> 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.

If so, then that statement proves that there is no largest element of N:
Letting S be N in particular, note that N is a subset of N (albeit not a 
"proper" subset).
Then the statement reads as the following for this particular choice S:
N has a largest element if, and only if, the complement of N has a least 

The compliment of N is the empty set.  To elaborate: the compliment of N 
is the set of all elements of N that are not elements of N.  No elements 
can both be and not be elements of N, so this set is empty.

The empty set does not have a least element.  In fact, it has no 
elements at all.

Therefore, N does not have a largest element.

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