The first transfinite cardinal. A collection of objects with cardinality aleph-null is the smallest collection such that there are exactly enough natural numbers to count them all. Yes, it blew my f-ing mind too. Compare to omega-null, the first transfinite ordinal.

Aleph-null bottles of beer on the wall,
Aleph-null bottles of beer,
You take one down, and pass it around,
Aleph-null bottles of beer on the wall.

Joke Nodes: Geek Jokes: aleph-null

2^aleph null is c, which is the cardinality of the set of real numbers.

In a sense hodgepodge is correct, and in a sense he is not. It would perhaps be more correct to say that a set with cardinality aleph-null is the smallest set such that there is no natural number to represent the number of elements in the set.

A set with cardinality aleph-null is a set which can be put in one-to-one correspondence with the set of natural numbers. These sets are called "countable" sets for this reason. Many important sets in mathematics are countable, such as:

One useful property of a countable set is the ability to put a non-dense ordering on it. This means that one can define "less than" such that there is no number "between" two adjacent numbers. This is very useful for certain kinds of proof, e.g. proof by induction.

As with all the transfinite numbers, aleph-null does not obey the same laws of algebra as finite numbers. Here are a few basic facts about algebra involving aleph-null:

k + aleph-null = aleph-null for any finite k

k aleph-null = aleph-null for any finite non-zero k

aleph-nullk = aleph-null for any positive k

However, kaleph-null = aleph-one for any finite k > 1

Yes, I know a lot of this is a bit wrong. I've ignored lots of horrible stuff about the Axiom Of Choice. This just gives a basic overview of the concept. If you want more grisly details, read Gorgonzola's write-up under cardinal.

This has been part of the Maths for the masses project

As 10998521 has pointed out, aleph0 is the cardinality of both the integers and the rationals. Essentially, that means that there are just as many counting numbers as there are fractions. This can be demonstrated as follows.

Observe the following division table:

/ | 1 | 2 | 3 | 4 | 5 | ...
1 | 1 |1/2|1/3|1/4|1/5|
2 | 2 | 1 |2/3|1/2|2/5|
3 | 3 |3/2| 1 |3/4|3/5|
4 | 4 | 2 |4/3| 1 |4/5|
5 | 5 |5/2|5/3|5/4| 1 |
. |
. |
. |

It's not difficult to see that this table contains all the positive rational numbers. Now, we can set up a one to one correspondence between the cells of the table and the integers: Start by labeling the upper left cell "1". Then the cell to its right (cell (1,2)) is "2". The one to the lower left of that (cell (2,1)) is "3". The next cell is (3,1), then (2,2), and so on, zigzagging across the table and skipping anything we've seen already. Now if we imagine copying the table but changing the sign of all the fractions and using the negative integers as labels, we see that the set of rationals has the same cardinality as the set of integers, namely aleph0.

As a side-note, the larger cardinality of the set of reals arises from the fact that there are uncountably many irrational numbers. Though I won't actually prove fact here, I will give a bit of a suggestion of why it might be true:

2aleph-null is the cardinality of the power set of the integers. That is, the set of all unordered subsets of the integers. (See Cantor diagonalization for a justification of |R| > |N|.) Now, if we pick one set out of this power set, put it in some order (mumblemumbleaxiom of choicemumblemumble) and throw in a decimal point, we have some real number which is distinct from any other real number which could be constructed from one of the other elements of the original set.

One quick correction: 2aleph-null is not necessarily aleph1. aleph1 is, by definition, the cardinality of the smallest uncountable set. 2aleph-null is the cardinality of the reals, commonly called c. So since |R| > |N| and |N| = aleph0, we have c ≥ aleph1. It has been proven that it is unprovable whether c > aleph1 or c = aleph1. This is known as the continuum hypothesis.


Cantor's Continuum Hypothesis was that 2aleph-0=aleph-1.

What hasn't been determined is whether or not this is true, hence the title "hypothesis."

What has been determined is that granting it true we resolve one model of mathematics and granting it false we resolve another.  Both models are equally true, in a vague sense.  It helps to remember we are only dealing with a priori frameworks and not physical things (or are we is it right to pose unanswerable hypothesis when describing unanswerable hypotheses??).

At this point we are faced with an aesthetic decision whether we want a simplified model or a more elegant one.

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