Cardinals: numbers used to describe the cardinality of sets. Georg Cantor introduced the use of the symbol aleph, the first letter of the Hebrew alphabet, to symbolize infinite cardinals.

The infinite cardinals are closely connected with one of the greatest puzzles in 19th and 20th century mathematics, namely, the ►continuum hypothesis. To help illustrate this puzzle, here is a summarized step-by-step introduction to set theory; in addition, we recommend reading the article on ►countability.

A set is a conceptual collection of elements — for example, numbers. Sets are indicated by curly braces; thus, the set consisting of the three numbers 3, 4, 5 is indicated as 3, 4, 5 }. You can think of the braces indicating a set as of a thought bubble holding together the three elements 3, 4, and 5.

A set may contain a finite or an infinite number of elements. In this dictionary, we are primarily interested in the latter, of course. The set of natural numbers, for example, is infinite (see ►countability). We indicate the infinite continuation of the sequence of natural numbers by three dots ...:

Now, the cardinal number (or simply cardinality) of a set is the number of its elements. (As a memory hook, consider that the more supporters a cardinal has the more powerful - or of higher cardinality - he is.) The set { 3, 4, 5 }, for example, has three elements, so its cardinality is 3.

A subset of a set A is a set B containing any selection of elements from those contained in A — including, possibly, all of those elements or even no elements at all. Thus, the set { 3, 4, 5 } has a total of eight possible subsets { }, { 3 }, { 4 }, { 5 }, { 3, 4 }, { 4, 5 }, { 3, 5 } and 3, 4, 5 }. The set containing zero elements { } is also called the "empty set".

A set of cardinality n has precisely 2n subsets. For there are exactly 2n possibilities to select n random elements from this set. In our 3, 4, 5 } example we obtain 23 = 8 subsets.

For each set A there is a larger set that is A's power set. The power set of a set A is simply the set of its possible subsets. In our 3, 4, 5 } example the power set would be the following set: { { }, { 3 }, { 4 }, { 5 }, { 3, 4 }, { 4, 5 }, { 3, 5 }, 3, 4, 5 } }.

If a set is of cardinality n, then, according to what was said above, its power set is of cardinality 2n. As we see, power sets are much more "powerful" than their original sets; they are of much higher cardinality. This will be significant for our subsequent considerations.

So much for general set theory. Now we turn to Cantor's infinite sets and their aleph numbers. Cantor indicated the cardinality of the set of ►natural numbers by means of the symbol א0 (pronounced as: aleph zero). This is the most important cardinal number, because it automatically belongs to all ►countably infinite sets insofar as any such set contains exactly the same number of elements as does the set of natural numbers. In some mathematical expositions, the number of natural numbers is indicated by naleph ("naive aleph", for the most basic kind of infinity).

Cantor suspected that sets with even "higher" levels of infinity would also be of higher cardinality. He consequently named the next higher possible cardinality א1 (aleph one). Depending on how many levels of infinity exist, there will be corresponding additional aleph numbers א2, א3, א4 etc., each of which is greater than its predecessor and corresponds to the next higher level of infinity. Let us think about which concrete sets these cardinals — which at this point are still merely hypothetical — might belong to.

The aleph numbers are subject to some strange rules of computing as a result of their infinity levels. Thus, for example, for each infinite cardinal we have the following rule א:

In words: If an element is added to an infinite set then the number of elements in that set does not change. Infinite plus one equals infinite. The same applies if we add two, three, or even infinitely many elements:

You'll find the reason for this rule if you look once again into the article on ►countability: If, for example, we add the set of even numbers to the set of odd numbers, thereby adding the numbers of their elements and hence their cardinalities, then we obtain the total set of natural numbers. However, since all three sets are countable, they all have the same cardinality; that is, each of the first two sets (the ones we added together) has exactly as many elements as the third (the one we got as a result). Hence, nothing changes with regard to the number of elements of an infinite set when we multiply it by 2, 3, 4 or any other finite number a:

Let us now boldly multiply the number of elements in our infinite set by an infinite number. To do so we simply replace each individual element within the set by the complete set itself. As we can figure, this amounts to multiplying the set's cardinal by itelf. Yet this, too, leaves the number of elements unchanged:

This formula basically represents the ►countability of fractions. For each natural number n corresponds to just as many fractions as there are natural numbers: n/1, n/2, n/3,. etc. Therefore, multiplication of a cardinal by itself corresponds to the transition from the set of natural numbers to the set of fractions. An infinite set retains the same number of elements even if it is multiplied by itself an indefinite number of times:

Of course, the question now arises whether any given arithmetic operation on א will yield א The answer is: No, not every one. For Cantor has shown:

2א is even larger than א! Let us recall: 2א is the cardinality of the power set of a set with cardinality א. And the cardinality of a power set is always greater than that of the original set — and, as Cantor has shown, this applies even to infinite sets. Thus, the cardinality of the power set of an infinite set is indeed even "more infinite" than that of the original set.

With this Cantor showed that there really is a sequence of aleph numbers א0, א1, א2, etc. For, since we can always derive a power set for any given infinite set, and even a power set for the power set, there is no upper limit regarding the aleph numbers. The sequence of alephs, and thereby that of the different levels of infinity, is itself infinite!

Besides the set of natural numbers with its cardinality א0  there is also the set of ►real numbers. These are all numbers with any given positions after the decimal point. We know that this set is not ►countable and hence that it is "more infinite" than the set of natural numbers. And now our sixty-four-thousand dollar question: What is the cardinality of the set of real numbers?

Cantor has shown that this cardinal amounts to 2א0, which is equal to the cardinality of the natural numbers' power set. This can be illustrated as follows: The power set of the set of natural numbers contains all conceivable subsets, each of which contains no more than a countably infinite number of elements. The set of real numbers contains all conceivable numbers, each of which has no more than countably infinite positions after the decimal point. Thus both sets are comparable with regard to the number of their elements.

Thus, we have now discovered two infinite cardinals belonging to concrete sets, namely א0, which is the cardinality of the set of natural numbers (and the smallest aleph number), and 2א0, which is the cardinality of the set of real numbers.

To conclude this excursion on cardinals, let us think about the obvious question whether 2 may be the second smallest infinite cardinal or, in other words, whether the following applies:

Or is there yet another infinite set whose cardinality is larger than א0, but at the same time smaller than 2א0? This seemingly innocuous question had dramatic consequences for Georg Cantor himself as well as for 20th century mathematics. The answer is quite surprising. You will find it in the article on the continuum hypothesis.


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