Already Published Paper in 2020
Title : Time & Relativity in Mathematics!! Godel Incompleteness Results,
Cantor Diagonalization &
Countability of Real Numbers,
Famous paradoxes e.g.
Richard, Russell, Skolem, Liar , Set Theory, ZFC, in Relativistic Mathematical Space-Time
Author : Pankaj Mani,FRM,CQF NewDelhi,India
By Pankaj Mani ,CQF,FRM(manipankaj9@gmail.com)
Abstract :
In this paper, we try to
revisit some of the most fundamental issues lying at the foundation of mathematics in space-time
relativistic perspective ,rather than conventional
absolute space. We are adding a new
dimension “Time” to the mathematics and review it in Space-Time
relativistic framework to resolve the major
foundational issues and making it in
line with physical world realities. We shall look at the famous Cantor’s
Diagonalization approach in to show
the Countability of Real Numbers and explain
Infiniteness in that perspective. We shall also look to resolve the famous paradoxes e.g. Richard, Russell,Liar, Skolem
. We shall also look at the the foundation of Set theory historically in Space-time to restore the issues that had led to ZFC
by elimination and restrictions. As a consequence, we shall also revisit Godel
Incompleteness theorems for Real Numbers
and also otherwise explain the “inconsistency” in the new framework. This could possibly lead to entirely new way of looking at conventional mathematics in broad sense.
Note: Space here means
that mathematical space in which the Numbers
are constructed physically ! Time is
the dimension aligned with that
mathematical Space in which Numbers are constructed ! The combined structure is Mathematical Space-Time or simply
Space-time throughout this paper.
Moreover one key implications
to mention is that Even Human Languages e.g. English,
French, Hindi etc. are fundamentally structurally incomplete or inconsistent that would have
huge implications in day to day life communications.
Introduction to the
idea :
There is an interesting
paper by eminent Computer Scientist Gregory Chaitin [Reference 1]“ How Real are the Real Numbers”
in which he talks about the fundamental issues with Real Numbers,
Borel’s idea of definition
of Real Numbers to the extremely powerful
Richard’s paradox. He points out how
Real Numbers in Mathematics are at –a distance from the the measurement point of view in Physics where we have not
been able to measure physically beyond few digits in the real world.
Hence, a question is raised, how real are the Real Numbers really or
Other mathematical entities in the physical real world. Infact, He talks interestingly that The Real Numbers are so
Unreal if we compare them with Real world realities of the Physical
Perspectives. In another
Computable Universe Hypothesis by eminent physicist Max Tegmark,MIT, he talks about trying some ways to deal with
Real Numbers issue as the fundamental issue to model
Physical Realities.. Reference[2].
Recently Stephen Wolfram
has launched Computer Project with
Physics in Computational Aspects. All
of them have some fundamental issues
with Humanly Created Mathematics.
The fundamental issue is that Humans have created a
Mathematical System which is at
conflict with Nature’s way of working. This is causing so many fundamental issues. Real Numbers being one
of the prominent examples assuming
infinite divisibility of Space/Particle . Mathematically,
say a Real Number say Sqrt 2 and 1
are relatively defined where the former has unending decimal representation but Physically both have same finite Physical existence
depicting a finite length. Evidently, the problem is
the human defined system of Mathematics
here Real Numbers. and Infact ,Physical theories have evolved from early age Galileo to Newton to Einstein to
QM, but Mathematics still resides in old age
where the ancient mathematicians came up with
the system by deriving their at that time
contemporary ideas about the
physical universe.
Since Euclid to Einstein to Riemann all had raised doubt
about the Continuous nature of Space
at different times. . If there exits
the most basic scale of the existence
of physical object say the Planck
Scale, then everything in this Universe is
a multiple of that most basic unit and hence all are basically Natural Numbers fundamentally and naturally but Human
defined system of mathematics doesn’t seem to be
coherent with all those.
In this paper, I would try to address and resolve these fundamental issues
/problems in a
completely new way of looking at the mathematical entities that could make it aligned with the physical world realities. So, here, I try to look at them in
Physical world perspective of reality ! I’ll try to expose a the hidden dimension to view the entire Mathematics in line with the
Physical world reality that has often been ignored.
