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?
In an approximated form, consider the 3 dimensional space:
Here, represents an infinitesimal element basis vector, which is nonzero, yet smaller than any real number (which has the basis vector 1). Similarly, 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 and is trivial. Multiplication can be constructed such that , where is an orthogonal basis vector. Furthermore, one can make the choice that . However, the division of z’s with multiple nonzero elements can be trickier. Furthermore, the meaning of is unclear.
Consider this simple first example of division:
One might expect to get , but a denominator still remains. To remove the denominator would require an infinite series:
One may want to find the inverse of :
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 and to be unique, making the algebra non-commutative. This is an intriguing philosophical notion!