The issue entered a new phase when Georg Cantor in 1874 introduced what is now called naive set theory and used it as a base for his work on transfinite numbers.
One position was the intuitionistic mathematics that was advocated by L. E. J. Brouwer, which rejected the existence of infinite objects until they are constructed.
Hilbert's goal of proving the consistency and completeness of set theory or even arithmetic through finitistic means turned out to be an impossible task due to Kurt Gödel's incompleteness theorems.
Historically, the written history of mathematics was thus classically finitist until Cantor created the hierarchy of transfinite cardinals at the end of the 19th century.
Leopold Kronecker remained a strident opponent to Cantor's set theory:[2] Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.
Aristotle especially promoted the potential infinity as a middle option between strict finitism and actual infinity (the latter being an actualization of something never-ending in nature, in contrast with the Cantorist actual infinity consisting of the transfinite cardinal and ordinal numbers, which have nothing to do with the things in nature): But on the other hand to suppose that the infinite does not exist in any way leads obviously to many impossible consequences: there will be a beginning and end of time, a magnitude will not be divisible into magnitudes, number will not be infinite.
Towards the end of the 20th century John Penn Mayberry developed a system of finitary mathematics which he called "Euclidean Arithmetic".