If we look at the evolution of terms
in mathematics e.g. Set or
Numbers etc. We would find that their
construction have not been done without time
dimension. They have been constructed at different stages in the time
dimension. This is extremely vital
when we go at the foundation of paradoxes, although we have been ignoring this otherwise in normal calculation.
Traditionally, what we have been
doing is to just presuming that the entire mathematical concepts exist at the same time absolutely, by
overlooking their procedure of construction, that has actually occurred in time dimension. It’s just like physics
here.. Not to forget, one of the
major related problem P vs NP in
Theoretical Computer Science talks about the time
involved to run an algorithm.
This paper looks into those vital hidden time
dimension which is very much required along with space to construct
the notion of mathematical basis e.g. Set
,Numbers. Infact, Unlike
conventional approach, where mathematicians typically just talk about the Space aspects of Mathematics in Absolute
sense, I would like to focus on the
Time Aspects also leading to inclusive Space-Time view of these mathematical entities in Relativistic
Sense. We shall see, this foundation
change in view would try to resolve many important results e.g.
Countability, Uncountability,
Infinity , Paradoxes in Space- Time
perspective that was derived using
Cantor Diagonalization method, which is the backbone of so many important derived results and the Cantor based set theory. Historically many legendary mathematicians have spoken against
the Cantor based set Theory!
These traditional
results at the foundation of arguably
one of the the most important
discoveries of mathematics and logic
e.g. Godel Incompleteness Theorems.
We can look at them in entirely new perspective in Space-Time to possibly address its limitations and expand them further in broader areas
as well We would find that many
fundamental limitations of Mathematics were long lasting historically so far because of the
ignoring Relativistic and Time dimension that is coherent with the evolution of Physical realities.
Most importantly, The Newly
Proposed Set theory ZFC was proposed as the foundation
of Set theory by eliminating those paradoxes
in erstwhile absoluteness of Set
rather than resolving them foundationally. We
would show that those paradoxes are no more existing when we include the
Time dimension of mathematics rather say in Relativistic rather than the need to propose ZFC . So, this
way, it also tries to solve the
problem fundamentally rather than eliminating
them.
Traditional Proof of Cantor Diagonalization Method to Prove the Uncountability of Real Numbers.
Set T of Infinite
Sequence of all the binary
digits exists
If
s1, s2, … , sn, … is any enumeration of elements from T,
then there is always an element
s of T which
corresponds to no sn in the enumeration.
The proof starts with an enumeration of elements from T.
Next, a sequence s is constructed by choosing the 1st digit
as complementary to the 1st digit
of s1 (swapping 0s for
1s and vice versa), the
2nd digit as complementary to the 2nd digit of s2, the 3rd
digit as complementary to the 3rd
digit of s3,
and generally for every n,
the nth
digit as complementary to the nth digit
of sn.
s1=(0,0,0,0,0,0,0,...)
s2=(1,1,1,1,1,1,1,...)
s3=(0,1,0,1,0,1,0,...)
s4=(1,0,1,0,1,0,1,...)
s5=(1,1,0,1,0,1,1,...)
s6=(0,0,1,1,0,1,1,...)
s7=(1,0,0,0,1,0,0,...)
...
s=(1,0,1,1,1,0,1,...)
By construction, s differs
from each sn, since their nth digits differ (highlighted in
the example). Hence, s cannot occur in the enumeration.
Based on this theorem, Cantor then
uses a proof of contradiction to show
that: The set T is uncountable.
The proof starts by assuming that T
is countable. Then all its
elements can be written as an enumeration s1, s2, ... , sn,.... Applying the previous theorem to this enumeration produces a
sequence s not belonging to the enumeration. However, this
contradicts s being an element of
T and
therefore belonging to the enumeration. This contradiction implies that the
original assumption is false. Therefore, T is uncountable.
Space-Time Proof of Countability of Real Numbers using
Cantor Diagonalization:
The above proof seems to miss the more fundamental deep
aspect while using the method of contradiction. It previously concluded that the
assumption of “T is countable” is false but what could also be meant at the deeper level it’s not actually about
the UnCountability of the Set T rather that the assumption of the absolute existence of the set T
is itself objectionable.
