If you are ultrafinitist then by definition the set N does not exist... (nor any infinite set countably or not).

If you pose the assumption of a biggest number for N, you come to a contradiction because you use the successor operation which cannot admit a biggest number.(because N is well ordered any successor is strictly bigger and the successor operation is always valid *by definition of the operation*) So either the set N does not exists in which case it makes no sense to talk about the biggest number in N, or the set N does indeed exists and it makes no sense to talk about the biggest number in N (while it makes sense to talk about a number which is strictly bigger than any natural number). To come back to the proof by contradiction you gave, the assumption (2) which is that BIGGEST+1 is in N, is completely defined by the mere existence of BIGGEST. If BIGGEST exists and well defined it entails that BIGGEST+1 is not in N (but this invalidate the successor operation and hence the mere existence of N). If BIGGEST in contrary does not exist (as such, means it is not the biggest) then BIGGEST+1 is in N by definition of N. Regards, Quentin 2009/6/4 Torgny Tholerus <tor...@dsv.su.se>: > > 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 > > > > -- All those moments will be lost in time, like tears in rain. --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---