Infinitesimals and Infini-tesa-tesimals

[Phrak]

Infinitesimals and "Infini-tesa-tesimals"

Positive infinitesimals are defined as greater than zero, and less than 1/n, where n is any number 1,2,3... The set of negative infinitesimals is the same, but where negative infinitesimals are less than zero and greater than 1/-n.

Infinities are the reciprocal of infinitesimals.

Call the set of real numbers A₀. Call the set of infinitesimals A_-1. Call the set of infinities A₁.

Now, I'd like to have more sets with elements smaller than A_-1 but greater than zero, and sets with elements larger than A₁ for the positive elements of each set:-

The set of sets being …, A_-2, A_-1, A₀, A₁ , A₂, … .There’s an immediate problem as there isn’t any room left between zero and A_-1 to fit A_-2.

Can this be resolved and a consistent arithmetric with the operators (+,-,*,/) be obtained?

 

[Char. Limit]

Ehehe... why not define the second infinitesimals as "greater than 0, and less than $1/A_n$, where A_n denotes any member of the set $A_1$?

 

[tiny-tim]

Hi Phrak!

If I remember correctly, the infinitesimals are defined as equivalence classes of the set of countable sequences converging to zero …

so for example the class of sequences converging as fast as 1/n³ is smaller than the class of sequences converging as fast as 1/n².

A_-2 would presumably be defined as equivalence classes of the set of countable sequences of members of A_-1 …

that looks to me like the same set.

(but I can't quite see what the definition would be, and I certainly can't prove it )

 

[Hurkyl]

Infinities are the reciprocal of infinitesimals.

That depends on the particular number system. This statement is true for the hyperreals and the projective reals, but not for extended reals, for example.

Call the set of real numbers A₀. Call the set of infinitesimals A_-1. Call the set of infinities A₁.

The real number 0 is infinitessimal, for the record.

Also, for the hyperreal numbers, you are missing many elements -- for example, any element of the form r + e where r is a nonzero real and e is a nonzero infinitessimal.

The set of sets being …, A_-2, A_-1, A₀, A₁ , A₂, … .

To do something like this, you can't let A_-1 be "all infintiessimals". There are several reasonable things you could do...

Can this be resolved and a consistent arithmetric with the operators (+,-,*,/) be obtained?

But it's not clear exactly what properties you want these sets to have.



Consider the set of all rational functions in x. This can be ordered by declaring x to be positive and infinitessimal.

(If that's not clear, this ordering can also be defined as being the lexicographic order on the Laurent series of the rational function)

It's fairly natural to define a collection of sets such as

B_n = all rational functions whose Laurent series has leading term xⁿ​

(this is indexed reverse to how you were indexing your A's)

These sets are precisely the equivalence classes of nonzero rational functions under the equivalence relation

x ~ y iff x/y is neither infinite nor infinitessimal​

In the hyperreals, we could define a similar equivalence relation. The set of equivalence classes is not discrete though -- between any two classes there exists another class.

 

[Phrak]

Ehehe... why not define the second infinitesimals as "greater than 0, and less than $1/A_n$, where A_n denotes any member of the set $A_1$?

Yes, that's one candidate, but it seems to have circular definitions. Using lower case Latin similarly subscripted as elements of the sets A_k and sticking to the positives to avoid lengthy arguments,

then a_-1 is now redefined as greater than zero and greater than any a_-2.

There's also a possible pathology. Define a second set of sets ..., B_-3, B_-2, B_-1, B₀ where

B_-3 = A_-2
B_-1 = A_-1
B₀ = A₀
​

One seems free to add or delete any number of sets of A and still fulfill a recursive definition of A_-n-1 such that a_-n-1>0 and a_-n-1<a_-n, where $n \leq 0$.

 

[Phrak]

Hi Phrak!

If I remember correctly, the infinitesimals are defined as equivalence classes of the set of countable sequences converging to zero …

so for example the class of sequences converging as fast as 1/n³ is smaller than the class of sequences converging as fast as 1/n².

A_-2 would presumably be defined as equivalence classes of the set of countable sequences of members of A_-1 …

that looks to me like the same set.