That means, There can’t exist
any such Absolute “Set T” statically
in the dimension of Time nullifying
the fundamental concept of “Set”.
Rather, here the Set is applicable not in absolute rather relative sense with respect
to say time dimension in which numbers
are constructed.
Numbers fundamentally come into existence in
the dimension of time dynamically. They have
been assumed to exist independent of time. That holds true for Natural,
Real etc. numbers.
They have Relative
Existence depending upon Observer like in Physics
!
It’s because of this assumption of “Absoluteness” of Set,
this result of Uncountability comes into picture.
But in our relativistic perspective, Such “absolute set T can’t exist, they can
exist in the dimension of time
relativistically. That means, all the elements of the sets are created in
different dimension of time rather
their absolute existence at the same time for the observer. That also
means that Uncountability and Countability have
Relativistic existence.
Infact Construction of T is like the Set in Russell’s
paradox. We shall deal with the Russell’s paradox
in the next section in Relativistic Terms.
The foundational Concept of Set based on the
Principle of Simultaneity is not aligned with
Physics based realities. That Construction of a Set without Time
dimension leads to all the Problems and its incompatibility with the Physics.
So, here goes on taking the
traditional steps of Cantor Diagonalization of Binary digits At time t= 0 say the following sequence exists i.e. T(0)
s1=(0,0,0,0,0,0,0,...)
s2=(1,1,1,1,1,1,1,...)
s3=(0,1,0,1,0,1,0,...)
s4=(1,0,1,0,1,0,1,...)
s5=(1,1,0,1,0,1,1,...)
s6=(0,0,1,1,0,1,1,...)
s7=(1,0,0,0,1,0,0,...)
...
At t= 1 , T(1), the new
s is added to T(0).
s=(1,0,1,1,1,0,1,...)
If the entire
proof is viewed in this perspective, the end result of Cantor Diagonalization
is Not the Absolute Uncountability
rather a Relative Countability w.r.t. in
the Time dimension. The absolute Uncountability arose because the
absolute existence of the Set T was presumed to exist but that’s not true and can be experimentally /practically
realized because any time one tries
to make an absolute confined existence, it will lead to the conclusion that
after this there will be no Natural Number
or any Number beyond that.
If
we look at the process
deeply, the contradiction arises because the definition of T varies
in the dimension of time and the absolute confined existence of all
the infinite possibilities at the
same time i.e. independent of observer time doesn't hold true and possible.
In fact to start
with, T at t=
0 and as a new s is
created from the previous T at t=0, that becomes
part of the new
T at
t= 1. and both the T at different times are not the same .They vary dynamically.
The new enumeration “s” is excluded/external
to T(0) but included / internal to T(1). Hence, in Space-Time relativistic sense it has no contradiction but if time is removed
and only Space is
taken then, it leads to the contradiction as how s could be both external and
internal at the same time!!
This is the fundamental
cause .
The earlier contradiction arose because the time evolution of T is
ignored and the absolute existence
of all the infinite elements of T is presumed independent of time in space. But fundamentally , it's in space -time as in
physics. In other words, Infiniteness
of Numbers or Numbers in general is a dynamic
constructive process in space-time. not only
space.
T (0)was Countable at t=0 but T(1) also
countable at t=1 (after the new s enumeration comes up by taking earlier one as the absolute set of all infinite
ones) . But if time dimension is hidden, it looks contradictory !
The Countability assumption appeared to be violated because earlier it was presumed to be absolute existence of T in space independent of time. But that's not possible regarding T. But the issue
is more fundamental that lies in Relativistic Space-Time existence of Numbers
and Infiniteness of elements for that matter.
T(at t=0) is
assumed to be Countable ..but as the new enumeration s is appended, T(t) is
also Countable. But Since time dimension is hidden/ignored, T overall becomes
Uncountable because of Contradiction. But in Space-Time with
time dimension included, both T exist at different time and hence co contradiction and hence no Uncountability.
