To put it shorter:

Can we construct a function defined on all set of hyperreals or subset of them ( or superreals )which would preserve the property:

REAL f( subset of hyperreals) = such that

f(h(x)^h(y)) = f( h(x)) * f(h(y)) for all x, y>=0

Where h(x), h(y) is the subset of hyperreals relevant to the problem and the same for all x, y- what ever x, y, the subset of hyperreals we can use to construct such function does not change.

This requires that it is possible to establich one to one correspondence between subset of hyperreals and real numbers, or, at least, 2 real numbers 1 and 0. In my simple thinking, it requires only 2 hyperreals besides each real to do that- as indicated in previous post.

I wonder also how to extend this to hyper imaginaries as there is not reason they can not exist, so they must exist.

If true, such function would perform the necessary transformation

required from 3_log.

If so , that would allow simple generalization to n_log, (n,m)_log by just adding more scales of numbers , like 2_hyperreals which are formed by 2 neighbouring hyperreals , between them,

3_hyperreals formed from 2 neighbouring 2_hyperreals etc.

The nice thing is, surreal numbers of Conway are constructed exactly in such way, so that field has been already covered very deeply, with all ordinalities , cardinalities calculated and proved.

Ivars

Can we construct a function defined on all set of hyperreals or subset of them ( or superreals )which would preserve the property:

REAL f( subset of hyperreals) = such that

f(h(x)^h(y)) = f( h(x)) * f(h(y)) for all x, y>=0

Where h(x), h(y) is the subset of hyperreals relevant to the problem and the same for all x, y- what ever x, y, the subset of hyperreals we can use to construct such function does not change.

This requires that it is possible to establich one to one correspondence between subset of hyperreals and real numbers, or, at least, 2 real numbers 1 and 0. In my simple thinking, it requires only 2 hyperreals besides each real to do that- as indicated in previous post.

I wonder also how to extend this to hyper imaginaries as there is not reason they can not exist, so they must exist.

If true, such function would perform the necessary transformation

required from 3_log.

If so , that would allow simple generalization to n_log, (n,m)_log by just adding more scales of numbers , like 2_hyperreals which are formed by 2 neighbouring hyperreals , between them,

3_hyperreals formed from 2 neighbouring 2_hyperreals etc.

The nice thing is, surreal numbers of Conway are constructed exactly in such way, so that field has been already covered very deeply, with all ordinalities , cardinalities calculated and proved.

Ivars