Nonconstructive proof in the context of Axiom of choice

⭐ Core Definition: Nonconstructive proof

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. Constructivism also induces a different meaning for some terminology (for example, the term "or" has a stronger meaning in constructive mathematics than in classical).

↓ Menu

HINT:

In this Dossier

⭐ Core Definition: Nonconstructive proof
Nonconstructive proof in the context of Proof by contradiction
Nonconstructive proof in the context of Proof mining

Nonconstructive proof in the context of Proof by contradiction

In logic, reductio ad absurdum (Latin for "reduction to absurdity"), also known as argumentum ad absurdum, (Latin for "argument to absurdity") apagogical argument, or proof by contradiction is the form of argument that attempts to establish a claim by showing that following the logic of a proposition or argument would lead to absurdity or contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof.

This argument form traces back to Ancient Greek philosophy and has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate. In mathematics, the technique is called proof by contradiction. In formal logic, this technique is captured by an axiom for "reductio ad absurdum", normally given the abbreviation RAA, which is expressible in propositional logic. This axiom is the introduction rule for negation (see negation introduction).

View the full Wikipedia page for Proof by contradiction

↑ Return to Menu

Nonconstructive proof in the context of Proof mining

In proof theory, a branch of mathematical logic, proof mining (or proof unwinding) is a research program that studies or analyzes formalized proofs, especially in analysis, to obtain explicit bounds, ranges or rates of convergence from proofs that, when expressed in natural language, appear to be nonconstructive.This research has led to improved results in analysis obtained from the analysis of classical proofs.

View the full Wikipedia page for Proof mining

↑ Return to Menu

Nonconstructive proof in the context of Axiom of choice

Nonconstructive proof Study page number 1 of 1

Play TriviaQuestions Online!

Skip to study material about Nonconstructive proof in the context of "Axiom of choice"

⭐ Core Definition: Nonconstructive proof

Nonconstructive proof in the context of Proof by contradiction

Nonconstructive proof in the context of Proof mining