Infact at any
moment, they are finite also. This is
because T(t) is never Complete list
for any value of T ..ie T(0) or T(t). The assumption of Completeness of T is not valid
and also the cause of Contradiction ! Hence, T (t) is
actually Finite in Space-Time. Its when we remove Time dimension, it becomes Infinite !
In other
words, Countability, Uncountability ,Set etc. are Relativistic in Space-Time
rather Absolute . This time version would similarly hold true for
any sequence including real numbers.
Infiniteness
as Finiteness in Mathematical Space-Time.
Infact also,
Inifniteness of Numbers exist because of hidden dimension of time. As we view
in the dimension of time,
at any moment of time the mathematical infiniteness could also be finite in
Space-time. Say for example Natural Numbers are assumed to be Infinite because
they are presumed to be absolutely existing at the same time
without time but in reality construction of Natural Number occurs in time dimension. At any particular moment which will always be finite, finite number of natural numbers would
exist not all of them at the same time. All the natural numbers cant exist at the same time in the set. They need to be constructed by the
observer relatvistically just like in
physics and at any moment in time, there will be construction finite number of times only.
So, Infinite(at time time t )= Finite.
Cantor came
up with different levels of Inifnity and reciprocally(just
by dividing 1 by different levels of
infinity could also prove to different levels of Inifinitesimals/0) These all issues arise because
of lack of time dimension. But the construction of number requires
time dimension.
Key Result **(A)
Hence the Conclusion is With Time dimension in
Space-Time at any particular time t ,
any sequence( including Real Numbers
) is Countable and Finite . It’s when time is ignored, they appear to be Uncountable & Infinity in Space. We need to look at them
in Space-Time Reality as in Physics.
Constructivism of Set,Numbers etc. can’t happen
without time.
Infinitness (
at time t ) = Finiteness i.e. Inifniteness in Space becomes Finiteness in
Space- Time.
The same issues arise in context
of Russell's paradox
.The paradox arises
because the Master
Set is assumed to exist in
space independent of Time dimension. But the truth is they have hidden time
dimension also I.e. space-time. The same set has time evolution also in space.Contradiction
arises because the same Set varies in time and they appear contradictory in the absolute space. In reality, they exist in Space-Time not Space.
Henceforth, this implies that Numbers and Set
etc should be seen as in Space-Time at fundamental levels.
For example Set at
T=0 and T= t . Real Number at T= 0 and at
T= t .
Then entire
unending decimal parts of the Real Numbers cant exist at the same time T= 0. It can always be extened further and this can
be viewed in Space-Time where at any
particular T= t, the Real Number
is actually Rational
Number. Let's take an example
sqrt(2)= 1.414.........
So, here
at Sqrt2 (at T= 0) = 1.4
Sqrt2 (at T= 1)
= 1.41
Sqrt2 (at T=
2) = 1.414
and so on....in the
time dimension.
Sqrt2 cant exist completely in Space at any particular time T..It can always be extended further and that would be in the Space-Time.
Borel view of Real Numbers (Chaitin's paper) :A
real number is really real only if it
can be expressed, only if it can be
defined using a finite number of words.It is only real if it can be named or
specified as an individual
mathematical object. And in order
to do that we must necessarily
employ some particular language e.g. English/French. Whatever the choice of language,
there will only be countable
infinity of possible
texts.
Infact his view can be addressed in space-time
perspective. Real Numbers
will always be expressive in finite words at
a particular T= t in Space-Time.
This space-time view will help resolve the issues of
Countability and Uncountability of different
type of Numbers. Even Real Number at a particular T= t will always be
Countable. Even the Concept of
Infinity will become Finite at T= t. So, many things will get more coherent
practical view and compatible with Physics.
Hence, many fundamental issues
regarding Real Numbers
will get sorted
out in Space-Time.
The previous
assumption that the Set contains all the possible elements absolutely at the same time independent of time leads to contradiction..
The new element
is not in the previous
list Set at T= 0 but became
the new element
of the derived version of
the Set which evolved at T= t. So,
it's all about the evolution of the
Set in the dimension of time.
Earlier the new element was not present at T=0 in the Set but it came in the list at
T=t . But contradiction is because it
is thought that how the new element outside the Set became the part of the previous Set. But it is happening
at different Time as dimension.. not at the same time .This is the
point of contradiction!!
