How big is infinity? The answer depends upon the infinity in question. There are different sizes of infinity, but we can compare them through a concept called countability.

A countable infinity is one which can be counted using the natural numbers (all of the whole numbers greater than zero): \(1\), \(2\), \(3\), \(4\), \(5\), \(...\), etc. The set of natural numbers is itself infinite in size, but if we pick any particular natural number, whether it's \(10\) or \(10^{9}\), we can eventually count our way up to that number in a finite amount of time (though perhaps not in a human lifetime). This is what is meant by countable, and any other infinite set which can be mapped in a one-to-one correspondence with the set of natural numbers will also be considered to be "countably infinite".

For instance, the set of integers (\(...\), \(-3\), \(-2\), \(-1\), \(0\), \(1\), \(2\), \(3\), \(...\)) is infinite yet countable. Using the natural numbers, begin counting the integers as follows, jumping back and forth between positive and negative as you progress away from \(0\):

\(1: 0\)

\(2: 1\)

\(3: -1\)

\(4: 2\)

\(5: -2\)

\(6: 3\)

\(7: -3\)

\(...\)

The numbers on the left are the natural numbers marking each position in the count, and the numbers on the right are the integers which are being counted.

We observe the following pattern: if \(n\) is a positive integer, then it will be at the \(2n\) position in the count, and if \(n\) is a negative integer, then it will be at the \([2(-n)]+1\) position in the count. \(0\) is in position \(1\) of the count. Thus, every integer is accounted for at some point during the count, and we have a one-to-one correspondence between the natural numbers and the integers.

This result should be surprising, as the natural numbers are actually a subset of the integers. For every natural number, \(n\), the integers contain both \(n\) and \(-n\). So although, intuitively, the integers contain two elements for each single element within the natural numbers, the cardinality, or size, of their infinities is the same.

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The countability of the rational numbers:

Arrange the rational numbers (numbers which can be expressed as a fraction, where the numerator and denominator are integers) in a table such that the number of each row corresponds to the numerator and the number of each column corresponds to the denominator.

A countable infinity is one which can be counted using the natural numbers (all of the whole numbers greater than zero): \(1\), \(2\), \(3\), \(4\), \(5\), \(...\), etc. The set of natural numbers is itself infinite in size, but if we pick any particular natural number, whether it's \(10\) or \(10^{9}\), we can eventually count our way up to that number in a finite amount of time (though perhaps not in a human lifetime). This is what is meant by countable, and any other infinite set which can be mapped in a one-to-one correspondence with the set of natural numbers will also be considered to be "countably infinite".

For instance, the set of integers (\(...\), \(-3\), \(-2\), \(-1\), \(0\), \(1\), \(2\), \(3\), \(...\)) is infinite yet countable. Using the natural numbers, begin counting the integers as follows, jumping back and forth between positive and negative as you progress away from \(0\):

\(1: 0\)

\(2: 1\)

\(3: -1\)

\(4: 2\)

\(5: -2\)

\(6: 3\)

\(7: -3\)

\(...\)

The numbers on the left are the natural numbers marking each position in the count, and the numbers on the right are the integers which are being counted.

We observe the following pattern: if \(n\) is a positive integer, then it will be at the \(2n\) position in the count, and if \(n\) is a negative integer, then it will be at the \([2(-n)]+1\) position in the count. \(0\) is in position \(1\) of the count. Thus, every integer is accounted for at some point during the count, and we have a one-to-one correspondence between the natural numbers and the integers.

This result should be surprising, as the natural numbers are actually a subset of the integers. For every natural number, \(n\), the integers contain both \(n\) and \(-n\). So although, intuitively, the integers contain two elements for each single element within the natural numbers, the cardinality, or size, of their infinities is the same.

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The countability of the rational numbers:

Arrange the rational numbers (numbers which can be expressed as a fraction, where the numerator and denominator are integers) in a table such that the number of each row corresponds to the numerator and the number of each column corresponds to the denominator.

There will be an infinite number of rows and columns, but every rational number will appear somewhere in the table. For instance, given the number \(\frac{3672}{529}\), we will find this number in the 529th column of the 3647th row. The entries of this table can then be counted, starting with \(\frac{1}{1}\), by progressing through in a diagonal fashion along the red arrows. Any repeated entries are skipped, such as \(\frac{2}{4}\) which would already have been counted in its reduced form of \(\frac{1}{2}\). Proceeding through the table in such a way, all of the rationals can be counted and we find that the rational numbers, like the integers, share the same infinite size as the natural numbers.

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The uncountability of the real numbers:

Now let's look at an infinite set of numbers which is not countable: the real numbers. The real numbers are the union of the rational numbers and the irrational numbers, and they account for all numbers which may be found on the continuum of the real number line. One can think about this set as all numbers which may be expressed by a decimal representation, though the decimal itself may be infinitely long.

The extent of the real numbers is so vast that in exploring their countability we don't even need to look at the entire set in order to show that their infinitude is larger than that of the natural numbers. In fact, we merely have to consider the real numbers between \(0\) and \(1\).

