THIS PROJECT IS UNDER THE NEXT PHILOSOPHICAL PRINCIPLES
1) CONSCIOUSNESS IS INFINITE. CONVERSELY THE INFINITE IS A FUNCTION AND PROPERTY OF THE CONSCIOUSNESSES.
2) BUT THE PHYSICAL MATERIAL WORLD IS FINITE.
3) THEREFORE MATHEMATICAL MODELS SHOULD CONTAIN ONLY FINITE ENTITIES AND SHOULD NOT INVOLVE THE INFINITE.
THIS PROJECT THEREFORE IS CREATING AGAIN THE BASIC OF MATHEMATICS AND ITS ONTOLOGY WITH NEW AXIOMS THAT DO NOT INVOLVE THE INFINITE AT ALL.
Our perception and experience of the reality, depends on the system of beliefs that we have. In mathematics, the system of spiritual beliefs is nothing else than the axioms of the axiomatic systems that we accept. The rest is the work of reasoning and acting.
The abstraction of the infinite seems sweet at the beginning as it reduces some complexity, in the definitions, but later on it turns out to be bitter, as it traps the mathematical minds in to a vast complexity irrelevant to real life applications.
1) CONSCIOUSNESS IS INFINITE. CONVERSELY THE INFINITE IS A FUNCTION AND PROPERTY OF THE CONSCIOUSNESSES.
2) BUT THE PHYSICAL MATERIAL WORLD IS FINITE.
3) THEREFORE MATHEMATICAL MODELS SHOULD CONTAIN ONLY FINITE ENTITIES AND SHOULD NOT INVOLVE THE INFINITE.
THIS PROJECT THEREFORE IS CREATING AGAIN THE BASIC OF MATHEMATICS AND ITS ONTOLOGY WITH NEW AXIOMS THAT DO NOT INVOLVE THE INFINITE AT ALL.
Our perception and experience of the reality, depends on the system of beliefs that we have. In mathematics, the system of spiritual beliefs is nothing else than the axioms of the axiomatic systems that we accept. The rest is the work of reasoning and acting.
The abstraction of the infinite seems sweet at the beginning as it reduces some complexity, in the definitions, but later on it turns out to be bitter, as it traps the mathematical minds in to a vast complexity irrelevant to real life applications.
The two different systems of axioms , of the Traditional Mathematics, and of the Digital Mathematics as defined in my previous axiomatic systems, or as can be defined by similar axiomatic systems by other authors, do coexist peacefully. Actually without the work done in the traditional mathematics with the infinite, the pages and theorems of the axiomatically different Digital mathematics , would not be possible to fill easily. But I consider the traditional mathematics, more ARTISTIC compared to the digital mathematics that are eventually in applications more REALISTIC and appropriate. It is like in designing cars, first you need a quick sketch, that gives the general shape fast (this is the mathematics with the infinite). Then you need more engineering perfect design, which might be more laborious. Or you can conceive the Digital Mathematics a daughter discipline of the Traditional Mathematics. I do consider the Computer science too as a daughter science of Traditional Mathematics, and that is the way to view Digital Mathematics too. With the exception the computer science is a daughter married to the Electric Engineering-husband, while Digital Mathematics is still .....living with the father or mother discipline of traditional mathematics........
The digital natural numbers are of course a finite segment of the traditional natural numbers.
But there is a special axiomatic system for them as well.
The digital natural numbers have a particular size Ω that gives the maximum feasible natural number.
I do not claim here that the axioms of the digital natural numbers, below, are independent, in other words none of them can be proved from the others. As the elements are finite, there may be such a case. But I am strongly interested a) at first that are non-contradictory, and b) second that are adequate many, so as to describe the intended structure. later simplified and improved in elegance versions of the axioms may be given.
We define the natural numbers in two scales (and later precision levels) that are two unequal initial segments of the natural numbers N(ω)< N(Ω) .The number ω is called the Ordinal size ω of the local system of natural numbers Ν(ω) while the Ω is the cardinal size of the global system of natural numbers. ω<Ω. If we start with integers n1, n2, n3 from N(ω), then their addition and multiplication, have the commutative semiring properties but without closure in N(ω), but with values in N(Ω). We call the N(ω), the local segment while the N(Ω) the global segment.
1) The number 1 is a natural number and belongs both to N(ω), and N(Ω) .
6) Axiom of sufficient large size. If we repeat the operations of the commutative semiring starting from elements of the local version N(ω), ω-times, the results are still inside the larger set N(Ω). This is also called by the term that the Ω is seemingly infinite relative to the ω
This last Peano axiom of induction is useful only if the natural numbers are formulated within a formal logic (the axiom itself as a formal proposition is in 2nd order formal logic) that its size Ω(l) is less than the size of the objective system of natural numbers Ω. Otherwise for sufficient large
Any two models Μ1 Μ2 of the digital natural numbers Ν(Ω),Ν(ω) of equal size ω, Ω are isomorphic.
I may be noticed that digital natural structures are finite structures. But the theory of finite structures and finite models of axiomatic theories does not exist in its entirety within digital mathematics, as it is utilizing the infinite and countable infinite which does not exist, in digital mathematics.
For axiomatic theories in general see
http://en.wikipedia.org/wiki/Axiomatic_system
The consistency of axiomatic systems in Digital mathematics is one of the less important problems to deal with, because of all models of them being finite.
