On the infinitesimally small

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R(dx) is |R|, as it is isomorphic to R(x), the field of rational functions, which has a spanning set of size |R| and a linearly independent set of size |R|, making its dimension |R|. The standard definition of dimension in field extension does not apply in this case.
  • #1
vshiro
Consider R, or rather the equivalence class of fields isomorphic to R, endowed with an order type of the same kind. Let us consider a new object, dx, a quantity such that dx > 0, but for all a > 0 in R, dx < a. (We are considering dx, something outside of R, but without the context of a larger universe set.)
Now let add dx to R, and field-ify it. That is, we are considering the infinite field extension R[dx]. Furthermore we inherit the order of R, but extend it to include dx. The way I think about the order type is that, suppose we can sort of magnify R until we see a, b, and a, b have no more elts of R between them (there is no c with a<c<b)... we can't actually do this, but entertain this idea for a moment. Adding dx to R is like fitting another whole copy of R between a and b: we have between them a < a + k*dx < b for all k in R. But then we magnify this new copy of R until we get to a, b and add another copy of R (dx^2) and so on, and do this for all nonexistent pairs a, b satisfying there is no c st a<c<b...
My question is, what is the "dimension" of R[dx]? It is not 2.. The above thought experiment suggests that it is |Z| (aleph-1), but I don't believe that is true. It may well be |R|, but I suspect the answer is much more sinister...

Note that the standard definition of dimension in field extension doesn't apply.
 
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  • #2
What definition of dimension are you wanting to use?
 
  • #3
Now let add dx to R, and field-ify it. That is, we are considering the infinite field extension R[dx]

Do you mean R(dx)? R[dx] is merely a ring in this case, because dx cannot be written as a root of a polynomial in R[x].


My question is, what is the "dimension" of R[dx]?

For R[dx], it's easy. R[dx] is isomorphic to R[x], which is countably infinite dimensional over R; its basis is
{1, x, x^2, x^3, x^4, ... }


R(dx) is isomorphic (ignoring the order) to R(x), the field of rational functions. Finding a vector space basis of R(x) over R is a lot trickier, but we can solve our problem without it:

Consider S = {p(x)/q(x) | p and q are in R[x] and q(x) is not 0}

Clearly S is a spanning set of R(x) (it includes every element of R(x)!), and |S| = |R|, so the dimension of the extension can be no more than |R|.

However, consider T = {1/(x - a) | a in R}

linearly independant, so the dimension of the extension can be no less than |T| = |R|.


So, R(dx) is a field extension of R with dimension |R|


Note that the standard definition of dimension in field extension doesn't apply.

Why?


Hurkyl
 

What is the concept of infinitesimally small?

The concept of infinitesimally small refers to quantities that are incredibly small and approach zero, but are not actually equal to zero. It is a fundamental concept in mathematics and is used to understand the behavior of functions and equations at specific points.

How is the concept of infinitesimally small used in calculus?

In calculus, the concept of infinitesimally small is used to describe the behavior of a function at a specific point. By taking the limit of a function as the input approaches a specific value, we can determine the slope of the function at that point and make predictions about its behavior.

What is the relationship between infinitesimally small and the derivative?

The derivative of a function is defined as the limit of the function as the input approaches zero. This means that the derivative is a measure of the function's behavior at an infinitesimally small scale. In other words, the derivative describes how a function changes at specific points, which is directly related to the concept of infinitesimally small.

Can infinitesimally small quantities be used to represent physical objects?

No, infinitesimally small quantities cannot be used to represent physical objects. While they are used in mathematical and scientific models to describe the behavior of physical systems, they do not have a physical existence. In reality, there is always a smallest unit of measurement, and quantities cannot approach zero indefinitely.

What are some real-world applications of the concept of infinitesimally small?

The concept of infinitesimally small is used in various fields such as physics, engineering, and economics. In physics, it is used to describe the behavior of particles at a quantum level. In engineering, it is used to optimize designs and understand the limits of materials. In economics, it is used to model market trends and make predictions about economic behavior.

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