(but I can't quite see what the definition would be, and I certainly can't prove it )

Hi, Tiny.

There seems to be many different ways to define infinitesimals. I'd thought that I borrowed the notion from the definition of hyperreal numbers, but now I can't find it.

 

[Phrak]

That depends on the particular number system. This statement is true for the hyperreals and the projective reals, but not for extended reals, for example.

The real number 0 is infinitesimal, for the record.

Hmm.. I left that out. The number sets I have in mind are {…, A_-2, A_-1, A₀, A₁ , A₂, …, 0}, where zero is neither real nor infinitesimal but stands alone.

Also, for the hyperreal numbers, you are missing many elements -- for example, any element of the form r + e where r is a nonzero real and e is a nonzero infinitessimal.

Me: "The set of sets being …, A-2, A-1, A0, A1 , A2, … ."

To do something like this, you can't let A_-1 be "all infintiessimals". There are several reasonable things you could do...

[...]

But it's not clear exactly what properties you want these sets to have.

I have in mind:

1) Each set A_j should obey the transfer principle with the real numbers.
2) Expressions such as a₀+a₁ would be irreducible,
3) 1/a_j = a_-j,
4) a_j/a_k = a_j-k
5) a_ja_k = a_j+k
​

My focus has been to get to 4) and 5) with consistency.I don't often get around to mathematics in general so I can't really decode much of the rest you have to say without frequent visits to Wikipedia, where this doesn't always work.

Consider the set of all rational functions in x. This can be ordered by declaring x to be positive and infinitessimal.

(If that's not clear, this ordering can also be defined as being the lexicographic order on the Laurent series of the rational function)

It's fairly natural to define a collection of sets such as

B_n = all rational functions whose Laurent series has leading term xⁿ​

(this is indexed reverse to how you were indexing your A's)

These sets are precisely the equivalence classes of nonzero rational functions under the equivalence relation

x ~ y iff x/y is neither infinite nor infinitessimal​

In the hyperreals, we could define a similar equivalence relation. The set of equivalence classes is not discrete though -- between any two classes there exists another class.

 

[Phrak]

Ehehe... why not define the second infinitesimals as "greater than 0, and less than $1/A_n$, where A_n denotes any member of the set $A_1$?

actually, I'm beginning to like this idea, in view of this:

There's also a possible pathology. Define a second set of sets ..., B_-3, B_-2, B_-1, B₀ where

B_-3 = A_-2
B_-1 = A_-1
B₀ = A₀
​

It seems to imply non-uniqueness over many of these transfinite numbering systems, and if so, a dimensioning function is required that may require fractional subscripts of orders of infinity. However, I imagine that I'm not making any sense to anyone else at this point.

 

[Char. Limit]

actually, I'm beginning to like this idea, in view of this:



It seems to imply non-uniqueness over many of these transfinite numbering systems, and if so, a dimensioning function is required that may require fractional subscripts of orders of infinity. However, I imagine that I'm not making any sense to anyone else at this point.

If you're not making sense to the rest of us, then can you say it in simple Mathematics?

 

[Phrak]

If you're not making sense to the rest of us, then can you say it in simple Mathematics?

I've been considering how to put your question into symbols.

Defining the two sets, above, in better terms:

A = {0,A}, where A = {..., A_-2, A_-1, A₀, A₁, A₂, ...}

B = {0,B}, where B = {..., B_-2, B_-1, B₀, B₁, B₂, ...}

The differential of x, i.e. dx¹ is a first order infinitesimal of x and an element of A_-1. x, itself, is an element of A₀.

All numbers constructed of A are vectors of the form

$$\stackrel{n}{\sum} x_n dx^{-n}$$

Likewise for B:

$$\stackrel{m}{\sum} y_m dy^{-m}$$

One example that makes a connection between elements of the sets B and A is

m = f(n) = n².

For fractional differentials such as dx^3/14,

n = phi/14 and m = phi/5, for example.

The generalization is of course that n and m should have a relation over the field of reals or complex numbers. I hope this makes sense.