I've never seen an ultrafinitist definition of the natural numbers. The usual definition via Peano's axioms obviously rules out there being a largest number. I would suppose that an ultrafinitist definition of the natural numbers would be something like seen in a computer (which is necessarily finite). The successor operation would be defined such that Successor (Biggest) = 0 or -Biggest.

Brent Quentin Anciaux wrote: > 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 >> >> > > > > --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---