Full Idea
Since between any two rational numbers there is an infinite number of rational numbers, we could consider that we have infinity in three dimensions: positive numbers, negative numbers, and the 'depth' of infinite numbers between any rational numbers.
Gist of Idea
Between any two rational numbers there is an infinite number of rational numbers
Source
Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5)
Book Reference
Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.17
A Reaction
This is before we even reach Cantor's staggering infinities (Ideas 8662 and 8663), which presumably reside at the outer reaches of all three of these dimensions of infinity. The 'deep' infinities come from fractions with huge denominators.
Related Ideas
Idea 8662 The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend]
Idea 8663 Raising omega to successive powers of omega reveal an infinity of infinities [Friend]