Belette asks about the history of the continuum problem. I'm not an expert, and the subject is huge, but there are a couple of interesting books I recommend. One is Paolo Mancusu's Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century, which covers a lot of the history of the 'indivisibles' question in the seventeenth century and before. The other is Ewald's excellent source book From Kant to Hilbert which covers the period in the nineteenth century when a lot of the advances were made, both in the theory of the continuum and in mathematical logic (although the two subjects overlap considerably at this point).
In the fourteenth century and afterwards the main debate was not so much about whether the continuum could be composed of indivisibles (points), but whether indivisibiles could exist at all. Was the continuum composed of indefinitely divisible lines alone, or a mixture of lines and points? Ockham's discussion of the continuum is here in chapter 45 of part I of the Summa, where he argues against the existence of points, lines etc.
On Cantor's contribution, the idea of transfinite number is often mentioned, but I believe Frege predates him with (The Foundations of Arithmetic). Cantor's main contribution was the idea that the number of the reals was different from the number of the natural number. His argument for this, as I commented here, is unusual and remarkable – possibly the most unusual and remarkable thing in all logic and mathematics - in that nothing appears to predate it.