Do Abstract Mathematical Axioms About Infinite Sets Apply To The Real, Physical Universe?

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Abstract
In mathematics, if one starts with the infinite set of positive integers, P, and want to compare the size of the subset of odd positives, O, with P, this is done by pairing off each odd with a positive, using a function such as P=2O+1. This puts the odds in a one-to-one correspondence with the positives, thereby, showing that the subset of odds and the set of positives are the same size, or have the same cardinality. This counter-intuitive result ignores the “natural” relationship of one odd for every two positives in the sequence of positive integers; however, in the set of axioms that constitute mathematics, it is considered valid. In the physical universe, though, relationships between entities matter. For example, in biochemistry, if you start with an organism and you want to study the heart, you can do this by removing some heart cells from the organism and studying them in isolation in a cell culture system. But, the results are often different than what occurs in the intact organism because studying the cells in culture ignores the relationships in the intact body between the heart cells, the rest of the heart tissue and the rest of the organism. In chemistry, if a copper atom was studied in isolation, it would never be known that copper atoms in bulk can conduct electricity because the atoms share their electrons. In physics, the relationships between inertial reference frames in relativity and observer and observed in quantum physics can't be ignored. Furthermore, infinities cause numerous problems in theoretical physics such as non-renormalizability. What this suggests is that the pairing off method and the mathematics of infinite sets based on it are analogous to a cell culture system or studying a copper atom in isolation if they are used in studying the real, physical universe because they ignore the inherent relationships between entities. In the real, physical world, the natural, or inherent, relationships between entities can't be ignored. Said another way, the set of axioms which constitute abstract mathematics may be similar but not identical to the set of physical axioms by which the real, physical universe runs. This suggests that the results from abstract mathematics about infinities may not apply to or should be modified for use in physics.
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First archival date: 2021-09-27
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