∞Exploring Infinity ∞ By Christopher Imm Johnson County Community College

1. The “Z mod n”, group,. This is the set formed by the remainders of the integers divided by a positive integer n. 2. The set of all Natural numbers,, {1,2,3,4,…}. 3. The set of all positive rational numbers,, the ratio of two integers in simplified form. 4. The set of real numbers, What are the sizes of the following sets?

Example 1 The “Z mod n”,, group is formed by the remainders of the integers divided by a positive integer n. This is intuitive by asking how many elements would be in the set. Counting the number of elements in the set, the size of this is set is n. What are the possible remainders upon division by n?

Example 2 What is the size of the set,, {1,2,3,4,…}? This may be a bit harder to visualize, our first question may be, is infinity allowed to represent the size of a set? If so, how do we represent this?

Example 3 What is the size of the set of, the positive ratio of two integers in simplified form? Again the answer seems to be infinity, however, this set seems a bit “larger” than the one in example 2. Why? Because you can think of many positive rational numbers between the first two natural numbers… For example the sequence: This means there are an infinite number of numbers between the first two numbers of another infinite set, the natural numbers.

Example 4 What is the size of the set of the real numbers on the interval [0,1]? The answer is infinity, again, however, the underlying question is, how can we compare these infinite sets?

Introducing Georg Cantor A German mathematician born in St. Petersburg, Russia in 1845.

Cantor introduced different “sizes” of infinity Cantor devised a system from the Hebrew letter aleph, called aleph numbers. Cantor devised a system from the Hebrew letter aleph, called aleph numbers. These numbers were also called Cardinal numbers or Cardinals, for short. These numbers were also called Cardinal numbers or Cardinals, for short. The “smallest” infinite set was described as Cardinal aleph-naught or aleph-null. The “smallest” infinite set was described as Cardinal aleph-naught or aleph-null. It was denoted as It was denoted as The set of the natural numbers,, as mentioned in Example 2, have Cardinal The set of the natural numbers,, as mentioned in Example 2, have Cardinal

Discerning between sizes of infinity Cantor used the idea of a bijection between a set and the natural numbers,, to describe all sets Cardinality Cantor used the idea of a bijection between a set and the natural numbers,, to describe all sets Cardinality A bijection is a one to one, onto mapping between two sets. A bijection is a one to one, onto mapping between two sets. Sets of this type were sometimes called countably infinite or countable. Sets of this type were sometimes called countably infinite or countable. The positive rational numbers,, in example 3, are another example Cardinality The positive rational numbers,, in example 3, are another example Cardinality

There is a bijection between and The idea of the 1-1, onto correspondence follows in the diagram below.

Cont. If any fractions not in reduced form are eliminated and we follow the arrow, the set of is in one to one and onto correspondence with. Therefore is countable. and two good examples of sets of Cardinality and two good examples of sets of Cardinality

The Song To the tune 99 bottles of beer on the wall. ♪Aleph-null cups of coffee on the wall, Aleph-null cups of coffee. Take one down, pass it around, Aleph-null cups of coffee on the wall. ♫ (And repeat…)

Larger sizes of infinity. Cantor realized that there were sets larger than aleph- naught. The next Cardinal number he defined was aleph-one. Cantor realized that there were sets larger than aleph- naught. The next Cardinal number he defined was aleph-one. These sets were called the Cardinality of the continuum represented by the real numbers, These sets were called the Cardinality of the continuum represented by the real numbers, It was denoted as or c for the continuum. It was denoted as or c, for the continuum. The set of real numbers,, or the subset on the interval [0,1] as mentioned in Example 4, have Cardinality The set of real numbers,, or the subset on the interval [0,1] as mentioned in Example 4, have Cardinality Sets of this size are referred to as uncountable. Sets of this size are referred to as uncountable.

Cantor’s diagonalization argument Prove that the set S = is uncountable.

Cont. Cantor’s original diagonal argument was done with a binary representation of the real numbers in decimal form. Thus, the new decimal representation was chosen to be the complement of each diagonal element, forming a new number not in the set. Cantor’s original diagonal argument was done with a binary representation of the real numbers in decimal form. Thus, the new decimal representation was chosen to be the complement of each diagonal element, forming a new number not in the set.

