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time to the mathematical treatment of physics, most notably general relativity which has since been axiomatised 5 . Similar attempts have been made with the standard model, though this is far from complete. Other physical theories, with a somewhat longer history have been broken down to this, most basic level, below I will discuss both mechanics and probability calculus as well as their applications in more advanced physics. Probability calculus is again worth defining, as the calculation of the probability of an event. This study that lies on the border between mathematics and physics can be expressed as just three, familiar, axioms: 0 ܲሺ ܧ ሻ 1 ܲሺܵሻ ൌ 1 ܲሺ ܧ ଵ ܧ ଶ … ሻ ൌ ܲሺ ܧ ሻ ∞ ୀଵ , ݂݅ ܧ ଵ ܧ , ଶ … ܽ ݎ ݁ ݉ ݑݐݑ ݈݈ܽ ݕ ݁ ݔ ݈ܿ ݏݑ ݅ ݒ ݁ These may seem insufficient, to prove all that you have learnt about probability calculus, however, it has been shown to be consistent and independent, and has managed to prove all statements currently known to be true, although its completeness cannot be proved in accordance with Gödel’s incompleteness theorems. Furthermore, to demonstrate how powerful these seemingly inert statements can be, I will show how they can prove the inclusion, exclusion principle. A/B is deϐined as A ת B ′ visually: ܣ ൌ ሺ /ܣ Bሻ ሺA ת Bሻ , which are mutually exclusive ܤ ൌ ሺ /ܤ Aሻ ሺA ת Bሻ , which are also mutually exclusive ܲሺ ܣ ሻ ൌ ܲ൫ሺA/Bሻ ሺA ת Bሻ൯ ൌ ܲሺA/Bሻ ܲሺA ת Bሻ – 3rd axiom (if you include an infinite number of empty sets following A and B which all have probability 0, which I have not shown for simplicity) ܲሺ ܤ ሻ ൌ ܲ൫ሺB/Aሻ ሺA ת Bሻ൯ ൌ ܲሺB/Aሻ ܲሺA ת Bሻ -3rd axiom ܲሺ ܣ ሻ ܲሺ ܤ ሻ ൌ ܲሺA/Bሻ ܲሺA ת Bሻ ܲሺB/Aሻ ܲሺA ת Bሻ ൌ ܲሺሺA/Bሻ ሺA ת Bሻ ሺB/Aሻሻ ܲሺ ת ܣ Bሻ – 3rd axiom ൌ ܲሺ ܣ ܤ ሻ ܲሺ ܤ ת ܣ ሻ – Which follows from the definition of A/B From here, the inclusion, exclusion principle follows inductively. 6 ■ This system has clearly got a mathematical cleanness, but what about its relevance A/B B/A A ת B
5 An Axiom System for General Relativity Complete with respect to Lorentzian Manifolds 6 http://www.proofwiki.org/wiki/Inclusion-Exclusion_Principle
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