An immediate motivation of Cantor to work on what became set theory was his earlier work on trigonometric series. To solve a problem in that domain he considered the set (a closed set) of zeros of such a function, then the derived set of this set, the derived set of this set and so on. This is all still classical, but then had to go a step beyond that to consider first the intersection of all these sets, and then the derived set of that set and so on.

So he came to consider transfinite ordinals.

This is discussed in various places, including "Set Theory and Uniqueness of Trogonometric Series" by Kechris or "Uniqueness of trigonometric series and descriptive set theory, 1870–1985" by Roger Cooke (Archive for History of Exact Sciences, 1993)

The original paper is (I think) "Ueber die Ausdehnung eines Satzes ais der Theorie der trigonometrischen Reihen (Math. Annalen, 1872)"

Another motivation was his earlier work on number theory. Using what is now called a diagonalisation argument he was able to prove results on the existence of transcendental numbers. This is in his 1874 paper "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers")

In brief, the original motivation was to have better tools for making progress on existing problems.

An immediate motivation of Cantor to work on what became set theory was his earlier work on trigonometric series. To solve a problem in that domain he considered the set (a closed set) of zeros of such a function, then the derived set of this set, the derived set of this set and so on. This is all still classical, but then had to go a step beyond that to consider first the intersection of all these sets, and then the derived set of that set and so on.

So he came to consider transfinite ordinals.

This is discussed in various places, including "Set Theory and Uniqueness of Trogonometric Series" by Kechris or "Uniqueness of trigonometric series and descriptive set theory, 1870–1985" by Roger Cooke (Archive for History of Exact Sciences, 1993)

The original paper is (I think) "Ueber die Ausdehnung eines Satzes ais der Theorie der trigonometrischen Reihen (Math. Annalen, 1872)"

Another motivation was his earlier work on number theory. Using what is now called a diagonalisation argument he was able to prove results on the existence of transcendental numbers. This is in his 1874 paper "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers")

In brief, the original motivation was to have better tools for making progress on existing problems.

An immediate motivation of Cantor to work on what became set theory was his earlier work on trigonometric series. To solve a problem in that domain he considered the set (a closed set) of zeros of such a function, then the derived set of this set, the derived set of this set and so on. This is all still classical, but then had to go a step beyond that to consider first the intersection of all these sets, and then the derived set of that set and so on.

So he came to consider transfinite ordinals.

Another motivation was his earlier work on number theory. Using what is now called a diagonalisation argument he was able to prove results on the existence of transcendental numbers. This is in his 1874 paper "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers")

In brief, the original motivation was to have better tools for making progress on existing problems.

An immediate motivation of Cantor to work on what became set theory was his earlier work on trigonometric series. To solve a problem in that domain he considered the set (a closed set) of zeros of such a function, then the derived set of this set, the derived set of this set and so on. This is all still classical, but then had to go a step beyond that to consider first the intersection of all these sets, and then the derived set of that set and so on.

So he came to consider transfinite ordinals.

This is discussed in various places, including "Set Theory and Uniqueness of Trogonometric Series" by Kechris or "Uniqueness of trigonometric series and descriptive set theory, 1870–1985" by Roger Cooke (Archive for History of Exact Sciences, 1993)

The original paper is (I think) "Ueber die Ausdehnung eines Satzes ais der Theorie der trigonometrischen Reihen (Math. Annalen, 1872)"

Another motivation was his earlier work on number theory. Using what is now called a diagonalisation argument he was able to prove results on the existence of transcendental numbers. This is in his 1874 paper "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers")

In brief, the original motivation was to have better tools for making progress on existing problems.