This book is for (1) students interested in methodology and the foundations of mathematics – e.g. studying physics, engineering, economics, psychology, thus a broad group who use mathematics – and (2) teachers of mathematics who are sympathetic to the idea of bringing set theory and number theory into mathematics education.The book presents:(A) Constructivism with Abstraction, as a methodology of science.(B) Particulars about infinity and number theory, within foundations and set theory.(C) Correction of errors within mathematics on (B), caused by neglect of (A).Other readers are (3) research mathematicians, who would benefit from last correction, but who must mend for that they are not in the prime target groups.Set theory and number theory would be crucial for a better educational programme:(i) They greatly enhance competence and confidence.(ii) They open up the mind to logical structure and calculation also in other subjects.(iii) They are fundamental for learning and teaching themselves.The axiomatic system for set theory ZFC is shown to be inconsistent. Mathematics has been in error since Cantor 1874 because of neglecting above methodology of science.