Thank you very much. I realized I made some false statements as well. It seems likely that reliance on (not P -> Q and not Q) -> P being a tautology is the easiest proof of there being no largest natural number.

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Brent Meeker wrote: > Brian Tenneson wrote: > >> >> >>> 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. >> >> > > Let S={even numbers} the complement of S, ~S={odd numbers} ~S has a > least element, 1. Therefore there is a largest even number. > > Brent > > > > > > --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---