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In constructive mathematics, the limited principle of omniscience (LPO) and the lesser limited principle of omniscience (LLPO) are axioms that are nonconstructive but are weaker than the full law of the excluded middle. They are used to gauge the amount of nonconstructivity required for an argument, as in constructive reverse mathematics.
Markov's principle (also known as the Leningrad principle [1]), named after Andrey Markov Jr, is a conditional existence statement for which there are many equivalent formulations, as discussed below. The principle is logically valid classically, but not in intuitionistic constructive mathematics. However, many particular instances of it are ...
The main practical use of weak counterexamples is to identify the "hardness" of a problem. For example, the counterexample just shown shows that the quoted statement is "at least as hard to prove" as Goldbach's conjecture. Weak counterexamples of this sort are often related to the limited principle of omniscience.
Likewise, God cannot make a being greater than himself, because he is, by definition, the greatest possible being. God is limited in his actions to his nature. The Bible, in passages such as Hebrews 6:18, says it is "impossible for God to lie". [9] [10] A good example of a modern defender of this line of reasoning is George Mavrodes. [11]
The usual statement of the Sylvester–Gallai theorem is not valid in constructive analysis, as it implies the lesser limited principle of omniscience, a weakened form of the law of excluded middle that is rejected as an axiom of constructive mathematics. Nevertheless, it is possible to formulate a version of the Sylvester–Gallai theorem that ...
California’s plan that provides insurance to homeowners who can’t get private coverage needs $1 billion more to pay out claims related to the Los Angeles wildfires, the state Insurance ...
Only 4% of U.S. pet owners currently insure their pets, often due to persistent myths about how these policies work. Separate fact from fiction around pet policies — and how they can save you a ...
In a package of bills announced Monday, the Dems want to adjust the legal standard to make it easier to sue a campus that has allowed discriminatory harassment to go unchecked as well as ensure ...