Abstract
Abstract Recent educational studies in mathematics seek to justify a thesis that there
is a conflict between students intuitions regarding infinity and the standard theory of
infinite numbers. On the contrary, we argue that students intuitions do not match but to
Cantor’s theory, not to any theory of infinity. To this end, we sketch ways of measuring
infinity developed at the turn of the 20th and 21st centuries that provide alternatives
to Cantor’s theory of cardinal and ordinal numbers. Some of them introduce new kinds
of infinite numbers, others simply define new arithmetic for Cantor’s infinite numbers.
With regard to these new approaches, we argue that there are various intuitions of actual
infinity which can find an adequate theory.
We also present pre-Cantorain theories of actual infinity developed within the tradition
of geometrical optics. They corroborate our claim on various intuitions concerning actual
infinity.
