As The First Generation of Infinite Set Theory is based on present classical infinite theory system, it is bound to be unable to get rid of the confusion of "potential infinite and actual infinite" contents, which will inevitably lead to the "potential infinite and actual infinite" confusion of working ideas and results in the quantitative cognitions and studies to infinite mathematical things in infinite sets. For example, making up freewheeling al kinds of different definitions of “infinite” (more infinite, less infinite; more more infinite, more more more more infinite, more more more more more infinite, more more more more more more infinite, ...) and construct those mathematical things of “the theory of transfinite numbers, double abstraction theory, Power Set Theorem, elements of a set with the property T = {x|x📷x}, the Continuum Hypothesis, ...”. Such kind of mathematical things are inconsistent with the definition of "infinite" and the definition of "set", and no one knows whether they belong to " potential infinite" or " actual infinite". These are the factors leading to the paradoxes. So, in "first generation of infinite set theory", people can use Generation Principle to make many infinite concepts of " transfinite numbers, trans-trans-finite numbers, trans-trans-trans-finite numbers, ... ; and can use Power Set Theorem to make many other infinite concepts of “super infinite, super-super infinite, super-super-super infinite, ...". For hundreds of years, people have been trying so hard to study and fabricate various "infinite concepts", study and fabricate various formal logic, formal languages and "assembly line" operations related to those "contradictory and colorful concepts of infinite".
In the old generations of mathematical analysis, we encountered different concepts of "infinite" -------- potential infinite, actual infinite (non-infinite infinite), more infinite, less infinite, low-order infinite, high-order infinite, more-more infinite, more-more-more infinite, more-more-more-more infinite ... Different "infinite" concepts produce different “infinite related number forms" : "potential infinite" produces " potential infinitesimal number form, potential infinity number form", " actual infinite" produces " actual infinitesimal number form, actual infinity number form" the “low-order infinite" produces " low-order infinitesimal number form, low-order infinity number form", "high-order infinite" produces "high-order infinitesimal number form, high-order infinity number form", .... Here we meet exactly the same situation as that in "first generation of infinite set theory". For more than 2500 years, people have been not paying enough attention to the foundation of "infinite related number forms" and carrying out necessary qualitative cognizing and studies to them but trying very hard to study and make up all kinds of formal logics, formal languages and "assembly line" to deal with those conflicting "infinite related number forms" produced by “different concepts of "infinite". But the reality is that the infinite related paradoxes family members are on the rise.
Is it true that everyone can have his (her) own definition of infinite in our science?
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