Help on infinitesimal calculation

[seratend]

Hi everybody,

I am trying to get addtionnal data on "infinitesimal numbers" dx. I am not sure about the terminology, I have heard it a long ... long time ago during a lecture (my memory may be wrong, so may be I was sleepind and it was during a dream? ).
I think (memory) that the definition may be something like "whatever e>o, 0<|dx|<e" where dx is the "object" satisfying such definition (i.e. the "infinitesimal number" definition may be the "smallest" open set in a topology?).

Does anyone can provide some information about this?
Thanks a lot in advance.

Seratend.

 

[matt grime]

Non-standard or Robinsonian analysis. Google is your friend. (Your topology thing isn't correct - (dx)^2 is even smaller than dx.)

 

[seratend]

Non-standard or Robinsonian analysis. Google is your friend. (Your topology thing isn't correct - (dx)^2 is even smaller than dx.)

Thanks for this initial input. I will try to get some information. Googling almost always displays the classical differential analysis topics and that is not what I am looking for.
I have an initial look, and I am surprised; I didn't think in going into the hyper real numbers space !

Your topology thing isn't correct - (dx)^2 is even smaller than dx.)

Well, I have already underline that my definition may be not consistent or correct at all (It is all what I can remember of, and my memory is poor).
However, assuming the usual topology on |R, dx in this definition may be viewed as a set rather than a number. In that case, It can be rewriten as the intersection of all the open sets containing x for example. In addtion, I do not assume that this "limit" is unique.

Seratend.

 

[matt grime]

Your topological interpretation really isn't correct. The intersection of all open sets that contain x is the (closed) one point set that contains x. The infinitesimals are basically the space of (laurent) series in one variable with coeffs in R with the natural lexicographical ordering.

 

[seratend]

Your topological interpretation really isn't correct. The intersection of all open sets that contain x is the (closed) one point set that contains x.

yes, evident for the limit! (I should have deduced it before posting it , it is a shame ). So I need to change the definition is case of the topology context.

The infinitesimals are basically the space of (laurent) series in one variable with coeffs in R with the natural lexicographical ordering.

So do you mean (I am afraid with the "basically") that I have a bijection between the points of the vectorial space (complete space? if there is a norm and what norm?) of the Laurent series and the infinitesimals that comply with the definition (e>o 0<|dx|<e, for any inifinitesimal dx )?
(your lexicographic ordering is first i ordering of zi (i belongs to Z) then the ordering of coefficient ai? - that reminds me something about the ordering pb of the infinitesimals).

Thanks in advance, It helps me a lot.

Seratend.

Now, rediscovering the laurent series.

 

[matt grime]

I never mentioned anything about convergence or topology or norms - the series are the formal power series (in dx) (with only finitely many non-zero coeffs) so a typical element looks something like:

a+ b(dx) +c(dx)^2

and we know dx<r for all real r

c(dx)^2<b(dx) for all real c and b.

but this is a nonstandard way of thinking about nonstandard analysis (something that I know less than zip about as is rather clear from my posts here) and was just an attempt to describe the underlying set of the surreal numbers, and approximately how they are ordered.

 

[seratend]

I never mentioned anything about convergence or topology or norms - the series are the formal power series (in dx) (with only finitely many non-zero coeffs) so a typical element looks something like:

a+ b(dx) +c(dx)^2

and we know dx<r for all real r

c(dx)^2<b(dx) for all real c and b.

but this is a nonstandard way of thinking about nonstandard analysis (something that I know less than zip about as is rather clear from my posts here) and was just an attempt to describe the underlying set of the surreal numbers, and approximately how they are ordered.

I begin to understand the topic, even if I still not have found a good/simple paper on it. I have only speaken about topology and norms as a possible mapping to introduce this object (through a kind of bijection). However, this is not important, as I have seen, thanks to your initial input, that there are several possibilities to introduce these amazing objects (in order to keep a set).

Ok, you were speaking about the Laurent series as one possible representation of the hyper real numbers (i.e. an hyper real number represented by a countable set of real numbers may be represented by a generic Laurent serie – currently I don’t know if it is a single or a class of series).
I also begin to understand that this extension of the |R set plays with the boundaries of the zermello-fraenkel set theory axioms (we are near the set of all sets).

So I think I can say that my initial definition is somewhat correct (e>0, 0<|dx|<e where e is a real number, < is the ordering relation that is the usual order relation for real numbers). Thus, I have no solution if I consider that dx is a real number, but if I take a "larger" set, well constructed in order to get a set (ZF axioms), I can have a solution (in fact I understand an infinite, non countable number of solutions).

Do you know some interesting properties about the topology and the norm we can build on this new set (as an extension of the usual topology and norm on the |R set)? (I am still looking for a good and compact lecture about this topic).

Seratend.

Trying to swim across the hyper real number space .