Re: non standard finite numbers

Hi there,

The basic idea behind non standard numbers is quite simple:

the numbers go like this:

1, 2, 3, ... 10^10 (10,000,000,000), ..., 10^(10^10) (has 10,000,000,000 zeroes)
10^(10^(10^...10^10 times))), ....

then beyond all those you have some numbers that can never be reached in
that way. They are in all other respects the same as the other numbers
and all the same theorems apply - you can add, subtract and multiply
them, they are prime or composite, 3 limit, 5 limit, 7 limit etc
and so forth.

The reciproccals of the non standard finite numbers are the
infinitesimals.

Now, the way Abraham Robinson introduced these was rather complex.
That's because he started with a theory that basically said they
couldn't exist, but he found you could introduce them all the same
using a special kind of construction.

At that time all mathematicians though they were impossible.
He gave them respectability by showing that you can prove all the ordinary
calculus results using them, and get _exactly_ the same results.
To do this he introduced his star transform to go back and forth
between the ordinary numbers and the infinitesimals.

I think someone wrote a kind of introductory calculus
text book (high school, to first year undergraduate level)
using infinitesimals as the basis if I remember correctly,
so there is this interest in presenting non standard analysis
in a simpler way - you don't need to mention the star transform
except perhaps to say in the introduction to the book
that such a technique exists and shows that you
get the same results this way as by the standard method..

However, if you are interested in studying the non standard numbers
on their own terms you don't need to use the star transform at all.
Instead you can use a technique pioneered by Mycielski, or in set theory,
pioneered by Vopenka. Mycielski's one is the easiest to explain.

What you do is to start with the five peano axioms - that there is a
number 1, that every number has a successor, that numbers that have
the same successor are identical, that 1 isn't the successor of any
other number, and mathematical induction, which is the powerful one
- that if something is true for n = 1, and if
its truth for n implies its truth for n+1 for any n, then its
true for all n.

All you do is to add the idea of a feasible number, and two
extra axioms to describe how it works. One is that if n is feasible,
so is n+1. The other is that there exists a number which isn't feasible.

Finally, an important point, you _don't_ add F into the induction
axiom schema - you can't use induction with it.

That gives non standard finite numbers already. They aren't the
Abraham Robinson ones. I prefer to call them seemingly finite
numbers.

You can prove that the resulting system is consistent if
the Peano axioms are consistent, which is the best one
can do as Godel showed that you can never prove consistency
of the Peano axioms.

You can't do much with this predicate F, for instance you can't
yet prove that the product of two feasible numbers is always
feasible, because that would need induction.

So then the idea is to gradually add in more properties for
F, until you get a system powerful enough for what you want.
You can actually add in full induction for F so long as
you do it in a separate axiom and don't combine the two,
and always keep clear whether you are applying induction
on the feasible numbers only, or on the entire set of finite
and seemingly finite numbers.

You can use a similar approach in set theory. I mean
mathematical set theory here - perhaps best thought of
as a study of the notion of membership, and what it
means for one mathematical object to be a member of another
one.

That is also based on induction, again, you have feasible sets,
and two induction axioms, one for the feasible sets,
and one for all the finite sets. Plus, there ae a few
more details to figure out with set theory.

That then in a nutshell is what I was working on.

The relevance for harmonic series etc. is kind of philosophical.
It relates to rational intonation rather than just intonation.

It means that even the golden ratio, say, can be represented
by a ratio, in fact, infinitely many ratios all the same
up to an infinitesimal. So you will have 3 limit versions of
the golden ratio too, 5 limit ones, and so on, depending
on which of the infinitesimally identical values you choose.

Robert

Hi there,

Sorry, of course you also have to add in the axiom
that 1 is feasible, otherwise you might have no feasible
numbers to study!

Robert

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:

> Sorry, of course you also have to add in the axiom
> that 1 is feasible, otherwise you might have no feasible
> numbers to study!

As an algebraist I prefer the ultrafilter construction anyway. :)

Hi Gene,

> As an algebraist I prefer the ultrafilter construction anyway. :)

I like it too - it's a neat thing and beautiful.

There's nothing to stop you doing a similar thing the other way too
- start within the finite / seemingly infinite numbers to construct the
standard finite numbers and the Cantorian ordinals, in fact
Vopenka, who is a model theorist I believe, did some work
on models of ZF in the alternative set theory.

Robert