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 00 is this least element or say 01 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
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Received on Wed Jun 03 2009 - 11:02:34 PDT