I was very interested in the problem of extending to the reals (that is actual the "incomplete" predecessor function over the naturals)

I've asked the same question on MSE but it was a bit ignored...I hope because it is trivial!

http://math.stackexchange.com/questions/...its-fracti

The question is about extensions of to the reals with some conditions

I just noticed that I've made a lot of errors in my MathSE question, I'll fix it in this post (and later on mathSE)

From successor and inverse successor we can define the subtraction in this way

and

with , that is a modified predecessor function we could define its iteration using an "esotic subtraction" that is "incomplete" for naturals and is "complete" for reals (like we are cutting all the negative integers)

In this way we have

How we can go in order to extend to real ?

For example what can we know about ?

for example

if we put

then if

if is not a natural number

!?? what is going on here?

If is natural

in this case we should have that .

What do you think about this?

I've asked the same question on MSE but it was a bit ignored...I hope because it is trivial!

http://math.stackexchange.com/questions/...its-fracti

The question is about extensions of to the reals with some conditions

Quote:A- only if

B- is not discontinuous

I just noticed that I've made a lot of errors in my MathSE question, I'll fix it in this post (and later on mathSE)

From successor and inverse successor we can define the subtraction in this way

and

with , that is a modified predecessor function we could define its iteration using an "esotic subtraction" that is "incomplete" for naturals and is "complete" for reals (like we are cutting all the negative integers)

Quote:

In this way we have

Quote: and

How we can go in order to extend to real ?

For example what can we know about ?

for example

if we put

then if

if is not a natural number

!?? what is going on here?

If is natural

in this case we should have that .

What do you think about this?