Lawrence and Georgina,
Continuity vs. discreteness is an important issue in physics. Common sense distinguishes between what is considered as a number of countable indivisible items and what is modeled as an endlessly divisible and therefore uncountable liquid. Peirce defined a continuum as something every part of which has parts. Euclid called a point something that does not have parts. Accordingly, genuine continua like a volume, an area, or a line can just be approximated by a finite number of points, no matter how densely the points are thought to be arranged. Any finite set of points is zero-dimensional.
Do the natural numbers constitute a set? If we clarify that being infinite is a property, then we have to attribute this property to the rule of counting. The natural numbers are rather something to be set as large as desired. There is no largest number. The natural numbers are countable without restriction while the expression infinity means all of them and does not denote a quantum but something else, something uncountable.
From the possibility of bijection between 1, 2, 3, ... and 1, 4, 9, ... Galilei correctly concluded that the relations smaller, equal, or larger are not valid for (the entities of) infinite quantities but only for finite ones.
Georg Cantor started with the silly idea to count all numbers. When he used bijection to redefine countability and declared the natural numbers countably infinite, he did not accept that while any natural number is countable there is no countable entity of all natural numbers. Nonetheless, he correctly found out that every rational number is also countable.
In order to count uncountable numbers too, he introduced the most cardinal mistake into mathematics: Cardinality. Instead to simply distinguish discrete and accordingly countable numbers from uncountable real numbers, he draw the wrong conclusion that there must be more real numbers than natural ones. Common sense tells us that this does not work with the original meaning of being absolutely infinite. Cantor, whose mother was Catholic, tried in vain to convince cardinal Franzelin that there is an Infinitum creatum sive Transfinitum below the Infinitum aeternum increativum sive Absolutum. Up to now, the notion infinity is murky in set theory. In a letter to Dedekind, Cantor himself excluded what he called "absolut unendliche" alias "inkonsistente Vielheiten".
Dedekind's cut postulates - without any possibility for a proof - nothing else than trichotomy for the real numbers too: "... every point of one piece is located left from every point of the other one."
This is clearly at odds with the impossibility to resolve a line into a finite number of points. How differ so called real numbers from the rational ones? How can they fill the gaps between any two rational numbers?
Let me tell it as simply as possible: Rational numbers - irrational ones as well as incorporated rational ones - must be irreal in the sense that they are not countable. This means, they have to be represented like p/q with p as well as q infinite, in other words with infinite precision. Otherwise, for instance pi would differ from exact value. This means, even the tiniest difference between two rational numbers must be filled with a continuum which is not yet adequately described by the hugest number of points. Continuity is a different quality.
Imagine a letter with an uncountably lengthy post code. It will never arrive. While the difference between subsequent rational numbers can be made as small as you like, the difference between subsequent real numbers must be absolutely zero.
Of course, only approximations to such really real numbers can be numerically handled. On the other hand, several apparent problems including Schroedinger's cat vanish when we do not forget that mathematics is always treating the reals as if they were rationals. Weyl called them aptly a sauce.
Forget 0)[0 and 0](0 and 0)0(0 and the possibility to choose at will. The correct solution is to understand: For really reals, there is no difference between [ and ( at all.
Goedel questioned the hypothesis c = aleph_1 = 2 ^(aleph_0). He considered c much larger. So far there is no tenable basis for a quantitative comparison between the two qualities.
Regards,
Eckard