Why is The number of integers are infinite, and the integers are in infinite set?
Seems like something simple. ive been looking up the cantor diagrams cause i think this proves it. but alas im am unsure how to interpret it.
Why is The number of integers are infinite, and the integers are in infinite set?
Seems like something simple. ive been looking up the cantor diagrams cause i think this proves it. but alas im am unsure how to interpret it.
"Infinite" is defined differently in different situations. In set theory, we usually define "an infinite set" as follows. The set of all natural numbers is the minimal inductive set containing zero. Every member of this set is a natural number. An infinite set is then defined as a set for which there exists no bijection with any natural number.
Although the set of natural numbers is infinite, each of its members is finite. There is a big "gap" between the infinite set and each of its finite members, in a sense.
Conceptually, infinite means "beyond exhaustive", and it captures the idea that "there is no largest natural number", for example. But there are other ways to define it which do not require prior construction of objects. A Dedekind-infinite set is defined as a set for which there is an injection from that set onto a proper subset. For example, since the set of all natural numbers is infinite, we can pair them up so that all numbers are accounted for in one copy but not the other: {(0, 0), (1, 1), (2, 4), (3, 9), (4, 16), ... }. However, this definition has a slight unusual twist. Without AC, it is possible for a Dedekind-finite set to be infinite! This can happen because we can describe infinite sets that are not well-ordered, but for which we have no method to define an auto-injection.