This lack of time dimension leads to the
contradiction that new element real
number not in the previous Set is the part of the previous
Set .This is because of assumption of Absoluteness of the Set in Space having all the elements at
the same time and not taking into
consideration the Time dimension
aspect. The Set is not static rather
it is changing over time. But the fact is Relativistic nature of the Set in Space-Time.
Skolem
Paradox : Relativistic aspect in Space-Time.
.
Skolem, the mathematician has pointed out the
Relativistic aspect of Set theory, Counatability, Uncountability as a resolution to his Skolem Paradox.
A brief
description of the problem :
Skolem (1922) pointed out the seeming contradiction between the
Löwenheim–Skolem theorem on the one
hand, which implies that there is a countable model of Zermelo's axioms, and
Cantor's theorem on the other hand,
which states that uncountable sets exist, and which is provable from Zermelo's
axioms. "So far as I know," Skolem writes, "no one has called attention
to this peculiar and apparently paradoxical state of affairs.
By virtue of the axioms we can prove the existence of higher cardinalities... How can it be, then, that
the entire domain B [a countable
model of Zermelo's axioms] can
already be enumerated by means of the finite positive integers?" (Skolem
1922, p. 295, translation by Bauer-Mengelberg)
More specifically, let B be a countable model of Zermelo's axioms. Then there is some set u in B such that B satisfies the first-order formula saying that u is uncountable. For
example, u could be
taken as the set of real numbers in B. Now,
because B is
countable, there are only countably
many elements c such that c ∈ u according to B, because
there are only countably many
elements c in B to begin with. Thus it appears that u should be countable. This is Skolem's paradox.
Skolem went on to explain why there was no contradiction. In the
context of a specific model of set theory,
the term "set" does not refer to an arbitrary set, but only to a set
that is actually included in the
model. The definition of countability requires that a certain one-to-one
correspondence, which is itself a
set, must exist. Thus it is possible to recognise that a particular set u
is countable, but not countable in a particular model of set
theory, because there is no set in the model that gives a one- to-one correspondence between
u and the natural numbers
in that model.
Skolem used the term "relative" to describe this state of
affairs, where the same set is included in two
models of set theory, is countable in one model, and is not countable in the
other model. He described this as the
"most important" result in his paper. Contemporary set theorists
describe concepts that do not depend
on the choice of a transitive model as absolute. From
their point of view, Skolem's
paradox simply shows that countability is not an absolute property
in first order logic. (Kunen
1980 p. 141; Enderton 2001 p. 152; Burgess 1977 p. 406).
Skolem described his work as a critique of (first-order) set theory,
intended to illustrate its weakness as a foundational system:
"I believed that it was so clear that axiomatisation
in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would,
for the most part, not be very
much concerned with it. But in recent
times I have seen to my surprise
that so many
mathematicians think that these axioms of set theory
provide the ideal foundation for mathematics;
therefore it seemed to me that the time had come for a critique."
(Ebbinghaus and van Dalen, 2000, p. 147)
Space- Time Relativistic Resolution to Skolem Paradox.
The Set B is Countable at time t=0 in one frame with respect to the Set u .
But at =1 , the Set u appears
Uncountable at t=1 in another frame in which it is Real Number .
And
as we have already shown using the Cantor Diagonalization in Space-Time that Real
Numbers are Countable and Finite at any point in time t in Space- Time. So, in
the Space-time perspective, the Set u is also Countable at time t and Set B is already Countable. Hence,
there is no more contradiction in Space- Time.
Russell’s Paradox Resolution in Space-Time
Relativistic perspective
According to naive set theory,
any definable collection is a set. Let R be the set of all
sets that are not members of
themselves. If R is not a member of itself, then its definition dictates
that it must contain itself,
and if it contains itself,
then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox.
