Godel Results Extended to Real World : Redevelopment of Mathematics in Real World
[04/09, 12:24] Pankaj: Imagine the whole of mathematics as a
huge sack, and inside are all the possible things math can do. It’s a mighty
big sack, indeed. What Gödel proved is that, first, there exists in this sack a
set of things which cannot be proven or disproven, such as axioms. Second,
there is no possible way to prove these axioms from within that sack. It’s
impossible for math, on its own, to prove its own axioms.
Essentially, it’s a problem of self-reference. It’s an issue
seen, too, in Russell’s paradox about sets. More famously, the liar paradox
imagines a sentence like, “This sentence is false.” When you examine it
closely, it creates a logical circularity. If the sentence is true, then it’s
false; but then if it’s false, it’s true. It’s enough to make a robot’s brain
explode
[04/09, 12:24] Pankaj: Gödel applied a similar logic to the
whole system of mathematics. He took the sentence, “This statement is
unproven,” and converted it into a number statement about numbers (with a code
system known as “Gödel numbering”). He discovered that this proposition cannot
be proven within that system.
Going even further than this, Gödel concluded that in every
system that’s rich enough to allow for arithmetic, there will be a proposition
within it that cannot be proven by it’s own tools. We need some kind of “meta
language” to prove the rules by which a system operates. It’s a bit like how we
can’t see our own eyes or draw around the hand that’s holding the pencil.
[04/09, 12:25] Pankaj: Infact this within and outside the
system would work for every non-conscious system not just arithmetics as
demonstrated by Godel...It’s fundamental characteristics of Universe
[04/09, 12:26] Pankaj: Natural Numbers are the abstract
concepts not proper nouns
https://bigthink.com/hard-science/kurt-godel-foundations-mathematics-unproven/#Echobox=1693633100
Gödel applied a similar logic to the whole system of
mathematics. He took the sentence, “This statement is unproven,” and converted
it into a number statement about numbers (with a code system known as “Gödel
numbering”). He discovered that this proposition cannot be proven within that
system.
Going even further than this, Gödel concluded that in every
system that’s rich enough to allow for arithmetic, there will be a proposition
within it that cannot be proven by it’s own tools. We need some kind of “meta
language” to prove the rules by which a system operates. It’s a bit like how we
can’t see our own eyes or draw around the hand that’s holding the pencil.
Human Language Incomplete/Inconsistent to describe the
Truth..
Everything is Quantum wave having duality...Existing
mathematics doesn't have duality...it's fixed .. Incompatibility
Godel : Our formal reasoning is incomplete and some Truths are unprovable.
Theory of Everything in Mathematics would never be there
based on formalism...there has to be complementary approach ..duality
[01/09, 11:55] Pankaj: The key question left unanswered by
Gödel: Is this an isolated phenomenon, or are there many important mathematical
truths that are unprovable?
“((This Statement)=B is false)” = A
This statement A is true
when B is false.
Paradox arises in Godel context because it’s considered A is
true of A is false.. This requires
higher order time/ relativistic analysis
which current mathematics is not able to. Mathematics needs to come 4D
Dimension to handle Context, ambiguity ..Even Human Language syntax is not
so....leading to miscommunication !! Important for even AI/ML
Godel Proof is originally from English Type Paradox
Any mathematical system can’t prove its own axioms on which
it has been built though true...not just Arithmetical system as shown by
Godel...Every math Equations or anything can be expressed in pure English human
language and then prove the result Universally...
The mathematical system /any language system not able to
handle self Referential paradoxes from Real World due to time and relativity
missing and hence context . Mathematics needs to extend its dimension in 4D
Time and Relativity compatible to Physics or Real World.
Unified Theory of Physics could be Incompatible with this mathematics
due to these issues.
Kurt Gödel’s incompleteness theorem demonstrates that
mathematics contains true statements that cannot be proved. His proof achieves
this by constructing paradoxical mathematical statements. To see how the proof
works, begin by considering the liar’s paradox: “This statement is false.” This
statement is true if and only if it is false, and therefore it is neither true
nor false.
Now let’s consider “This statement is unprovable.” If it is
provable, then we are proving a falsehood, which is extremely unpleasant and is
generally assumed to be impossible. The only alternative left is that this
statement is unprovable. Therefore, it is in fact both true and unprovable. Our
system of reasoning is incomplete, because some truths are unprovable.
Gödel’s proof assigns to each possible mathematical
statement a so-called Gödel number. These numbers provide a way to talk about
properties of the statements by talking about the numerical properties of very
large integers. Gödel uses his numbers to construct self-referential statements
analogous to the plain English paradox “This statement is unprovable.”
Strictly speaking, his proof does not show that mathematics
is incomplete. More precisely, it shows that individual formal axiomatic
mathematical theories fail to prove the true numerical statement “This
statement is unprovable.” These theories therefore cannot be “theories of
everything” for mathematics.
The key question left unanswered by Gödel: Is this an
isolated phenomenon, or are there many important mathematical truths that are
https://www.scientificamerican.com/article/what-is-goumldels-proof/
[25/08, 22:16] Pankaj: Godel Results : Inconsistency is due
to the lack of Time and Realtivity in
Mathematics -& Human aspects... Uncertainty in Mathematics
[25/08, 22:22] Pankaj: Truth is beyond probable : Godel
result
[25/08, 22:23] Pankaj: What mathematicians can prove depends
on their starting assumptions, not on any fundamental ground truth from which
all answers spring.
[25/08, 22:25] Pankaj: Undecidable questions have even
arisen in physics, suggesting that Gödelian incompleteness afflicts not just
math, but — in some ill-understood way — reality.
