## Set Theory and Logic   ## Thought Experiments About Infinite Sets and Subsets Produce Experimental Artifacts: Relationship to Their Use in Physics

Authors: Roger Granet

Here, the conclusion in set theory that the size of an infinite set is the same as the size of an infinite subset derived from it is questioned.This is done not to try and invalidate any mathematical results because mathematics is an abstract field and does not necessarily have to accurately describe the physical world but in order to prompt the reexamination of the use of this result in physics, which does have to accurately describe the real, physical world and the relationships between its components. The rationale is as follows. First, it is suggested that thought experiments are still experiments and should follow the rules for good experimental technique, which include the need to study a system in a setting as close as possible to the "natural setting" to try and avoid experimental artifacts. Now, starting with the single set of the positive integers, one wants to compare the total number of integers to the total number of even integers within the "natural setting" of the single original set. 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 doing this, no elements are left over, and, therefore, the original set and the subset extracted from it are said to be the same size. However, extracting the evens and putting them into a separate subset dramatically alters the original single set system. This is analogous to a biologist extracting the nucleus from a cell, studying the nucleus and remaining parts of the cell in isolation and assuming that the results obtained are the same as in the original intact cell. They often are not. Does extracting the even integers out into a subset alter the results compared to those that would be obtained in the natural single set system? Yes. In the single set system, the positive integers march lockstep and in- phase with the odd integers from one to infinity, meaning that there is a built-in relationship in this system of one positive integer for every two total integers, which means that there are only one-half as many positive integers as total integers. This is a different result than that obtained after the subset extraction method, which means that the result produced by this method is an experimental artifact. This should be unacceptable in a well done experiment even if it is a thought experiment. It is suggested that this artifact may be related to some of the problems associated with infinities in physics.