Symbolically:
In 1908, two ways of avoiding the paradox were proposed,
Russell's type theory and the Zermelo set theory. Zermelo's axioms
went well beyond Gottlob Frege's axioms
of extensionality and unlimited set abstraction; as the first constructed axiomatic set theory, it evolved
into the now-standard Zermelo–Fraenkel set theory (ZFC). The essential difference between Russell's and Zermelo's solution to the paradox is that
Zermelo altered the axioms of set
theory while preserving the logical language in which they are expressed, while
Russell altered the logical language
itself. The language of ZFC, with the help of Thoralf Skolem, turned out to be
first-order logic.
Example:
Most sets commonly encountered
aren't members of themselves. For example, the set of all squares in the
plane. This set is not itself
a square in the plane, thus it is not a member
of itself (of its own
definition). Let us call a set "normal" if it is not a member of itself,
and "abnormal" if it is a member
of itself. The set of squares
in the plane is normal. On the other hand, the complementary set that contains everything which is not
a square in the plane is
itself not a square in the plane, and so should be one of its
own members and is therefore abnormal.
Now we consider the set of all normal sets, R, and try to determine whether R
is normal or abnormal. If R were normal, it would be contained in the set
of all normal sets (itself), and therefore be
abnormal; on the other hand if R were abnormal, it would not be contained in
the set of all normal sets (itself),
and therefore be normal. This leads to the conclusion that R
is neither normal nor abnormal:
Russell's paradox.
Space-Time perspective :
We considered R as the set of all the
normal sets to decide
whether normal or abnormal.
Case 1 : If R is normal(i.e. not a member of itself as per first definition) , then it’s abnormal
as per the second definition.
Case2 : If R is abnormal(i.e.
member of itself as per the first definition, then it’s normal as per
the second definition.
This leads to the conclusion R is neither
Normal nor Abnormal.
The set of all squares is not a
square itself making
it abnormal. If R is
normal at t=0 , then R will be abnormal
at t =1
If R is abnormal
at t=0 then R will be normal at t= 1 as per the defintition
1st definition occurs at t=0 and 2nd definition occur
at t= 1 i.e. different time not simultaneously.
R being Normal and Abnormal are
occurring at different point in time not simultaneously. Infact
, in Space-Time, this is possible and not
contradictory.
That’s how it has been defined
by us the definition of normal and abnormal.
It has been hidden in time
dimension, how they have been defined. that’s what the condition “If” & “Then”
represent. If and then occur
at different time dimension.
In other words, “Normal”
and “Abnormal” terms are relativistically existing in Space-
Time rather Absolute terms leading to pardox in Space.
In Space, this appears
contradictory when time dimension is ignored.
Hence, if we look at
Russell’s paradox in Space-Time perspective, it will not be a paradox
any more. That’s how it can be resolved .
The Concept of Set is Relativistic in Space-Time . It has Time dimension
included, which is often hidden.
As it has also been pointed out that the existence of Sequence T, (the set of
all possible infinite binary digits
sequence as explained before in the
Cantor Diagonalization ) is similar to Russell’s Set.
Liar Paradox :
A says “ I am
lying”
Case
1 :If A is speaking Truth, then He is a Liar.
Case2 : If A is speaking Lie, then He is Not a Liar.
Lets again look at relativistic aspects
hidden in Space-Time.
“I
am lying” occurs at time t=0 and “I am saying that I am lying” occurs at time t
=1 . Both the statements are occurring at different time in Space-Time. So, he is Liar
at say time t=0 but Not a Liar at time t=1 or opposite.
In other words, He is lying for something else but He is Not
lying overall or opposite. So, they are contextually and Relatively Liar or Not at
different Times in the space-time
So, it’s not a paradox
in Space-Time once we add Time dimension
. As earlier , it’s because
of ignoring Time dimension that it becomes
a paradox !
Richard
Paradox Resolution in Space-Time &
Godel Incompleteness Theorems
for Real Numbers:
Godel Incompleteness Results :
Cantor-diagnolization to the list of
all the possible mathematical definitions for individual real numbers in English
!!
Conventional Description:
“Thus there is an infinite
list of English phrases (such that each phrase is of finite
length, but the list itself is of infinite length) that
define real numbers unambiguously. We first arrange this list of phrases
by increasing length,
then order all phrases of equal
lexicographically (in dictionary order,
e.g. we can use the ASCII code, the phrases
can only contain codes 32 to 126), so that the ordering is canonical . This yields an infinite list of the corresponding real numbers: r1, r2,....................... Now define
a
new real number r as follows. The integer part of r
is 0, the nth decimal place of r
is 1 if the nth decimal place of
rn is not 1, and the
nth decimal place of r is 2 if the nth decimal
place of rn is 1.
