordinal numbers described above describe the same cardinality (i.e. aleph-null). Note that none of these numbers are greater or less than one another. They simply describe the order in which they appear in the infinite set.

Given that background, it is now important to move on to the infinities greater than aleph-null. Aleph-null is the cardinality of the natural numbers, which is a countable set. Now it important to take a glance at the cardinality of the set of real numbers, which is an uncountable set. Cantor’s diagonal argument will be used to illustrate the cardinality of the set of real numbers. An infinite set of real numbers is provided in the table below:

0.	2	4	3
0.	7	4	9
0.	2	5	7
0.	5	3	9

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In this scenario, we start in the largest place of the first number, then go to the second-largest place of the second number, then go to the third-largest place of the third number, etc. This forms of a diagonal line, hence the name “diagonal argument”. If the number we encounter is less than 9, a 1 is added, but if the number is less than 9, a 1 is subtracted. If this is done on every single number in the infinite set provided above, a new number is generated which is different from every number in one decimal place, meaning that a new number has been generated in this infinite list. From this, it can be obviously deduced that the fact that new real numbers can be generated indefinitely proves that the set of real numbers resists a one-to-one correspondence with the set of natural numbers, and thus has a cardinality greater than aleph-null. The cardinality of the set of real numbers and natural numbers is infinity, but the infinity that the cardinality of the set of real numbers is equal to is quantitatively greater than the infinity that the cardinality of the set of natural numbers is equal to.

Like the set of real numbers, another uncountable set is the power set of aleph-null. The power set of a set is all the possible subsets that can be made from the given set. The power set of a set is equal to 2 to the power of however many elements are in the set. For example, the power set of a given set with 5 elements is 2⁵ = 32, meaning that 32 subsets can be made from the given set of 5 elements. Note that the given set and the empty set also constitute the subsets which can be made from the given set. The power set of aleph-null is 2^ℵ₀ and is equal to an infinity which is greater than ℵ₀. This can also be proven with Cantor’s diagonal argument.

An infinite amount of subsets can be made from the natural numbers which is demonstrated in the table below:

1	2	3	4	5	6	7	8…
Y	N	Y	N	Y	N	Y	N
N	N	Y	Y	N	N	N	N
N	Y	N	N	N	N	N	N
Y	Y	Y	Y	Y	Y	Y	Y
N	N	N	N	N	N	N	N
Y	Y	N	Y	Y	N	N	Y
N	N	Y	N	N	Y	Y	N

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Y is an abbreviation of “yes” and means that the number is included in the subset. Likewise, N is an abbreviation of “no” and means that the number is not included in the subset. It can be seen that this table is infinite because an infinite number of subsets can be generated from a set full of infinite elements. If every subset can be matched with a natural number, it means that the power set of aleph-null is equal to aleph-null, but this is not the case. If we go diagonally down the table (i.e. the leftmost box of the topmost subset, the second leftmost box of the second topmost subset etc.) and change every “yes” to a “no” and every “no” to a ‘yes”, a new subset which was not previously present in the infinite table would be generated, and would differ from every subset on the infinite table in one element or number. New subsets like these which were never previously present on the infinite table created beforehand can be generated indefinitely. From this observation, it is evident that the power set of the natural numbers does not have a one-to-one correspondence with the natural numbers, meaning that the power set of aleph-null is greater than aleph-null. The power set of aleph-null is an infinity which is quantitatively greater than aleph-null itself.

Given the infinity ℵ₀ and its power set, it may seem as if a ceiling has been hit. “are there any larger infinities?” one may ask. The answer to that question is yes, and they can be reached through what is known as the axiom schema of replacement. The axiom schema of replacement serves an irreplaceable purpose in the set theory of Ernst Zermelo and Abraham Fraenkel. An axiom is a mathematical or logical proposition which is assumed to be true self-evidently without any need for proofs. For example, the following are five axioms of Euclid, a 4th century BCE Greek mathematician:

