Aleph NullDate: 01/22/98 at 19:08:14 From: Jonathan Subject: Aleph null What does aleph null represent? Date: 01/27/98 at 16:19:34 From: Doctor Joe Subject: Re: Aleph null Dear Jonathan, Before I deal with Aleph Null formally, I need to go through some stuff about sets, first the finite ones, then the infinite ones; and last of all I shall explain how you measure the size of these sets. A set is a collection of objects. An object is called an element. For instance, a class of K students is regarded as a set of students. Some sets are finite (meaning there is a finite number of elements in those sets) while some are infinite (for instance the set of all integers). If a class of K students has 12 members(or elements), then K has size 12. Technically, we say that the set has order 12 or we say that the cardinality of the set is 12. For infinite sets, we are also interested in whether we can somehow count the objects. Of course, we can never finish listing them. But what we may do (as best we can) is count the elements by assigning the natural numbers in a one-to-one way, so as not to skip any members of the set. This concept leads us to the formulation of countability. A set is said to be countable if either it is finite or it can be put on a one-to-one correspondence with the set of natural numbers {1,2,3,...}. For instance, we say that the set of integers is countable since the following listing demonstrates that there is a one-to-one correspondence between the set of integers and the set of natural numbers: 0 1 -1 2 -2 3 -3 ... | | | | | | | 1 2 3 4 5 6 7 ... If a set X has a one-to-one correspondence with the set of natural numbers N, then X is said to have the same cardinality as N. And the cardinality of N is denoted by the first cardinal number Aleph Null. -Doctor Joe, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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