Abstract
Here, I question the assumption in set theory that the size of an infinite set is the same as the size of an infinite subset derived from it as well as the validity of using this result in physics. Consider a thought experiment in which one wants to study the properties of the single set of the positive integers within the framework of that single set system, and one wants to compare the total number of positive integers in the set to the total number of even integers. The traditional experimental processing method extracts the even integers, puts them into a separate subset and pairs off the subset's and set's members one-to-one with a function. After finding no elements left over, the original set and the subset extracted from it are said to be of the same size. However, this method dramatically alters the original single set system and ignores the inherent relationships of that system in which every even integer is always accompanied by an adjacent odd integer. When one takes this relationship into account, one finds that there are twice as many positive integers as even integers within the single set system. The traditional processing method produces an experimental artifact. This should be unacceptable in a well done experiment. While this may be acceptable in the abstract field of mathematics, one cannot simply study a subset of a real, physical system in isolation without considering the inherent relationships between the subset and the rest of the system. And yet, the mathematics of infinities is based on doing just that and is used extensively in physics. I suggest that physics should require the use of proper experimental technique, especially in its logical foundations.
Roger Granet