The Song To the tune 99 bottles of beer on the wall. ♪ Aleph-one cups of coffee on the wall, Aleph-one cups of coffee. Take infinity down, pass infinity around, Aleph-one cups of coffee on the wall. ♫ (And repeat…)

The Cantor Set To create the Cantor set, take the interval [0,1] on the real number line, call this the initial stage or C 0. To create the Cantor set, take the interval [0,1] on the real number line, call this the initial stage or C 0. In the first stage, C 1, we remove the middle third of the segment. In the first stage, C 1, we remove the middle third of the segment. For each additional stage continue to remove the middle third of each segment, call the nth stage C n. For each additional stage continue to remove the middle third of each segment, call the nth stage C n. The Cantor set is C where, The Cantor set is C where,

Graphical Representation of C The first few “stages” below, C 0, C 1, C 2 :

What is the “length” of C? To figure this out, consider these questions: How many segments are taken away in each stage? How many segments are taken away in each stage? What is the length of each segment taken away in each stage? What is the length of each segment taken away in each stage? How can we represent the sum of all the segments taken away in each stage? How can we represent the sum of all the segments taken away in each stage? What is the limit of this sum as n- What is the limit of this sum as n->∞.

Cont. The number of segments taken away at each stage can be represented by The number of segments taken away at each stage can be represented by The length of each segment taken away at each stage can be represented by The length of each segment taken away at each stage can be represented by The total length taken away at each stage can be represented by the series The total length taken away at each stage can be represented by the series The total length taken away is given by: The total length taken away is given by:

Conclusion Since the Cantor set is constructed by a set of length one and the sum of the segments taken away is one, the Cantor set has a length of zero. Since the Cantor set is constructed by a set of length one and the sum of the segments taken away is one, the Cantor set has a length of zero. Length of sets are referred to as measure. Length of sets are referred to as measure. Thus, the Cantor set has measure 0. Thus, the Cantor set has measure 0.

What is the Cardinality of the Cantor set? To discover this, ask some other questions. What are some elements remaining in the Cantor set? What are some elements remaining in the Cantor set? Is there a convenient representation for the entire Cantor set? Is there a convenient representation for the entire Cantor set? Is there a bijection between the Cantor set and the natural numbers, ? Is there a bijection between the Cantor set and the natural numbers, ?

What are some elements remaining in the Cantor set? All the endpoints of the remaining intervals. All the endpoints of the remaining intervals. For example: 0, 1/3, 2/3, 1, 1/9, 2/9, 7/9, 8/9, and so on… For example: 0, 1/3, 2/3, 1, 1/9, 2/9, 7/9, 8/9, and so on…

Is there a convenient representation for the entire Cantor set? Notice that all elements in the Cantor set are powers of 1/3. A unique way to represent all elements is to use base 3 or the ternary representation. Exs: 0=0 3, 1/3=0.1 3, 2/3=0.2 3, 1=0.222… 3, 1/9=0.01 3, 2/9=0.02 3,7/9=0.21 3,8/9=0.22 3, and so on…

The answer to this question is no, by viewing the ternary representation of the Cantor set, C. The answer to this question is no, by viewing the ternary representation of the Cantor set, C. What about the real numbers, What about the real numbers, Consider the real numbers on the interval [0,1]. Consider the real numbers on the interval [0,1]. Try to find a surjective (onto) mapping, f, from C to the real numbers on [0,1]. Try to find a surjective (onto) mapping, f, from C to the real numbers on [0,1]. To this end, represent the real numbers on [0,1] in base 2 or binary. To this end, represent the real numbers on [0,1] in base 2 or binary. Is there a bijection between the Cantor set and the natural numbers, ?

