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Limited Principle Of Omniscience
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 . The LPO and LLPO axioms are used to gauge the amount of nonconstructivity required for an argument, as in constructive reverse mathematics. They are also related to weak counterexamples in the sense of Brouwer. Definitions The limited principle of omniscience states : :LPO: For any sequence a_0, a_1, ... such that each a_i is either 0 or 1, the following holds: either a_i=0 for all i, or there is a k with a_k=1. The lesser limited principle of omniscience states: :LLPO: For any sequence a_0, a_1, ... such that each a_i is either 0 or 1, and such that at most one a_i is nonzero, the following holds: either a_=0 for all i, or a_=0 for all i, where a_ and a_ are entries with even and odd index respectively. It can be proved constructively that the law of the exc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Mathematics
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation. There are many forms of constructivism. These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of Shanin and Markov, and Bishop's program of constructive analysis. Constructivism also includes the study of constructive set theories such as CZF ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. As used in modern logic, an axiom is a premise or starting point for reasoning. As used in mathematics, the term ''axiom'' is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms (e.g., ) are actually ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of The Excluded Middle
In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, Exclusive or, either this proposition or its negation is Truth value, true. It is one of the so-called Law of thought#The three traditional laws, three laws of thought, along with the law of noncontradiction, and the law of identity. However, no system of logic is built on just these laws, and none of these laws provides Rule of inference, inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law (or principle) of the excluded third, in Latin ''principium tertii exclusi''. Another Latin designation for this law is ''tertium non datur'': "no third [possibility] is given". It is a tautology (logic), tautology. The principle should not be confused with the semantical principle of bivalence, which states that every proposition is either true or false. The principle of bivalence always implies the law of excluded middle, while the converse i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructive Reverse Mathematics
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones. The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory. Reverse mathematics is usually carried out using subsystems of second-order arithmetic,Simpson, Stephen G. (2009), Subsystems of second-order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, doi:10.1017/CBO9780511581007, ISBN 978 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Counterexample
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or ''pure existence theorem''), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof. A constructive proof may also refer to the stronger concept of a proof that is valid in constructive mathematics. Constructivism is a mathematical philosophy that rejects all proof methods that involve the existence of objects that are not explicitly built. This excludes, in particular, the use of the law of the excluded middle, the axiom of infinity, and the axiom of choice, and induces a different meaning for some terminology (for example, the term "or" has a stronger meaning in cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov's Principle
Markov's principle, 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 nevertheless provable in a constructive context as well. History The principle was first studied and adopted by the Russian school of constructivism, together with choice principles and often with a realizability perspective on the notion of mathematical function. In computability theory In the language of computability theory, Markov's principle is a formal expression of the claim that if it is impossible that an algorithm does not terminate, then for some input it does terminate. This is equivalent to the claim that if a set and its complement are both computably enumerable, then the set is decidable. In intuitionistic logic In predicate logic, a predicate ''P'' over some ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |