User blog comment:Edwin Shade/The Continuum Hypothesis/@comment-5529393-20171118032719

Intuition can be deceiving. The rationals Q are dense (continuous by your terminology), like the reals, and if if you take the set of rationals between any two numbers, it will be equivalent to the set of rationals between any two other numbers, and so their size will be the same as well. So, you might think that, since Q is like R, and not like Z, then Q will have the same size as R, and be larger than Z. But the opposite is the case.

To see that Z and Q have the same size, note that they both can be "counted"; i.e. both can be bijected with the positive integers Z+. For Z, counting is simple: 0,1,-1,2,-2,3,-3,... . For Q, it's a little more complicated. Every rational number is the ratio of a pair of integers p/q, with q != 0. So if we graph all the integer points (p,q) with q != 0 on the plane, there are ways we can count them; one way is to "spiral out" from the origin, counting as you go. Some pairs of integers represent the same rational number, for instance 1/2 = 2/4. No problem, just skip over repeats. So we can count the rationals as well, and Q and Z have the same cardinality as Z+. But, from Cantor's proof we know that R has a larger cardinality than Z+, so R has a larger cardinality than Q.

There are other sets whose cardinality might surprise you. For example, if we take the set of reals between 0 and 1, remove the middle third (all the points between 1/3 and 2/3), remove the middle thirds of the remaining two parts (all the points between 1/9 and 2/9, and between 7/9 and 8/9), and so on up to infinity, what remains is a completely disconnected set of points. So this set is not "continuous" at all... but it has the same cardinality as the reals.

I think it would help you to learn what constitutes a proof and what doesn't. Simply appealing to intuition does not constitute a proof; it's not just "not a formal proof". Good luck on learning more about the subject though.