Re: Why do we have the natural numbers?

george wrote:

> re
> > > > > > 0 = {}
> > > > > > 1 = {{}}
> > > > > > 2 = {{{}}}
> > > > > > 3 = {{{{}}}}
> > > > > > . . .
> > > > > >
> > > > > > That's easier and better.
> george wrote,
> > > No, it's stupider.
>
> > C-B
> > At least you agree that the way the books do it is stupid.
>
> No, I don't. THIS way is stupid.
> The way the books do it nowadays is smart.
> Von Neumann was very smart.
> His way has BOTH depth[n]=n AND card[n]=n.
> And it also fits into NON-set-theoretic logical
> axiomatizations like PA, where +1 is application of
> a successor function.
>
> Sometimes you have to concede that the books
> really do know what they are talking about.

In the final analysis they are all equivalent. N serves as the
simplest example of an r.e. set, and I prefer the simplest
representation within Set Theory for that.

The primitive in N is x => x+1. The primitive in Set Theory is x =>
{x}.

C-B

.

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