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Infinity

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Infinity, often denoted ∞, is the concept of a quantity increasing without bound. Infinity is not a number; it is beyond every number. The axioms and concepts applying to finite numbers do not apply to infinity. For example, 1+1≠1, but ∞+1=∞.

When a series or quantity increases without bound, it is said to go to infinity. For example, 1+1+...+1 = ∞. Similarly, the limit as x goes to 0 of 1/x = ∞. Note that it is technically not proper to say 1/0=∞, because one could just as well say that 1/0 = -∞.

Often it is useful to refer to some thing by a number. When mathematicians discuss a set of things, if they succeed in labeling each item in the set with a different integer, then the set is countable. In some cases, there are "too many" items in the set to assign each one to a different integer -- such a set is uncountable. (Sometimes mathematicians can assign each item in an uncountable set to a different real number).

Classification of infinities

In modern mathematics, it is recognized that there is no single concept of 'infinity' - there are many different infinities. We can divide these into some broad classes. Firstly, there are the cardinals, which are used to measure the sizes of sets; and then there are ordinals, which are used to measure the position of an item in an ordered list. For finite quantities, we can use the same numbers for both - a race can have 3 participants, and you can come 3rd in the race. But, with infinite quantities, this no longer applies, the same numbers can now be used for both purposes. Thus, the smallest infinite cardinal is aleph-null, but the smallest infinite ordinal is omega.

Cardinals

We say two sets have the same size if it is possible to put their respective members into one-to-one correspondence. A cardinal can be conceived as the equivalence class of all sets having that size.

Consider the set of all natural numbers, and its subset the set of all odd natural numbers. One would think that there are half as many odd numbers as natural numbers. Yet, while that is true when dealing with some finite subset of the natural numbers, when dealing with their entirety, it turns out that there are as many odd numbers as natural numbers. This can be shown by the fact that the natural numbers and the odd natural numbers can be put in a one-to-one correspondence: 0 -> 1, 1 -> 3, 2 -> 5, 3 -> 7, etc. (represented by the formula 2*n + 1). Thus, whereas for finite sets if A is a proper subset of B, then the cardinality of A must be less than the cardinality of B, for infinite sets this is no longer the case; an infinite set may be the same size as one of its proper subsets.

The cardinality of the natural numbers is known as aleph-null. There are as many integers as natural numbers; and as many rational numbers as natural numbers. But it can be shown, using Cantor's diagonalization argument, that the reals are strictly larger than the natural numbers - you cannot put the natural numbers into one to one correspondence with the reals. The cardinality of the reals is the cardinality of the power set of the natural numbers, which is known as the power of the continuum (c) or as beth-one. (Aleph-null is the same as beth-null; whether aleph-one is the same as beth-one is a mathematical question, which essentially has no answer - and we can even show it has no answer.)

The distinction between several major viewpoints in the philosophy of mathematics is in how they deal with uncountable sets and countable (but infinite) sets.

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