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Cantor-Finlayson Theory
Posted:
Sep 11, 2012 12:25 PM
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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 Cantor-Schroeder-Bernstein 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 one-to-one 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 Cantor-Finlayson 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 well-established. Set-builder 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 "non-real 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
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