[25/08, 22:33] Pankaj: Godel would be true everywhere in the
universe(real numbers etc if involve
time and relativity in mathematics (or day to day language) to show that
ultimately everything is natural number in that context at a particular time
relatively
[25/08, 23:21] Pankaj: Because we can generate Gödel numbers
for all formulas, even false ones, we can talk sensibly about these formulas by
talking about their Gödel numbers
[25/08, 23:36] Pankaj: Godel results infact is linked with
Language which are converted into Godel Number
[25/08, 23:37] Pankaj: Like Computation of Language
[25/08, 23:38] Pankaj: You might think you could just posit
some extra axiom, use it to prove G, and resolve the paradox. But you can’t.
Gödel showed that the augmented axiomatic system will allow the construction of
a new, true formula Gʹ (according to a similar blueprint as before) that can’t
be proved within the new, augmented system. In striving for a complete
mathematical system, you can never catch your own tail
[25/08, 23:38] Pankaj: We’ve learned that if a set of axioms
is consistent, then it is incomplete. That’s Gödel’s first incompleteness
theorem. The second — that no set of axioms can prove its own consistency —
easily follows
[25/08, 23:40] Pankaj: What would it mean if a set of axioms
could prove it will never yield a contradiction? It would mean that there
exists a sequence of formulas built from these axioms that proves the formula
that means, metamathematically, “This set of axioms is consistent.” By the
first theorem, this set of axioms would then necessarily be incomplete.
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But “The set of axioms is incomplete” is the same as saying,
“There is a true formula that cannot be proved.” This statement is equivalent
to our formula G. And we know the axioms can’t prove G.
So Gödel has created a proof by contradiction: If a set of
axioms could prove its own consistency, then we would be able to prove G. But
we can’t. Therefore, no set of axioms can prove its own consistency.
[25/08, 23:40] Pankaj: Inconsistent means
Contradictory/Paradoxical Truth of Nature..
Incomplete..means Truth is beyond Probability
[25/08, 23:42] Pankaj: Infact this can be proven for all
beyond Natural Number to everything including Real World daily life
practically...if time and Relativity are applied in mathematics
[26/08, 00:19] Pankaj: To make it consistent..need
relativity and time perspective
[26/08, 00:25] Pankaj: So, Godel : Either Mathematics need
to evolve like real world physics or else this mathematics can’t be able to
formulate the consistently the real world physical truths...
Linked to Renormalization...issues ..that also shows
Fundamental Inconsistency
[26/08, 00:30] Pankaj: Hilbert : Formal Mathematics is
Certain....Godel showed Uncertainty in Mathematics hence can’t be
formalized...Turing... Algorithm can’t prove everything!! Undecidability
[26/08, 00:34] Pankaj: AI/ML math can’t deal with real world
Uncertainty
[26/08, 10:51] Pankaj: Kurt Gödel’s incompleteness theorem
demonstrates that mathematics contains true statements that cannot be proved
[26/08, 11:13] Pankaj: Self-referential systement Extremely
Important
[26/08, 11:52] Pankaj: Axiomatic system not able to look at
higher order logic leading to Contradictory
[26/08, 12:22] Pankaj: In physical reality, size of unit 1
keeps on changing the observer distance to see...but in conventional maths unit
1 remains constant.. Renormalization..
Renormalization
[26/08, 12:41] Pankaj: It shows Traditional Mathematics
can’t handle uncertainty, higher order logic etc...
The Problem is Mathematics is Deterministically designed based
on axioms but it is not able to handle the Uncertainty in its design, Like
Higher Order Paradoxical or Different arithmetic result like 1+1 = 2 & >
2 both at different times. These higher
aspects are not being managed by
the existing Mathematics what Humans do in Real World...leading to the
fundamental reason behind Godel Result...It’s Fundamental Results in Universe
avoid for every system. Say for example 1+1 = 2 in one context at time t relatively
and at other time 1+1 >2 at different time t relatively o/contextually. Now
if time and relativity are hidden, it will lead to Contradictory as happens in
the Normal Mathematics in context of Godel Results .
Therefore to make the
mathematics consistent, it has to involve time and Relativity of Observer’s
etc. And somewhere it is also true based on our life experiences that any language is insufficient to express
everything and also leads to Contradictory/Paradoxical ..
Infact Truth is Paradoxical as long as written in the form
of even Humans Language because of
Similar Godel type Constraints even for Languages(English, French, Hindi etc ).
That’s the reason. And the effort to write the Truth in Non Paradoxical way in the form of Language is
utopian . No Language can do this. That’s where Non verbal understanding of
Human and their Consciousness come to Play. One
can understand in mind but languages could be structurally Inconsistent to express that . It
has Social and Legal applications as well. For example People often have
communication gap, misunderstanding etc... This can be mathematically proved
when Time and Relativity are included as every time relatively, any infinity
would be finite .
Languages have to deal with Uncertainty because they have
been structured like Deterministic !!
Proof Theory also believes in Determinism but The Reality is
not so Straightforward.
Numbers are not absolute...They are relative. What is Real
measurement in one frame can be made Natural in other frame but yes the
assumption of Simultaneous Existence of Infinity needs to be removed and the
role of Time and Relativity matters in Real Day to Day World.. That’s the
fundamental change required how we look at the Mathematics. Theoretical
Mathematics can do anything . But I am talking about Real World Mathematics.
In Real Physical World, everything can be a finite natural /finite
number in some frame unlike those theoretical Real Numbers..
Godel Incompleteness Theorems in Real World not
Theoretical Mathematical World
Real World Mathematics
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