What is infinity and its implications?

Answer 1

In math, infinity is not a unique concept.

There are different ways of looking at infinity.

One is to consider that something is true when a limit is taken of larger and larger numbers.

Example: if x>0 then 1/x > 0, but if x>N it follows that 1/x < 1/N and so we can make 1/x as small as we want. This is written as lim(1/x, x=infinity) = 0.

Example: if you throw dice, then the average is defined as sum(x_i, i = 1..n)/n. A limit theorem says that the expected value of the dice, defined by lim(sum(x_i, i = 1..n)/n, n=infinity) exists and equals sum(1/6 * i, i = 1..6) = 3.5.

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Another concept of infinity arizes when you start counting the elements of a set.

The number of elements of a set A is called its cardinality and is written as card(A) or |A|. The "number of elements" formulation works with finite sets, but not with infinite sets.

Therefore, more precisely, the cardinalities of two sets A and B are considered equal if and only if there exists a bijection f:A->B.

f:A->B is a bijection between A and B if:

- for all b in B there is an a in A such that f(a) = b.

- for all a1, a2 in A with a1<>a2, f(a1)<>f(a2).

This definition of cardinality defines an equivalence relation between sets and it causes sets to be classified as belonging to a class of sets with equal cardinality.

There is also a natural order in the cardinalities.

card(A) <= card(B) iff there is a subset C of B and card(A) = card(C)

card(A) < card(B) iff card(A) <= card(B) and not card(B) <= card(A)

For finite sets, the cardinality is a natural number.

An infinite set is a set with a cardinality larger than any natural number, or in other words, for each n>0 it has a subset A with card(A) = n.

For infinite sets interesting things happen.

Consider the powerset of a set A as the set of all subsets of A, written as P(A).

Theorem (Cantor): card(A) < card (P(A)) where P(A) is the powerset of A.

Hence A, P(A), P(P(A)), P(P(P(A))) etc. all have different cardinalities, ALSO for infinite sets A.

The cardinality of a set is also called a cardinal.

There are therefore an infinite number of infinite cardinals.

The set of infinite cardinals has a cardinality bigger than any of its elements! (it is a paradoxical notion).

There are many more surprising results about infinite cardinals.

The answer to your question could be:

If there is infinite, then there is no largest cardinal.

(Google "large cardinals", "large cardinal arithmetic")