Re: Noetherian (Well-founded) Induction

Any further comments and examples from anybody would be welcome.


ma_khan1...@xxxxxxxxx wrote:
Many thanks for your comments. I appreciate your help.


William Elliot wrote:
On Fri, 12 Jan 2007 ma_khan1981@xxxxxxxxx wrote:

Can anyone please check and confirm the following definitions:

See my reply Jan. 11 to "Name for this poset?"

Noetherian Poset: A poset (P,<) is said to be Noetherian if every
non-empty subset of P has a maximal element or equivalently every
ascending chain in P terminates.

Principle of Noetherian (Well-founded) Induction: Let (P,<) be a
Noetherian poset. Let Q be a subset of P with the property: "every y in
P, with y>x, is in Q => x is in Q". Then Q = P.

I shall be very thankful if someone can also give some examples (or
links) of easy proofs by Noetherian induction or any of its equivalent
formulations.

Induction follows directly by several logical contrapositions from the
definition and is little more that an exacting exercise in logical duals.

Well founded order is the order dual or reverse of the unusual Noetherian
order. Thus a set is well ordered iff it's well founded and totally
ordered. Noetherian orders thus are not a generalization of well ordered,
well founded is.

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