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.


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.

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