For polynomial decidable fragments of Classical Peano arithmetic (bounded arithmetic ) see
Buss thesis http://www.math.ucsd.edu/~sbuss/ResearchWeb/BAthesis/Buss_Thesis_OCR.pdf
At the end of this chapter, there is
a) Advantages-disadvantages of these new digital mathematics compared to the classical analogue, infinitary mathematics.
b) A fictional discussion in dialogue form of celebrated historic creators of the old mathematics praising or questioning the new mathematics compared to the old.
Speakers: Pythagoras, Euclid. Confucius, Diophantos, Fibonacci, Fermat, Gauss, Cantor, Hardy, Goodstein etc
It is given an detailed discussion with arguments counter arguments and refutations about the , proof of the terminating of the Goostein sequence. (see http://en.wikipedia.org/wiki/Goodstein's_theorem ). Independed ststements in Peano, are compared for example with the 5th axiom of parallels of Euclidean axiomatic geometry.
It is also discussed effect of increasing the order of the formal language of natural (axiomatic digital) arithmetic , and Cantor points out the correspondence to his higher infinite cardinals.
How von Neumann created all the natural numbers, from sets and the nothingness (empty set)
I do not claim here that the axioms of the digital natural numbers, below, are independent, in other words none of them can be proved from the others. As the elements are finite, there may be such a case. But I am strongly interested a) at first that are non-contradictory, and b) second that are adequate many, so as to describe the intended structure. later simplified and improved in elegance versions of the axioms may be given.
1 SIMILAR TO PEANO, AXIOMS
We define the natural numbers in two scales (and later precision levels) that are two unequal initial segments of the natural numbers N(ω)< N(Ω) .The number ω is called the Ordinal size ω of the local system of natural numbers Ν(ω) while the Ω is the cardinal size of the global system of natural numbers. ω<Ω. If we start with integers n1, n2, n3 from N(ω), then their addition and multiplication, have the commutative semiring properties but without closure in N(ω), but with values in N(Ω). We call the N(ω), the local segment while the N(Ω) the global segment.
We have here an initial relation among the
natural numbers which is called successor or next of
a natural number x and it is denoted by S(x).
1) The number 1 is a natural number and belongs both to N(ω), and N(Ω) .
2) There is
no natural number whose successor is 1.
3) If x is a
natural number of N(ω), its successor S(x) , is also a
natural number belonging in N(Ω).
4) If two
different numbers of N(Ω) , have the same successor, then they
are equal, Formally if S(x)=S(y) then x=y .
5) (Peano
axiom of induction) If a property or formal proposition P()
holds for 1 (that is P(1)=true) and if when holding for x
in N(ω) holds also for P(S(x)) with S(x) in N(Ω) ,
then it holds for all natural numbers of N(ω).
6) Axiom of sufficient large size. If we repeat the operations of the commutative semiring starting from elements of the local version N(ω), ω-times, the results are still inside the larger set N(Ω). This is also called by the term that the Ω is seemingly infinite relative to the ω
This last Peano axiom of induction is useful only if the natural numbers are formulated within a formal logic (the axiom itself as a formal proposition is in 2nd order formal logic) that its size Ω(l) is less than the size of the objective system of natural numbers Ω. Otherwise for sufficient large
Ω(L)>>Ω,
we may simply construct a lengthy proof of this axiom starting from
P(1) then P(2) ...and finally P(Ω), which then it is a theorem.
Any two models Μ1 Μ2 of the digital natural numbers Ν(Ω),Ν(ω) of equal size ω, Ω are isomorphic.
For axiomatic theories in general see
http://en.wikipedia.org/wiki/Axiomatic_system
The consistency of axiomatic systems in Digital mathematics is one of the less important problems to deal with, because of all models of them being finite.
For polynomial decidable fragments of Classical Peano arithmetic (bounded arithmetic ) see
Buss thesis http://www.math.ucsd.edu/~sbuss/ResearchWeb/BAthesis/Buss_Thesis_OCR.pdf
At the end of this chapter, there is
a) Advantages-disadvantages of these new digital mathematics compared to the classical analogue, infinitary mathematics.
b) A fictional discussion in dialogue form of celebrated historic creators of the old mathematics praising or questioning the new mathematics compared to the old.
Speakers: Pythagoras, Euclid. Confucius, Diophantos, Fibonacci, Fermat, Gauss, Cantor, Hardy, Goodstein etc
It is given an detailed discussion with arguments counter arguments and refutations about the , proof of the terminating of the Goostein sequence. (see http://en.wikipedia.org/wiki/Goodstein's_theorem ). Independed ststements in Peano, are compared for example with the 5th axiom of parallels of Euclidean axiomatic geometry.
It is also discussed effect of increasing the order of the formal language of natural (axiomatic digital) arithmetic , and Cantor points out the correspondence to his higher infinite cardinals.
How von Neumann created all the natural numbers, from sets and the nothingness (empty set)
| First few von Neumann ordinals | ||
|---|---|---|
| 0 | = Ø | |
| 1 | = { 0 } | = {Ø} |
| 2 | = { 0, 1 } | = { Ø, {Ø} } |
| 3 | = { 0, 1, 2 } | = { Ø, {Ø} , {Ø, {Ø}} } |
| 4 | = { 0, 1, 2, 3 } | = { Ø, {Ø}, {Ø, {Ø}}, {Ø, {Ø}, {Ø, {Ø}}} } |
References
1) G. H. Hardy A course in Pure Mathematics Cambridge 10th edition 1975
2) J H Conway On numbers and games Academic Press 1976
3) H. A. Thurston “The number system” Dover 1956
3) H. A. Thurston “The number system” Dover 1956