The proof is done by contradiction. Assume that we can use the natural numbers to count the real numbers between \(0\) and \(1\). If this is so, then we can make a list where the natural number marking the place in the count is to the left and the real number associated with that count is to the right. Note that all of the real numbers can be represented as a decimal of infinite length because if the decimal is terminating (eg., \(0.27\) or \(0.9\)), then we can add an infinite number of \(0\)'s after the last digit (ie., \(0.27000...\) and \(0.9000...\)).

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The uncountability of the real numbers:

Now let's look at an infinite set of numbers which is not countable: the real numbers. The real numbers are the union of the rational numbers and the irrational numbers, and they account for all numbers which may be found on the continuum of the real number line. One can think about this set as all numbers which may be expressed by a decimal representation, though the decimal itself may be infinitely long.

The extent of the real numbers is so vast that in exploring their countability we don't even need to look at the entire set in order to show that their infinitude is larger than that of the natural numbers. In fact, we merely have to consider the real numbers between \(0\) and \(1\).

The proof is done by contradiction. Assume that we can use the natural numbers to count the real numbers between \(0\) and \(1\). If this is so, then we can make a list where the natural number marking the place in the count is to the left and the real number associated with that count is to the right. Note that all of the real numbers can be represented as a decimal of infinite length because if the decimal is terminating (eg., \(0.27\) or \(0.9\)), then we can add an infinite number of \(0\)'s after the last digit (ie., \(0.27000...\) and \(0.9000...\)).

Let us now construct a new number in the following way. The digit \(\lbrack d_n\rbrack\) is the digit in the nth position after the decimal point of the real number located at the \(n^{th}\) position of the count. In the image, \(\lbrack d_1\rbrack\) of the number at count \(1\) is \(2\); \(\lbrack d_2\rbrack\) of the number at count \(2\) is \(0\); \(\lbrack d_3\rbrack\) of the number at count \(3\) is \(7\); and so on. To create our new number, if \(\lbrack d_n\rbrack\) in the \(n^{th}\) real number on the list is a \(0\), put a \(1\) as the \(n^{th}\) digit of the new construction, and if it is not \(0\), put a \(0\) in that position. The new number constructed from the list in the image would look like this: \(0.0100100...\)

If we pick any real number in the list, the newly constructed number will have at least one digit which is different from the digits in that number, as our construction ensured that the nth digit of the new number always differs from the \(n^{th}\) digit of the real number located at the \(n^{th}\) position in the count. This new number, therefore, is not found on the list. But we supposed that the list contained all of the real numbers, and our constructed number is clearly a real number. Therefore, our list both does and does not contain all of the real numbers. Here is our contradiction, and so we find that the assumption that we could count the real numbers between \(0\) and \(1\) using the natural numbers was incorrect. Thus, the real numbers are uncountable, and the infinite size of the real numbers is larger than the infinite size of the natural numbers (and that of the integers and the rational numbers).

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Sets of numbers sink into deeper and deeper infinities. How deep do these infinities go? Is there a limit to how large an infinity can be? Can we always find a more expansive infinity? What about searching in the other direction? Can we always find smaller infinities?

In a previous post, we explored the density of the rationals and the irrationals in the real numbers. We could always find an infinitude of numbers, no matter where we looked within the real numbers. How does this relate to the infinite size of the real number set? Is the fact that the real numbers compose a continuum relevant?

Can our experiences be constrained, or can we always find within them new aspects and nuances to explore? Are they infinite? What does that mean for the fundamental nature of experience?

Can you give a size to any infinities contained within the experiences, feelings, and ideas in your life. Can you compare those infinities? Do some of the infinities fit inside of one another? Which are disparate?

The most fundamental feature of infinity is its expansiveness. Its character is varied and expressed through multiple forms. Reflect on infinity as a dynamic entity.

If we pick any real number in the list, the newly constructed number will have at least one digit which is different from the digits in that number, as our construction ensured that the nth digit of the new number always differs from the \(n^{th}\) digit of the real number located at the \(n^{th}\) position in the count. This new number, therefore, is not found on the list. But we supposed that the list contained all of the real numbers, and our constructed number is clearly a real number. Therefore, our list both does and does not contain all of the real numbers. Here is our contradiction, and so we find that the assumption that we could count the real numbers between \(0\) and \(1\) using the natural numbers was incorrect. Thus, the real numbers are uncountable, and the infinite size of the real numbers is larger than the infinite size of the natural numbers (and that of the integers and the rational numbers).

- - - - - - - - - -

Sets of numbers sink into deeper and deeper infinities. How deep do these infinities go? Is there a limit to how large an infinity can be? Can we always find a more expansive infinity? What about searching in the other direction? Can we always find smaller infinities?

In a previous post, we explored the density of the rationals and the irrationals in the real numbers. We could always find an infinitude of numbers, no matter where we looked within the real numbers. How does this relate to the infinite size of the real number set? Is the fact that the real numbers compose a continuum relevant?

Can our experiences be constrained, or can we always find within them new aspects and nuances to explore? Are they infinite? What does that mean for the fundamental nature of experience?

Can you give a size to any infinities contained within the experiences, feelings, and ideas in your life. Can you compare those infinities? Do some of the infinities fit inside of one another? Which are disparate?

The most fundamental feature of infinity is its expansiveness. Its character is varied and expressed through multiple forms. Reflect on infinity as a dynamic entity.