Postulated conceptions in regards to non-standard mathematical systems

The prior blog post gives rise to the idea of a whole new scaffolding, new building blocks, for different mathematical systems, and I hope to now elucidate a few possible conceptions as to how this is done. The following systems will be described briefly and synopsized in the following paragraph, for each a full set of axioms should be developed and rigorously analyzed (until a devastating paradox is encountered). Many of the concepts may overlap from one system to another:

The first idea and maybe the simplest concept would be the idea of a dual identity systems. That is where both 1 and 0 are completely interchangeable. Immediately problems arise when it is looked at in the set of all real numbers. So this concept is easiest to accept within the set all positive Integers. This ostensibly creates a mathematical system in which the null set is non existent, because at any time you come up with zero you can also come up with one. Through the lack of the empty set we find that problematic decimals such as 0/0 x/0 and 0/x

In order to make the first idea somewhat more powerful and useful a secondary duality is necessary, certain means of converting number systems. For the interchangeability of 0 and 1 to remain coherent it must be intrinsically connected to another set for which this same duality does not exist. This gives us the ability to move freely between two sets, however since the duality only exists in the set of positive integers if we are to move freely between that system and say the system of all integers (positive and negative) we will have to take the absolute value of said integer and save the sign for until we return back to that system.

This leads to another, broader duality; the notion of moving between a constrained system and an unconstrained system. A constrained system is any non-infinite sum of numbers which can be mapped onto a conjoined system which ranges in the full spectrum of infinity. The constrained system is by nature of it's finite quality can never have a concept of the infinite and the infinite system is aware of the constrained system insofar as it conceives of it as infinitely small part (regardless of how large) of an infinite whole. Though the constrained system will never be able to acknowledge the existence of the infinite system which surrounds it, any equation which exceeds the boundaries of that finite system will allude to the greater whole.

Of course these are all very simplistic prefaces and many of these ideas require further development and a structure of axioms, proof, and logical continuities