The preceding two paragraphs are an expression in English that
unambiguously defines a real number r. Thus r must be one of the numbers rn.
However, r was constructed so that it cannot equal
any of the rn (thus, r is an undefinable This is the paradoxical contradiction.”
Space-Time
Resolution :
If we apply here the previously
explained view in Space-Time, Richard paradox gets resolved in Space- Time . The contradiction gets rooted out .
The fundamental point is the absolute assumption of the Set of all the infinite list in dependent of Time
dimension leads to the contradiction. As we view in Space-Time dimension, the Set of all English phrases encoded into Real
Numbers is evolving in Time. In other words
as Real Numbers would be Countable in Space-Time at T=t, it can be
marched with the Set of English Phrases corresponding to them. If at new value of T
=t’ if new r is constructed using the previous
series, it can be defined using the new English
phrase. The earlier issue was just because of hiding Time dimension that in Space, it seemed that
Real Numbers are Uncountable but English Phrases are Countable and hence the issue. But in Space-Time there is no
absolute existence and as shown in the Cantor
Diagonalization that the issue regarding Real Number Uncountability was sorted
out in Space- Time. Hence , Richard
Paradox also gets resolved
in Space-Time.
So, applying
the Key Result
A : That means, at any time t, the Real Numbers set is Countable and
Finite in Space-Time and Hence,it can be matched with
the English sequence one to one as both are Countable. The initial issue or
contradiction was because of ignoring time dimension.
This entire Space-Time
Relativistic concept can be applied to the Set of All Real Numbers say in the Semantic Language e.g. English in the Cantor Diagonalization Form . That’s what Richard
paradox talks
about. Hence, Similar Cantor
Diagonalization result will be valid for Reals Numbers and their English
equivalents . Kurt Godel
considered Richard’s paradox as the
Semantic equivalent of Godel’s Incomepleteness Results.
“Infact one major result to follow is
that Humanly Created langauges e.g. English ,French, Hindi and others could
also be fundamentally incomplete or inconsistent structurally ,which would have
huge implications for day to day life communications”
Hence, in context of Richard paradox resultion in Space-time, Real
Numbers can also be encoded in English Languages and then by applying Cantor
Diagonlization, we can establish the Incompleteness and Inconsistency
of Mathematics for Real Numbers
systems also. We mean as earlier Godel Incompleteness
results are confined to certain Natural Numbers, They can also be extended to
Real Numbers as well.
Important Conclusion : The Resolution of Richard
Paradox will also lead to the result
that Godel Incompleteness
Results which is basically proven for the set of Natural Numbers
will also be valid for the set of Real Numbers!!
Secondly, What we
call Inconsistency in context of
Godel Incompleteness Theorems in
general, is infact consistent in space-time framework as explained in the Cantor Diaonalization
framework. Two contradictory results in absolute space could befree of contradictions
in space-time at different points in time relativistically.
Space –Time
Relativistic View of Set Theory rather than ZFC
In set
theory , Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Frenekel , is an axiomatic
system that was proposed in the early twentieth century
in order to formulate a theory of sets free of paradoxes such as Russell’s paradox . Today, Zermelo– Fraenkel
set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the
most common foundation
of Mathematics Zermelo–Fraenkel set theory with the axiom of
choice included is abbreviated ZFC, where C
stands for "choice",[1] and ZF refers to
the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.
Zermelo–Fraenkel set theory is intended to
formalize a single primitive notion, that of a hereditary
well-founded set , so that
all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer
only to pure
sets and prevent its models
from
containing urelements (elements
of sets that are not themselves sets). Furthermore, proper classes (collections of mathematical objects defined by a property shared by their members where the collections are too big to be sets) can
only be treated indirectly. Specifically, Zermelo–Fraenkel set theory
does not allow
for the existence of a universal sets (a set containing all sets) nor
for unrestricted comprehensions , thereby avoiding Russell's paradox
Von-Neumann-Berneys – Godel Set
Theory(NBG) is a commonly used
conservative extension of Zermelo–Fraenkel set
theory that does allow explicit
treatment of proper classes.
