Infinitesimal Analysis?

If there exists a non-standard analysis which addresses an entity which is larger than any real number, perhaps there are an infinite number of such entities which are each larger than the previous. Could a proper analysis be developed, which would give new insight on the concept of infinity?

One can construct the hyperreal numbers, which have the real numbers as a subset. Below, I present an infinite dimensional vector space. Is this space an abstraction of the hyperreal numbers?

z = a_{-\infty}\cdot 0 + \dots + a_{-3}\cdot\varepsilon^3 + a_{-2}\cdot\varepsilon^2 + a_{-1}\cdot\varepsilon + a_0\cdot 1 + a_1\cdot\Lambda + a_2\cdot\Lambda^2 + a_3\cdot\Lambda^3 + \dots + a_\infty \cdot\infty

In an approximated form, consider the 3 dimensional space:

z = a\cdot\varepsilon + b\cdot 1 + c\cdot\Lambda

Here, \varepsilon represents an infinitesimal element basis vector, which is nonzero, yet smaller than any real number (which has the basis vector 1). Similarly, \Lambda is larger than any real number, yet not infinite.

Currently, I am trying to develop a self-consistent closed algebraic structure from this infinite dimensional vector space. Coming up with a well defined division operator is a challenge. Is this challenge related to the infinitesimals themselves, or the fact that I have a multidimensional vector space?

Addition and subtraction of two z_1 and z_2 is trivial. Multiplication can be constructed such that \varepsilon\times\varepsilon = \varepsilon^2, where \varepsilon^2 is an orthogonal basis vector. Furthermore, one can make the choice that 1/\Lambda = \varepsilon. However, the division of z’s with multiple nonzero elements can be trickier. Furthermore, the meaning of \sqrt\varepsilon is unclear.

Consider this simple first example of division:

z_1 = 1+\varepsilon

z_2=\frac{1}{z_1} = \frac{1}{1+\varepsilon} = \frac{1-\varepsilon}{1-\varepsilon^2}

One might expect to get 1-\varepsilon, but a denominator still remains. To remove the denominator would require an infinite series:

z_2 = \frac{(1-\varepsilon)(1+\varepsilon^2)}{1-\varepsilon^4} = \frac{(1-\varepsilon)(1+\varepsilon^2)(1+\varepsilon^4)(1+\varepsilon^8)(1+\varepsilon^{16})(1+\varepsilon^{32})}{1-\varepsilon^{64}}

= (1-\varepsilon)(1+\varepsilon^2)(1+\varepsilon^4)(1+\varepsilon^8)(1+\varepsilon^{16})(1+\varepsilon^{32})(1+\varepsilon^{64})(1+\dots

= \sum_{n=0}^\infty (-1)^n\cdot\varepsilon^n

One may want to find the inverse of z_3 = a+b\cdot\varepsilon:

z_4 = \frac{1}{z_3} = \frac{1}{a}\sum_{n=0}^\infty \left(\frac{-b}{a}\cdot\varepsilon\right)^{n}

Next, one may want to find the inverse of z with an arbitrary number of terms. From here, division of two arbitrary z’s would be established.

This example seems to imply that a multiplication and division operator cannot be constructed for the 3D vector space.

Next, I will think about if and how this differs from Georg Cantor’s ordinal numbers. Interestingly enough, he interprets 1 + \Lambda = \Lambda and \Lambda +1 to be unique, making the algebra non-commutative. This is an intriguing philosophical notion!



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