Or the Intensive and the Extensive Infinite „In order to know the nature of the numbers according to which „geometrical distance” to be defined, for example, it would be necessary to know what happens both at indefinitely tiny and indefinitely large distance. Even today, these questions are without clearcut resolution.” (Roger Penrose, The Road to Reality) Abstract Today's mathematics is based on the continuum hypothesis (CH) about infinitesimals. Although Cantor formulated CH for extensive infinity, it can also be reformulated for intensive infinity. Thus, we have the foundation of a richer set theory, including classical mathematical analysis, in which the two-element numbers can now model the potential infinities corresponding to our experience, and the actual forms of infinities are captured…

Gaps in Solutions of Polynomial Equations „Mathematics is not settled. Even concerning the basic objects of study, like numbers and geometric figures, our ignorance is much greater then our knowledge” (Jordan Ellenberg)1 The consequences of the modification of the mathematical notion of infinity are manifold, and I have already mentioned several of them, though I have not explained them all in detail. An important property of this notion of infinity is that it is closely related to Cantor's continuum hypothesis (CH) and with it to the axiomatic foundation of set theory, since with a quantified version of CH and its alternatives2 we can "produce" three different set theories, similar to the way we arrive at different geometries in geometry depending…

Musings „From time immemorial, the infinite has stirred men's emotions more than any other question. Hardly any other idea has stimulated the mind so fruitfully. Yet, no other concept needs clarification more than it does.” (David Hilbert)1 Contents Introduction Two-element numbers and the qualitative nature of infinity Two-element numbers and the concepts of counting and measurement Conclusions Appendix – Elementary properties of two-element numbers ____________________________ 1 David Hilbert, „On the infinite”, https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Philosophy/Philosophy.html  After registration, you may ask for a translation of the full text by email.