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
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