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 

Torgny Tholerus

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