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Two theorems stating inherent limitations of all but the most trivial formal systems for arithmetic of mathematical interest.
Originally posted by Incompleteness
Any TOE wll be a finite formal system.
Originally posted by Incompleteness
I agree but incompleteness doesn't imply a fractal like nature is present in nature.
But I disagree with your view that there exists more primitive axioms or systems that logic
But certain types of logic and mathematics are complete and thus sound to use without worrying about incompleteness like elementary geometry is complete and first order logic.
So logic can be proven using logic.
Gödel showed that within a rigidly logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that are undecidable or undemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can neither be proved or disproved. Hence one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions ... It appears to foredoom hope of mathematical certitude through use of the obvious methods. Perhaps doomed also, as a result, is the ideal of science - to devise a set of axioms from which all phenomena of the external world can be deduced.
All consistent axiomatic formulations of number theory include undecidable propositions ...
Gödel showed that provability is a weaker notion than truth, no matter what axiom system is involved ...
You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you'll only create a larger system with its own unprovable statements. The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.
It lets one wonder if our reality is not some transcendental object which can never be fully described but always allowing new things to be done within it.
But a TOE does. How else would it apply to everything, unless it was literally a single pattern (formula, etc.) that explains all phenomena?
But I disagree with your view that there exists more primitive axioms or systems that logic
at least as we know it, the physical universe I should say
not everything in existence HAS to be logical
Godel proved that any formal system that is powerful enough to be recursive can be forced to produce inconsistent results without breaking any rules. If you could make geometry recursive (I'm sure someone has) then you should be able to produce Godel's theorem with it. I'd actually like to see that myself. And any system that isn't powerful enough for recursion is just incomplete, like "elementary" geometry. You can have a closed system that is consistent, but it won't be very powerful and can't tell you much of anything about the universe.
And any system that isn't powerful enough for recursion is just incomplete, like "elementary" geometry
You can have a closed system that is consistent, but it won't be very powerful and can't tell you much of anything about the universe.
Douglas S. Robertson offers Conway's game of life as an example:[14] The underlying rules are simple and complete, but there are formally undecidable questions about the game's behaviors. Analogously, it may (or may not) be possible to completely state the underlying rules of physics with a finite number of well-defined laws, but there is little doubt that there are questions about the behavior of physical systems which are formally undecidable on the basis of those underlying laws. Since most physicists would consider the statement of the underlying rules to suffice as the definition of a "theory of everything", these researchers argue that Gödel's Theorem does not mean that a TOE cannot exist. On the other hand, the physicists invoking Gödel's Theorem appear, at least in some cases, to be referring not to the underlying rules.
Where do you get that?
First-order logic (FOL) is a formal deductive system used in mathematics, philosophy, linguistics, and computer science.A first-order theory consists of a set of axioms (usually finite or recursively enumerable) and the statements deducible from them given the underlying deducibility relation. Usually what is meant by 'first-order theory' is some set of axioms together with those of a complete (and sound) axiomatization of first-order logic, closed under the rules of FOL. (Any such system FOL will give rise to the same abstract deducibility relation, so we needn't have a fixed axiomatic system in mind.) A first-order language has sufficient expressive power to formalize two important mathematical theories: Zermelo–Fraenkel (ZFC) set theory and (first-order) Peano arithmetic. A first-order language cannot, however, categorically express the notion of countability even though it is expressible in the first-order theory ZFC under the intended interpretation of the symbolism of ZFC. Such ideas can be expressed categorically with second-order logic.
All consistent axiomatic formulations of number theory include undecidable propositions ... Gödel showed that provability is a weaker notion than truth, no matter what axiom system is involved ...
Originally posted by Incompleteness
Sorry made a typing error meant to say that I disagree with your view that there exists more primitive axioms or systems 'than' logic. Which refers to that logic can be used to prove logic. And logic is independent of subjectivity unless you are implying that 1 + 1 =3
Well there is a error in argument in assuming that elementary geometry is incomplete
In fact any consistent formal system that is non-recursive will be complete. For example basic addition is complete.
You can have a closed system that is consistent, but it won't be very powerful and can't tell you much of anything about the universe.
Again no. Since it is full and well possible that the TOE is complete and consistent.
I'm not sure you understood what I was talking about in my post. I have nothing to argue about. If you have personal opinions on what theories have shown, or what they haven't, etc., you are certainly entitled to them.
It's either incomplete or inconsistent, has to be one or the other, that's what Godel proved some decades ago.
1st Theorem (Syntactic Version): Assuming that the axiom system A has the following two properties: A is sufficiently strong to be a plausible candidate for an axiom system for arithmetic; in particular, it is at least strong enough to permit the derivation, by correct reasoning, of any true elementary arithmetic statement involving the operations of addition, multiplication and successor-taking, and incorporates the principle of natural induction; A is consistent - i.e. it is not possible to deduce from it a contradiction;
Addition is not complete. Do you really think all fields of science and engineering can be reduced to simple addition?
So unless Godel has been refuted and you're the only one that knows about it, I have to say you don't know what you're talking about.
All I have to say, is that if you think you have a formal, complete, and consistent system, produce it. You'll be the first, and believe me, much greater minds than yours or mine have been trying for decades.
Gödel, Escher, Bach: an Eternal Golden Braid (commonly GEB) is a Pulitzer Prize-winning book by Douglas Hofstadter, described by the author as "a metaphorical fugue on minds and machines in the spirit of Lewis Carroll".
On its surface, GEB examines logician Kurt Gödel, artist M. C. Escher and composer Johann Sebastian Bach, discussing common themes in their work and lives. At a deeper level, the book is a detailed and subtle exposition of concepts fundamental to mathematics, symmetry, and intelligence.
Through illustration and analysis, the book discusses how self-reference and formal rules allow systems to acquire meaning despite being made of "meaningless" elements. It also discusses what it means to communicate, how knowledge can be represented and stored, the methods and limitations of symbolic representation, and even the fundamental notion of "meaning" itself.
In response to confusion over the book's theme, Hofstadter has emphasized that GEB is not about mathematics, art, and music but rather about how cognition and thinking emerge from well-hidden neurological mechanisms. In the book, he presents an analogy about how the individual neurons of the brain coordinate to create a unified sense of a coherent mind by comparing it to the social organization displayed in a colony of ants.
en.wikipedia.org...
You said :
... that since we can use mathematics to describe every phenomenon ...
Cantor Dust
Also known as the Cantor set, possibly the first pure fractal ever found – by Georg Cantor around 1872. To produce Cantor Dust, start with a line segment, divide it in to three equal smaller segments, take out the middle one, and repeat this process indefinitely.
Although Cantor Dust is riddled with infinitely many gaps, it still contains uncountably many points. It has a fractal dimension of log 2/log 3, or approximately 0.631.
Source : DavidDarling.info
Cantor Dust
Source : Platonic Realms
by the definition of the incompleteness of the TOE that there will always be statements (when physically translated : experiments or observations) that are unprovable with the current TOE thus meaning no coherent form of determinism will ever be possible in such a universe.