Thank you very much.  I realized I made some false statements as well.

It seems likely that reliance on (not P ->  Q and not Q) -> P being a 
tautology is the easiest proof of there being no largest natural number.

Brent Meeker wrote:
> 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 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.
> 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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