> From what you said earlier, BIGGEST={0,1,...,BIGGEST-1}. Then
> BIGGEST+1={0,1,...,BIGGEST-1} union {BIGGEST} = {0,1,...,BIGGEST}.
> Why would {0,1,...BIGGEST} not be a natural number while
> {0,1,...,BIGGEST-1} is?
If {0, 1, ... , BIGGEST-1} is a natural number, then {0,1,...,BIGGEST} is
too, and then so is {0, 1, ... , BIGGEST+1}, etc. There's no such thing as
a largest natural number: that's the whole point of the construction. The
set of all natural numbers is an infinite set, unbounded above. The set N
has no largest element within it: it is the set of all finite ordinals. N
(usually called omega when treated as an ordinal) has no predecessor,
because it is formed by taking the limit of all the ordinals below it, *not*
by applying the successor function "x+ = x U {x}". This is the way
well-ordering works...it's not symmetric. So any set described {a, b, ... ,
z} in the standard way is not N.
N is not the successor of any natural number; rather, it contains them all.
This allows us to talk about (and prove things about) all natural numbers.
This isn't an arbitrary mathematical choice. Without infinite sets, we
would be unable to rigorously prove things by induction, which is necessary
for a wide array of basic arithmetical proofs. This is because a finite set
of natural numbers cannot be closed under successor (or addition or
multiplication, for that matter). If you relied on only finitely many
numbers, your functions could take natural numbers and hand you back
something that isn't a number at all. This makes even basic math untenable.
Taking the closure of {} under successor is the solution.
(There are non-standard models of the natural numbers that contain numbers
other than the elements of N, but these are not well-ordered.)
Anna
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