Vtyjkyuk

Am I wrong in thinking that saying one infinity is larger than another is slightly disingenuous? My argument being the infinity is actually more dense than 'larger'. Obviously, we are already thinking very abstractly when ascribing a bigger size to the set of all positive irrationals vs the set of all naturals.

Because both are defined as having no end, it is really a matter of having more numbers in-between.

Yes, I am afraid you are wrong in thinking so. Being 'more dense' is not a valid guide to size where infinity is concerned - for example, the rationals are obviously 'more dense' on the line than the integers, but one can specify a 1-to-1 mapping between them, so in that sense they are actually the same size. On the other hand, one CANNOT specify any such mapping between the IRrationals and the integers, so they are of different sizes [the irrationals being larger since one can map the integers INto them].

More broadly: Historically people have appealed to two different principles when judging relative sizes of discrete sets of objects. The first principle is 'If I can pair them off exactly, they are the same size'. The second principle is 'The whole is greater than the part - i.e the size of a set is bigger than the size of any proper subset'. The problem is that, when considering infinite sets, these two principles usually give different answers about relative sizes. It used to be that people cited this fact to conclude that the whole notion of 'size' for infinite sets was just nonsense. It was Cantor's brilliant insight that sense COULD be made of 'size of infinite set' if one simply accepted that the second principle doesn't actually apply.

Infinite sets are quite different from finite sets with respect to "sizes". Everyone knows that if $A$ is a proper subset of a finite set $B$ then $A$ has fewer members than $B$, in that you cannot match the members of $A$, one-to-one, with the members of $B$ without having at least one member of $B$ left out. But we can match the even positive integers with $all$ the positive integers by matching $ 2n$ with $n,$ for each $nin Bbb Z^+.$

We can even map $Bbb Z^+$ onto the set of all $rationals.$ But we cannot map $Bbb Z^+$ onto all the points of the open real interval $(0,1)$, for reasons that are not obvious.

I suggest an introductory book on set theory. The preface in a math text will usually say what the book's intended "audience " is, which can help you to choose..... I recall a short book by Suppes that had no pre-requisites. And for some ideas and some fun, try "Stories About Sets" by Vilenkin.