Some comments related to ZFC : As we have seen that ZFC was formulated
to sort out the traditional issues
e.g. paradoxes existing
the erstwhile set theory . ZFC basically
tries to just remove the weed out the problematic aspects of
sets rather than providing the solution by going deep into the reasons behind
those problems.
As in this paper, where I have been resolving those problems/paradoxes
etc. in space-time, we can have
entirely new aspects of “Set” in Space-time rathr than just disallowing the
problematic parts of Set in ZFC. The
problems arising was basically
because of the lack of hidden time dimension and absoluteness of Set. The Space –time Relativistic Set doesn’t have those issues
any more.
ZFC Problem for Real Numbers and Godel Incompleteness Theorems
A similar phenomenon occurs in
formalized theories that are able to refer to their own syntax, in Zermelo-Frenkel-Set
Theory (ZFC). Say that a formula φ(x) defines a real number if there is exactly one real number r such that φ(r)
holds. Then it is not possible to define,
by ZFC, the set of all Godel formulas that define real numbers. For, if it were possible to define this set, it would be
possible to diagonalize over it to produce a new definition of a real number, following the outline of Richard's
paradox above. Note that the set of
formulas that define real numbers may exist, as a set F; the limitation of ZFC
is that there is not any formula that
defines F without reference to other sets. This is related Tarski indefinability theorem.
Space-Time Resolution :
In the Space-time view, when we have a new view about Real Numbers, Set ,the above constraint can also be resolved for real
numbers as well. (which is not possible in ZFC). We have earlier demonstrated
a new Space-time view to look at Set
theory which resolves the fundamental constraints of paradoxes rather eliminating them to formulate ZFC.
The given we have shown the Countability and Finiteness of Real Numbers at any moment time and
also Set in Space-Time, Godel Incompleteness Theorems
would be applicable to Real Numbers as well in Space-TIme
I have just though
of the following idea which I am just
writing in Cantor’s theory
Higher Order of 0 or
Infinitesimals Derived from in
Cantor’s theory
In Cantor
framework as there
are different levels
of Infinity, there are also
different levels of 0.
By taking T as the Set
of all possible binary digits. Put 0 before all so that all the sequences come after the
decimal part. A New system,
Now Just
like Cantor constructed Cantor Diagonalization using the latest sequence,
Similarly, the New sequence can be constructed after Decimal which is not in the list.. That shows even there
are different levels of Infinitesimals( or 0)
just like Infinity
In other words,just take the reciprocal of the different
orders of Infinity.
Conclusion :
In this paper, we looked
at the foundational mathematical
entities e.g. Real Numbers,Set in Space-Time Relativistic
perspective(making it in line with Physical
world realities) to resolve major paradoxes e.g. Richard,Russell,Skolem
, Liar Paradox,ZFC,The Concept of
Infiniteness to Cantor Diagonalization to show the Countability snd Finiteness of Real Numbers .
The above results derived
and analyzed would lead to the resolution of many existing
issues lying the at the foundation of mathematics in future.
Moreover one
key implications to mention is that Even Human Languages e.g. English,
French, Hindi etc could be fundamentally incomplete or inconsistent
structurally that would have huge day to day life implications to communicate
with each other !!
References :
[1] Gregory Chaitin Paper
“ How Real are the Real Numbers
?”
https://arxiv.org/abs/math/0411418
[2] Max Tegmark “Computable Universe Hypothesis.” https://arxiv.org/abs/0704.0646
[3]
https://mathworld.wolfram.com/CantorDiagonalMethod.html
[4] https://mathworld.wolfram.com/RichardsParadox.html
[5]
https://www.dpmms.cam.ac.uk/~wtg10/richardsparadox.html
[6]
https://mathworld.wolfram.com/GoedelsFirstIncompletenessTheorem.html
[7]
https://plato.stanford.edu/entries/goedel-incompleteness/
[8]
Wikipedia
