Brian Tenneson skrev: > >> How do you know that there is no biggest number? Have you examined all >> the natural numbers? How do you prove that there is no biggest number? >> >> >> > In my opinion those are excellent questions. I will attempt to answer > them. The intended audience of my answer is everyone, so please forgive > me if I say something you already know. > > Firstly, no one has or can examine all the natural numbers. By that I > mean no human. Maybe there is an omniscient machine (or a "maximally > knowledgeable" in some paraconsistent way) who can examine all numbers > but that is definitely putting the cart before the horse. > > Secondly, the question boils down to a difference in philosophy: > mathematicians would say that it is not necessary to examine all natural > numbers. The mathematician would argue that it suffices to examine all > essential properties of natural numbers, rather than all natural numbers. > > There are a variety of equivalent ways to define a natural number but > the essential features of natural numbers are that > (a) there is an ordering on the set of natural numbers, a well > ordering. To say a set is well ordered entails that every =nonempty= > subset of it has a least element. > (b) the set of natural numbers has a least element (note that it is > customary to either say 0 is this least element or say 1 is this least > element--in some sense it does not matter what the starting point is) > (c) every natural number has a natural number successor. By successor > of a natural number, I mean anything for which the well ordering always > places the successor as larger than the predecessor. > > Then the set of natural numbers, N, is the set containing the least > element (0 or 1) and every successor of the least element, and only > successors of the least element. > > There is nothing wrong with a proof by contradiction but I think a > "forward" proof might just be more convincing. > > Consider the following statement: > Whenever S is a subset of N, S has a largest element if, and only if, > the complement of S has a least element. > > By complement of S, I mean the set of all elements of N that are not > elements of S. > > Before I give a longer argument, would you agree that statement is > true? One can actually be arbitrarily explicit: M is the largest > element of S if, and only if, the successor of M is the least element of > the compliment of S. >

I do not agree that statement is true. Because if you call the Biggest natural number B, then you can describe N as = {1, 2, 3, ..., B}. If you take the complement of N you will get the empty set. This set have no least element, but still N has a biggest element. In your statement you are presupposing that N has no biggest element, and from that axiom you can trivially deduce that there is no biggest element. -- Torgny Tholerus --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---