Find the function f :C->[0,1] We can express every number in C in its ternary representation only consisting of 0’s and 2’s (repeating). For example for the first few stages: 0=0 3, 1=0.222… 3 1/3=0.1 3 =0.0222… 3, 2/3= /9= = … 3, 2/9=0.02 3, 7/9= = … 3, 8/9=0.22 3, and so on…

Cont. Similarly, we can express all reals on the interval [0,1] in their binary representation, from Similarly, we can express all reals on the interval [0,1] in their binary representation, from 0=0 2, to 1=0.111… 2. Thus, replacing all 2’s in the numbers in C by 1’s, creates a surjective (onto) mapping from C to the real numbers in [0,1]. Define f as follows:

Conclusion Since, C is surjective to the reals on [0,1], Since, C is surjective to the reals on [0,1], C must have at least the cardinality of c or Aleph-1. However since C is a subset of the reals on [0,1], it must be at most that Cardinality as well. Thus C has Cardinality of c or Aleph-1. It is worth noting, f is not bijective (1-1). For example, It is worth noting, f is not bijective (1-1). For example, Hence, f (7/9)=f (8/9), however 7/9≠8/9. Hence, f (7/9)=f (8/9), however 7/9≠8/9.

Summary of the Cantor Set The Cantor set has measure 0. The Cantor set has measure 0. The Cantor set is uncountably infinite, with Cardinality of c or Aleph-1. The Cantor set is uncountably infinite, with Cardinality of c or Aleph-1. The Cantor set is an example of a set which you can take an uncountably infinite number of elements away from an uncountable set and still have an uncountably infinite set. The Cantor set is an example of a set which you can take an uncountably infinite number of elements away from an uncountable set and still have an uncountably infinite set. The Cantor set has the added property of being closed and bounded. The Cantor set has the added property of being closed and bounded.

Enter Waclaw Sierpinski A Polish mathematician born in Warsaw, Poland in 1882.

Sierpinski’s Carpet The process: To build Sierpinski's Carpet, S, start with a square with side length 1 unit, completely shaded. (Iteration 0, or the initiator) To build Sierpinski's Carpet, S, start with a square with side length 1 unit, completely shaded. (Iteration 0, or the initiator) Divide each square into nine equal squares and cut out the middle one. (the generator) Divide each square into nine equal squares and cut out the middle one. (the generator) Repeat this process on all shaded squares. Repeat this process on all shaded squares.

Graphical representation of S Interactive Sierpinski's Carpet Interactive Sierpinski's Carpet

The size of S. Start with an area of 1 square unit. Start with an area of 1 square unit. The number of squares taken away at iteration n is The number of squares taken away at iteration n is The size of each square taken away at iteration n is The size of each square taken away at iteration n is The total area taken away at iteration n is The total area taken away at iteration n is The total area taken away from S is The total area taken away from S is

Conclusion The area of Sierpinski’s Carpet is 0.

The Menger Sponge The process: To build the Menger Sponge, M, start with a cube edge 1 unit. (the initiator) To build the Menger Sponge, M, start with a cube edge 1 unit. (the initiator) Divide the cube into twenty-seven equal cubes and cut out the middle one. (the generator) Divide the cube into twenty-seven equal cubes and cut out the middle one. (the generator) Repeat this process on all remaining cubes. Repeat this process on all remaining cubes.

Graphical representation of M Visualizing the Menger Sponge Visualizing the Menger Sponge Visualizing the Menger Sponge Visualizing the Menger Sponge

The size of M Start with an cube of volume 1 cubic unit. Start with an cube of volume 1 cubic unit. The number of cubes at iteration n is The number of cubes at iteration n is The volume of each cube at iteration n is The volume of each cube at iteration n is The total volume at iteration n is The total volume at iteration n is The total volume of M is The total volume of M is

Extensions to higher dimensions The procedure then for creating an N dimensional pyramid can be summarized by the following rules. Start with an N-1 dimensional cube centered at the origin. Start with an N-1 dimensional cube centered at the origin. Pull the midpoint of the cube (origin) into the Nth dimension. Pull the midpoint of the cube (origin) into the Nth dimension. Make edges from the midpoint to each vertex of the N-1 cube. Make edges from the midpoint to each vertex of the N-1 cube. Make faces using the midpoint and each edge of the N-1 cube. Make faces using the midpoint and each edge of the N-1 cube. Using these rules the 4D pyramid (hyper-pyramid) is constructed by taking a 3D cube and pulling its midpoint into the 4th dimension. Using these rules the 4D pyramid (hyper-pyramid) is constructed by taking a 3D cube and pulling its midpoint into the 4th dimension.

A “4-D” Hyper-Gasket

The process can be continued by forming another pyramid with the hyper-cube, and so on…