Naïve set theory
By Tom Brown
The cardinality of a set would give how many elements it has. An "infinite set" can be characterised as a set being equivalent to a strict subset of itself. In other words when I remove members of the set the number of members stays the same. One has to think about this a bit.
Infinities, countable and beyond
A set that can be put in a one-to-one correspondence with the natural numbers is called countably infinite. This is the same as saying the elements of a set can be written out in a row.
Countable sets can always be sequences and typical examples are the positive numbers, the even numbers, squares or all of the integers. Surprisingly the set of rationals can be written out as in a row and a very clever pattern. Thus fractions are countable and can be exhausted in a sequence. In fact countably many countable sets is still countable.
A rational number is a fraction of integers (the word "ratio") here the definition comes in. We know also that a rational number has a repeating decimal expansion and visa-versa. The irrational numbers are all the remaining reals.
Of exceptional originality in a very fundamental, simple but ingenious proof the "Cantor diagonal argument" shows that the set of real numbers cannot be written out as a sequence.
For any sequence of real numbers a real number is constructed that does not belong to this set, they are more prolific, of a strictly higher degree of infinity and not countable in the sense as is of the integers and rational numbers.
Mathematics and Religion
Cantor's Theorem is just as simple a proof involving power sets and orders of infinity and demonstrates that there are in fact an abundance of infinities and such that no maximum cardinality exists.
A power set P(S ) of a set is the collection of all subsets of a given set S. Cantor proved that the power set's cardinality is always strictly greater than of S. This would mean that there cannot be a largest order of infinity.
Cantor had mental breakdowns and suffered from long periods of severe depression, his life story reminds of Van Gogh.
Very distressed at his discoveries and being a devout Roman Catholic actually corresponded, with the Pope himself, who fortunately approved of and quite liked Cantor's ideas on infinities and their interpretation and implications.
To Cantor this all was of a deeply serious religious nature.
Fundamental considerations and principles
Russel's paradox in a very clever argument in turn says in essence that there does not exist a "set of all sets" and leading to problems and very serious issues.
The axiom of completeness is an assumption that is crucial to all study of real and abstract analysis without it you won't get far. It says that any convergent sequence of real numbers' limit is a real number. The point is that it is a real number, the reals are "complete" and there are different equivalent formulations.
Very briefly the continuum hypothesis says that there is no cardinality strictly between that of the rationals and the continuum of real numbers. It has been shown that the hypothesis as well as its negation are consistent with the normally accepted axioms of the real numbers. One can use either but I think it often is assumed it probably simplifies matters.
Also you have the famous axiom of choice which is inherently extremely controversial.
These ideas appear simple but the discoveries were made only in the late 1800s that is very recently actually. It is a bit more complicated but this is the essence and is sufficient for now. The discoveries have led to profound insights in philosophy, and how we see mathematics, science and reality and the world around us.
The transfinite numbers are in a sense the new irrationalities, they stand or fall with the finite irrational numbers.
Georg Cantor (1845 – 1918)