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The idea that math is 'out there' is incompatible with the idea that it consists of formal systems." Tegmark's response [10]: sec VI.A.1 is to offer a new hypothesis "that only Gödel-complete (fully decidable) mathematical structures have physical existence. This drastically shrinks the Level IV multiverse, essentially placing an upper limit ...
In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly axiomatize that hypothesis, i.e. to prove an implication A → B, it is sufficient to assume A as a hypothesis and then proceed to derive B.
In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases ...
In this example, the first premise is a conditional statement in which "P" is the antecedent and "Q" is the consequent. The second premise "affirms" the antecedent. The conclusion, that the consequent must be true, is deductively valid. A mixed hypothetical syllogism has four possible forms, two of which are valid, while the other two are invalid.
In contrast, an indirect proof may begin with certain hypothetical scenarios and then proceed to eliminate the uncertainties in each of these scenarios until an inescapable conclusion is forced. For example, instead of showing directly p ⇒ q , one proves its contrapositive ~ q ⇒ ~ p (one assumes ~ q and shows that it leads to ~ p ).
Together with the axiom of choice (see below), these are the de facto standard axioms for contemporary mathematics or set theory.They can be easily adapted to analogous theories, such as mereology.
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