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Article Rating     # Is there an infinity between 0 and 1?

## Infinity

If a set is infinite , no matter how many members of the set are counted, there are still more that have not been counted.  Infinity can also be said to be larger than any arbitrary value. Infinity is not a number, so it can not be added, subtracted or multiplied. The symbol for infinity is like an 8 lying on its side: ∞ .

A set is finite if it does not go on forever. A set that is finite can be very large, such as the natural numbers from 1 to 1 billion. If it is possible to count all the members of a set, then it is finite. It would take a long time to count from 1 to 1 billion, but it is possible  .

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#### Denumerable Infinity

In 1874, Georg Cantor (1845 – 1918) published a proof that not all infinities are the same. One class of infinity is denumerable , meaning countable.

The set of integers is the definition of denumerable infinity. Any set that has a one to one correspondence with the set of integers is called denumerable.

The set of real numbers is not denumerable. Start with 1 . Then add a decimal point: 0.1 . Now keep adding a zero after the decimal point: 0.01 , 0.001 , 0.0001 , … . This can go on forever. Then it would need to be repeated for 2 , 0.2 , 0.02 , … . So the size of the set of real numbers is a nondenumerable infinity.

#### Subsets of Infinite Sets

A subset of an infinite set may be either finite or infinite. Take the set of integers. Since integer go on forever, the set of integers is an infinite set. The set < 1 >is a subset of the set of integers. It is a finite set, as it has exactly one member. The set of all even numbers is also a subset of the integers. However, the set of all even numbers is infinite, since it is impossible to find the highest even integer.

### References

1. McAdams, David E. . All Math Words Dictionary, infinity . 2nd Classroom edition 20150108-4799968. pg 99. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. Archimedes . The Sand Reckoner . Translated by Unknown. http://euclid.trentu.ca. Last Accessed 8/7/2018. http://euclid.trentu.ca/math/sb/3810H/Fall-2009/The-Sand-Reckoner.pdf . Buy the book
3. Davies, Charles; Bourdon, M. . Key to Davies’ Bourdon : with many additional examples, illustrating the algebraic analysis : also, a solution of all the difficult examples in Davies’ Legendre . pp 14-16. www.archive.org. A. S. Barnes & Co.. 1866. Last Accessed 8/7/2018. http://www.archive.org/stream/additionalbourd00davirich#page/14/mode/1up/search/infinity . Buy the book
4. Derek C. Goldrei . Classic Set Theory for Guided Independent Study . pp 1-2. Springer. August 1 1996. Last Accessed 8/7/2018. Buy the book

McAdams, David E. Infinity . 4/23/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/i/infinity.html .

### Revision History

4/23/2019: Updated equations and expressions to new format. ( McAdams, David E.)
3/20/2019: Clarified language. ( McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. ( McAdams, David E.)
8/6/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. ( McAdams, David E.)
8/11/2008: Miscellaneous revisions. ( McAdams, David E.)
6/7/2008: Corrected spelling. ( McAdams, David E.)
2/5/2008: Change keyword highlighting. ( McAdams, David E.)
8/7/2007: Initial version. ( McAdams, David E.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.

## How Big Is Infinity? 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^), 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):

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.

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>), 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), by progressing through in a diagonal fashion along the red arrows. Any repeated entries are skipped, such as (frac) which would already have been counted in its reduced form of (frac). 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.

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_nrbrack) is the digit in the nth position after the decimal point of the real number located at the (n^

) position of the count. In the image, (lbrack d_1rbrack) of the number at count (1) is (2); (lbrack d_2rbrack) of the number at count (2) is (0); (lbrack d_3rbrack) of the number at count (3) is (7); and so on. To create our new number, if (lbrack d_nrbrack) in the (n^

) real number on the list is a (0), put a (1) as the (n^

) 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^) digit of the real number located at the (n^) 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.  