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Then, by the well-ordering principle, there is a least element ; cannot be prime since a prime number itself is considered a length-one product of primes. By the definition of non-prime numbers, n {\displaystyle n} has factors a , b {\displaystyle a,b} , where a , b {\displaystyle a,b} are integers greater than one and less than n ...
In second-order logic, however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering theorem one may deduce the axiom of choice, but from the axiom of choice one cannot deduce the well-ordering theorem. [7] There is a well-known joke about the three statements, and their relative amenability to intuition:
In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total ordering on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the ordering is then called a well-ordered set (or woset). [1]
It can then be proved that induction, given the above-listed axioms, implies the well-ordering principle. The following proof uses complete induction and the first and fourth axioms. Proof. Suppose there exists a non-empty set, S, of natural numbers that has no least element. Let P(n) be the assertion that n is not in S.
The most important among them are Zorn's lemma and the well-ordering theorem. In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering theorem. Set theory. Tarski's theorem about choice: For every infinite set A, there is a bijective map between the sets A and A×A.
The well-ordering principle is equivalent to the axiom of choice. [11] Around 1895, Cantor began to regard the well-ordering principle as a theorem and attempted to prove it. [12] In 1895, Cantor also gave a new definition of "greater than" that correctly defines this concept without the aid of his well-ordering principle. [13]
If the order is a total order then it is called a well-order. In set theory, a set x is called a well-founded set if the set membership relation is well-founded on the transitive closure of x. The axiom of regularity, which is one of the axioms of Zermelo–Fraenkel set theory, asserts that all sets are well-founded. A relation R is converse ...
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction [1] used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. [2]