Towards a general mathematical theory of experimental science
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In this article we lay the groundwork for a general mathematical theory of experimental science. The starting point will be the notion of verifiable statements, those assertions that can be shown to be true with an experimental test. We study the algebra of such objects and show how it is closed only under finite conjunction and countable disjunction. With simple constructions, we show that the set of possible cases distinguishable by verifiable statements is equipped with a natural Kolmogorov and second countable topology and a natural σ-algebra. This gives a clear physical meaning to those mathematical structures and provides a strong justification for their use in science. It is our hope and belief that such an approach can be extended to many areas of fundamental physics and will provide a consistent vocabulary across scientific domains.
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