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For example: In Peano arithmetic, second-order arithmetic and related systems, and indeed in most (not necessarily formal) mathematical treatments of the well-ordering principle, the principle is derived from the principle of mathematical induction, which is itself taken as basic.
Unlike the standard ordering ≤ of the natural numbers, the standard ordering ≤ of the integers is not a well ordering, since, for example, the set of negative integers does not contain a least element. The following binary relation R is an example of well ordering of the integers: x R y if and only if one of the following conditions holds ...
In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are ...
Well-ordering theorem: Every set can be well-ordered. Consequently, every cardinal has an initial ordinal. Zorn's lemma: Every non-empty partially ordered set in which every chain (i.e., totally ordered subset) has an upper bound contains at least one maximal element. Hausdorff maximal principle: Every partially ordered set has a maximal chain ...
This proof used his well-ordering principle "every set can be well-ordered", which he called a "law of thought". [10] 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 ...
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]
In the context of the other Peano axioms, this is not the case, but in the context of other axioms, they are equivalent; [25] specifically, the well-ordering principle implies the induction axiom in the context of the first two above listed axioms and Every natural number is either 0 or n + 1 for some natural number n.
However, if the relation in question is already well-ordered, one can often use transfinite induction without invoking the axiom of choice. [4] For example, many results about Borel sets are proved by transfinite induction on the ordinal rank of the set; these ranks are already well-ordered, so the axiom of choice is not needed to well-order them.