It is often regarded as the creative source of the infinite symbol "∞", because if someone stands on the surface of a huge Mobius belt and walks along the "road" he can see, he will never stop. But this is an untrue rumor, because "∞" was invented before Mobius belt. ?

Arixtote (384-322 BC), an ancient Greek philosopher, believed that infinity could exist, because finite quantities were infinitely separable, but infinity could not be achieved.

/kloc-in the 20th century, a great Indian mathematician, Bascara, appeared, and his concept was close to that of theory. The symbol of putting 8 horizontally as ∞ to represent infinity was first used in john wallis's paper Arithmetic Infinity (published in 1655).

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There are different definitions of infinity in set theory. German mathematician Cantor proposed that the number of elements (cardinality) corresponding to different infinite sets has different "infinity".

The only way to compare different infinite "sizes" here is to judge whether "one-to-one correspondence" can be established and abandon Euclid's view that "the whole is greater than the parts". For example, integer sets and natural number set have the same infinite cardinality, because they can establish a one-to-one correspondence.

Natural number set is an infinite set with the smallest cardinal number, and its cardinal number is represented by the lower right corner of the Hebrew letter Alef.

It can be proved that the power set of any set (the set formed by all subsets) is greater than the original set. If the original radix is a, the radix of the power set is recorded as (2 to the power of a). This is the so-called Cantor theorem.

For two infinite sets, whether the bijection between them can be established can be used as a criterion to compare their sizes.

To be exact, we use the concept of cardinality to describe a set. For a finite set, its cardinality can be considered as the number of elements, but for an infinite set, cardinality can only be understood in this way (of course, it can also be said that the cardinality of an infinite set is the number of elements, but this number is no longer the meaning in everyday language).

If there is bijection (one-to-one correspondence) between set A and set B, it is considered that their cardinality is the same; If the subsets of A and B are bijective, then the cardinality of A is not greater than that of B, that is, A is injective to B and B is injective to A; When the cardinality of A is not greater than that of B, and the cardinality of A and B are different, A is considered to be less than B. ..

Under the framework of ZFC set theory, any set is well ordered, so the cardinality of two sets is always greater than, less than or equal to one of them, and there will be no incomparable situation. But if axiom of choice is not included, then we can only compare the cardinality of well-ordered sets.

For example, the cardinality corresponding to countable sets such as natural number set, integer sets and even rational number sets is defined as "Alev zero". The uncountable set is said to be larger than the countable set, and the real set is defined as "Alef One" because its cardinality is the same as the power set of natural numbers, which is the zeroth power of Alev's 2.

Since the power set of an infinite set always has a higher cardinality than itself, it can be proved that the number of infinite cardinality is infinite by constructing a series of power sets. However, interestingly, the number of infinite cardinal numbers is more than any cardinal number, so it is an "infinity" bigger than any infinity, and it can't correspond to a cardinal number, otherwise it will produce a form of Cantor's paradox.

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