1. Things which are equal to the same thing are also equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.

The axioms above are fairly simple, yet incredibly powerful ideas which can be thought of as the concrete foundation of mathematics. The above-listed statements are self-evidently true and are thus axioms. The axiom schema of replacement is an axiom which, put simply, is an axiom which says that if the set of natural numbers is replaced with any other object, the cardinality of the set remains the same. For example, if every number in the set of natural numbers is replaced with an apple, then the cardinality of the apples remains the same, it remains aleph-null. This axiom is the solution to the problem that the axiom of infinity poses. The axiom of infinity was propounded by Ernst Zermelo and was the axiom which declared that infinity is a number which exists and is equal to the cardinality of the set of natural numbers. However, if this is the case, then can the set of ordinal numbers from omega to “omega plus omega” (or omega multiplied by two) be defined? Due to the fact that omega is the order type of aleph-null, the set of numbers from ω to ω+ω must have a cardinality equal to ℵ₀. The axiom of infinity, however, establishes that infinity is the cardinality of the set of natural numbers, but does not establish that infinity is the cardinality of the set of ordinal number from ω to ω+ω. Hence, the answer to the question of whether the cardinality of such a set is defined by the axiom of infinity might seem to be no. A solution which may come to mind is that a new axiom can be made every time a set with an infinite cardinality which declares that that the new infinite set exists. However, this approach creates infinitely many unnecessary axioms. This is where the axiom schema of replacement comes in. As described previously, the axiom schema of replacement states that if the set of natural numbers is replaced with any other object, the cardinality of the set remains the same. This axiom schema of replacement is helpful in this situation because the set of natural numbers (1, 2, 3, 4, 5, 6 etc.) can be replaced with the numbers ω+1, ω+2, ω+3, ω+4, ω+5, ω+6 etc. The set still has the same cardinality because the natural numbers have been replaced by ω+1, ω+2, ω+3 etc. and according to the axiom schema of replacement, if the natural numbers are replaced by any other object, the cardinality of the set remains the same, meaning the the set of numbers between ω and 2ω have the same cardinality as the set of natural numbers (i.e. ℵ₀). Similar to how the infinite set between ω and 2ω has been defined, other infinite sets like these can be defined. The set {ω, 2ω, 3ω, 4ω, 5ω, 6ω…} can also be shown to have an infinite cardinality by replacing the elements in the set of natural numbers with the aforesaid set. Another infinite set {ω, ω², ω³, ω⁴, ω⁵, ω⁶…} can be defined as well. There are many infinite sets like the aforementioned sets that can be defined and even ordinal numbers as great as ω^ω^ω^ω^ω… can be generated. Interestingly, an ordinal number which comes after all of the ordinal numbers in the aleph-null-sized set after ω can be defined. This ordinal number can be called ω₁, and due to the fact that ω₁ comes after every single ordinal in the ℵ₀-sized set after ω, the order type of ω₁ corresponds to the cardinality ℵ₁. Using the axiom schema of replacement, new sets such as {ℵ₀, ℵ₁, ℵ₂, ℵ₃, ℵ₄…} and sets of their corresponding order types such as {ω, ω₁, ω₂, ω₃, ω₄…} can be generated. By using this axiom new greater and greater infinities are being generated. Greater infinities can be generated using the axiom schema of replacement as well. Enormous infinities such as “ω sub ω sub ω sub ω sub ω…” can be generated. Replacement can be used repeatedly to generate greater and greater infinities such as the one elucidated above. However, taking the power set is also a magical tool which helps us generate greater and greater infinities. After an infinity has been reached such as “ω sub ω sub ω sub ω…”, the power set of the set of all numbers until “ω sub ω sub ω sub ω…”, can be taken, which generates a greater infinity. Repeatedly using replacement and power sets can generate new enormous unprecedented infinities. However, there can exist such an infinity which is so great that it cannot be generated no matter the amount of replacement and power setting is used. Such an infinity is known as an inaccessible cardinal. There are distinct kinds of inaccessible cardinals which are used extensively in set theory for more advanced mathematics. These and more aspects of set theory will be discussed in the Part 2 of this article.