# Re: The seven step-Mathematical preliminaries

```Jesse Mazer skrev:
>
>
> > Date: Fri, 5 Jun 2009 08:33:47 +0200
> > From: tor...@dsv.su.se
> > Subject: Re: The seven step-Mathematical preliminaries
> >
> >
> > Brian Tenneson skrev:
> >>
> >> How can BIGGEST+1 be a natural number but not belong to the set of all
> >> natural numbers?
> >
> > One way to represent natural number as sets is:
> >
> > 0 = {}
> > 1 = {0} = {{}}
> > 2 = {0, 1} = 1 union {1} = {{}, {{}}}
> > 3 = {0, 1, 2} = 2 union {2} = ...
> > . . .
> > n+1 = {0, 1, 2, ..., n} = n union {n}
> > . . .
> >
> > Here you can then define that a is less then b if and only if a belongs
> > to b.
> >
> > With this notation you get the set N of all natural numbers as {0,
> 1, 2,
> > ...}. But the remarkable thing is that N is exactly the same as
> > BIGGEST+1. BIGGEST+1 is a set with the same structure as all the other
> > natural numbers, so it is then a natural number. But BIGGEST+1 is not a
> > member of N, the set of all natural numbers.
>
> Here you're just contradicting yourself. If you say BIGGEST+1 "is then
> a natural number", that just proves that the set N was not in fact the
> set "of all natural numbers". The alternative would be to say
> BIGGEST+1 is *not* a natural number, but then you need to provide a
> definition of "natural number" that would explain why this is the case.```
```
It depends upon how you define "natural number".  If you define it by: n
is a natural number if and only if n belongs to N, the set of all
natural numbers, then of course BIGGEST+1 is *not* a natural number.  In
that case you have to call BIGGEST+1 something else, maybe "unnatural
number".

--
Torgny Tholerus

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