
CantorFinlayson Theory
Posted:
Sep 11, 2012 12:25 PM


It's quite simple to found the integral calculus where the Equivalency Function is a simple object in the theory. How to arrive at the fundamental theorem of differentiation or fundamental theorem of calculus from the simple existence of this function that has constant infinitesimals that look just like dx or d, the differential, this helps.
http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus
Reading the Wikipedia, yes the general EF seems to work up indefinite integration then valid EF the validation of here why it represents continuous functions, that's why there's general and valid and general is valid. Basically, in the course of conversation, general EF (for the finite case) and here valid (for the infinite) equivalency functions:
EF_n(n) = n/d, n> d
These are also known as functions of fractions or functions of ratios.
EF(n) = n/d, n> d, d> oo
Then it's called the equivalency function, because it's the only function, or in piece wise composition, that maps each of the numbers that are the counting numbers to the unit interval of real numbers, it defines the real number segment. Then those two sets are equivalent because each set has a way to map its elements onto all of the other sets elements. That's the CantorSchroederBernstein theorem, that two sets are equivalent if they map onto all or surject onto the other. The sets biject to each other, there is a onetoone function, between elements of the two sets, that is all elements of each set. Here, they have equivalent cardinals, where the cardinality of the set is defined by the elements of the set and the functions between them, their cardinals are equal.
Then, the point of research in this, is that though , in the standard construction of the limit sequence as the definition of the real number, Eudoxus/Cauchy/Dedekind: The cardinals are defined as Cantor's cardinals where the powerset result shows (and EF shows for every other function) for each set, that all the possible combinations of the set is more than are in the set itself (defining less and more with cardinals). Then, because the construction of real numbers is as into the combinations of the numbers, they as a set have a different and totally ordered cardinal, a higher cardinal. Then, where if two sets have the same cardinal to a third then they are all equal, it would be a paradox that the line segment and natural numbers were equal but the real numbers in their construction and natural numbers were not. Where the theory with cardinals has a universal set it would be its own powerset, another paradox.
Then, in my new CantorFinlayson theory, basically it's the same Finlayson theory as I ever had about mathematics. This follows from a pretty or rather standard education. Now, as you might now know, that means for example, .999 = 1 and .999 =/= 1. Also, here 1 is the same as 1.0. So, just like in calculus when I heard them say "just like drawing the line, it is not drawing the line" and ".999... = 1", then "the real line is disordered", basically I simply recognize that the establishment is that they are and why. Set theory is quite simple, general and wellestablished. Setbuilder notation in builder notation and definitions is of course complete in the finite. Then, a rejection of cardinality for the features of the real continuous line isn't a rejection of set theory's cardinality. So, adding these features of the system to the theory, basically that is Zermelo Fraenkel set theory, with the consideration that for real analysis, there is a special case defined into the analysis, where there are other reasons for "nonreal functions" in real analysis. That's politesse where really Ross Finlayson theory is the Null Axiom Theory, or, an axiomless system of natural deduction.
So, just because it's a paradox to have EF exist, or here (all) functions of fractions, so is that the universe exists, and it does (in "standard", "modern" mathematics).
Then, I'll agree that it takes one infinity to represent a real number between zero and one or zero and any other number, that to write all the real numbers takes two infinities. This could be an infinity for the number and an infinity for the part on the other side of the radix, or from the number and one, this here isn't much different from standard where in the constructions these infinite structures are built into the framework. Also, as it maintains and develops standard real analysis, and maintains all finite combinatorics, it's generally useful.
Regards,
Ross Finlayson

