Atypical axioms and a further investigation into the concept of a dual identity system

To expound upon the first idea listed in the last blog entry, a notion of a dual identity system, certain things must be covered in greater detail and a set of axioms must be developed. But before those axioms are divulged the very basis of axioms must first be looked into.

To begin an axiom is supposed to be self evident, something so simplistic it cannot be argued against. Of course of Euclids original 5 axioms of geometry the parallels postulate was proven to be false in many circumstances which made it a less than satisfactory axiom. The other axioms speak of two points creating a straight line and circle defined by a centre and a radius, the simplest of stuff. But of course an axioms simplicity is also intrinsically a part of how familiar we are with the subject matter or maybe more appropriately, the subject matter and it's axioms become familiar over time (due to their role in common society and mathematical systems) and so are taken as self evident and described as the foundational blocks of the system (i.e. the 1 + 1 = 2 or 1 + x = successor (x) axiom of standard arithmetic), but in an abstracted realm things are not always what they seem. A straight line, though it is easy enough to imagine or represent geometrically is only a concept. There can be no truly straight line in reality so a truly straight line is only a function of the abstract, and when looked at thusly what exactly is the straight line going through in the abstract sense? Of course we are all very used to the conventions of straight and so on, and we take with it the pratfalls of human error and the imperfection of all corporeal objects and use the geometry to create perfect formulas for these abstract conventions which we have all grown so familiar with. But that is just the point, we are now so familiar with these axioms that there is nothing that can lay a shadow of a doubt into their validity, they have been worked rigorously for thousands of years and have proven their worth time and time again, and so they have locked in their place and are now intrinsic to the whole of civilized humanity (and then some).

So at present I wish to develop a few axioms which will be counterintuitive and seemingly strange for a system which is abstract in the fullest sense of the word and which bears no particular intentional relevance to society and the physical realm.

the axioms are postulated as follows:

let there exist a set of discrete positive numbers such that 0 = 1 

let this dual relationship be described as [0,1] 

there exists no true empty set 

Now a further postulate could be added which incorporates a greater deal of simplicity and proceeds as follows

all numbers greater than 1 have a coefficient of an immutable 1 

However I believe some of the beauty of this system lies in the fact that all numbers can be effectively equated to 0 which equates them to 1 as well, which creates a vast realm of possible solutions, though it may also add complication. If this postulate is abandoned than the following postulate is added as an axiom of the system :

all equations consisting of n elements have a minimum of n + 1  solutions not including [0,1] which is necessarily always a solution as well

This is due to the fact that all numbers have a natural coefficient of 1 attached to them and since 1 is now equal to 0 all numbers are equal to 0 and 1 simultaneously 

so for the simple arithmetical equation of 2x4 we have the following set of answers  = {2,4,8} and [0,1] which can be disregarded as trivial. It will, however, prove a valuable technique for now any element in an equation may be eliminated with ease. 

The obvious downside to this tangential axiom is that it will create a very large number of possible solutions for all equations, creating some deal of confusion. It could also lead to a sort of intrinsic meandering which is my hope for this particular system of mathematics, and which may in and of itself contain intriguing new avenues and beautiful solutions. Further it can be examined that now all equations can be broken down into minimum, median, and maximum solutions. 

Regardless of which of the final two postulates is adopted there is one seemingly obvious pratfall to this whole system, however I believe that said pratfall is actually the systems strongest and most elegant feature. The system quite clearly falls short as a means of precision calculating and it's glaring pratfall is the new vagueness given to all numbers there in. For instance the final postulate which states "equations consisting of n elements have a minimum of n + 1  solutions " gives rise to the question of what do we mean by n + 1 if within this system 1 = 0, therefore n + 1 solutions also equals n + 0 solutions, and furthermore , if said postulate is taken as an axiom of the system n = 0 and 1 as well, so within the language of the system n + 1 elements means essentially nothing. But than of course we are all well aware of what was meant by n + 1 elements because we are already familiar with standard arithmetic, number systems, etc, and the known definition of n + 1 elements is clear to us, despite the paradoxical multitude of solutions given.  The beauty of this systems lies in that paradox, the strength of our knowledge of what the maximum solution is and the simultaneous existence of contrary information which must be accepted as solutions to the problem as well. Of course the minimum and median solutions to the n + 1 postulate can be abandoned because we are speaking of mathematical objects using standard mathematical systems, which becomes the meta-language from which laws and axioms of the new system are derived.

To clear things up a bit it must be stated that this system will not suffice on it's own, but rather exists as an entity separate, though still related to, our standard mathematical systems. That is, in order to discuss the workings of this new system, we must use our familiar pre-established axiomatic systems. 

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