Gödel Numbering in Discrete Systems

Shawn

This is one of my rare threads in the Logic and Philosophy of Mathematics section. Please be aware that I am not a professional academic talking about logic; but, rather a layman trying to better understand the intricacies of logic and, in this case, Gödel numbering in discrete ordered systems. With that said, I look forward to some understanding if my terminology is incorrect, and if my point is ambiguous, or in other words, please bear with my shortcomings in logic.

I have tried to understand Gödel numbering in systems of existing logic. I made a point about Gödel's Incompleteness Theorem, saying the following;

Gödel's incompleteness theorem applies to formal languages with countable alphabets. So it does not rule out the possibility that the alphabet can add Gödel numbering to account for new variables and even rules, or according to Douglas Hofstadter's GEB, 'isomorphisms,' within the system itself.

I would like to expand on the type of isomorphisms I have in mind. Now, would it be possible to translate the information that one would utilize from Gödel numbering into a isomorphism, such as something so simple as algebra or linear algebra? I say this because I find it perplexing whether certain logical systems can be compressed, or according to my understanding - rather, a more concise format of information being encoded. I can't provide any examples at the moment; but, I would like to ask the interlocutor whether they think that by creating an isomorphism between a discrete system, such as the color encoding of RGB, into algebra or linear algebra, then can one achieve a more concise way (if not already done so) to encode enumeration of logical processes, in terms of decidability?

Sorry if what I might be saying is unclear, and I hope any questions can clarify and vagueness or ambiguity.

Thanks,
Shawn

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Count Timothy von Icarus

Not sure if this is what you had in mind, but in computation there are multiple systems that are computationally universal/Turing complete (e.g. Turing Machines, cellular automata, lambda calculus, etc.). So, when it comes to Kolmogorov Complexity, a measure of "algorithmic information," this measure is said to be equivalent to the shortest program that will output a string.

For example, ababababab has a short description, "print ab 5 times," whereas "a04bfsp" would necessarily have a longer description. But the languages do differ in meaningful ways, and one way is in determining the length of a program. It's an open question how best to deal with this, but the variance from switching languages is bounded (the invariance theorem). There are a ton of benefits to looking at how these systems are similar, and also how they are different (e.g. parallel processing and concurrent systems that allow for execution "out of order," the domain of process calculus.)

So that's one sort of example. And there is a real world application in that, when you try to turn the formalizing into a real computer, some systems are faster, or "more concise."

TonesInDeepFreeze

The basic subjects of the original post deserve to be stated clearly:

(1) Godel-Rosser is a conditional.

The antecedent is: T is a formal, consistent theory for a certain amount of arithmetic. (The 'certain amount of arithmetic' is best understood in context of each particular theory.) (And every formal theory has only countably many symbols in its language.)

The consequent is: T is incomplete.

So it's about theories in languages. I don't know what is meant by "discrete ordered systems" [emphasis added].

(2) Godel numbering assigns unique numbers to the symbols, formulas and finite sequence of formulas and in such a way that we can mechanically recover the symbol, formula or finite sequences of formulas from its Godel number. It's quite easily understandable, though it depends most saliently on the fundamental theorem of arithmetic.

And yes, if we add countably many symbols, we can devise another Godel numbering.

(3) An isomorphism is a 1-1 homomorphism. (Informally, a homomorphism is a structure preserving function.)

Starting here, the original post is quite unclear to me:

(4) "[...] add Gödel numbering to account for new [...] rules"

I don't know what that means.

Then the rest of the post is even more unclear.

ssu

I would like to expand on the type of isomorphisms I have in mind. Now, would it be possible to translate the information that one would utilize from Gödel numbering into a isomorphism, such as something so simple as algebra or linear algebra? — Shawn

Here's a dubious remark from a similar amateur math-logics fan. I think that the in algebra the similar problems do arise. After all, similar undecidability result rose in the simple question of computablity. You have to have extremely simple math (that limits out basically normal math) to not have Gödel's results taking effect.

Shawn

↪Count Timothy von Icarus

Hey, thanks for responding to my post.

What I had in mind in Gödel numbering, was whether by doing so in terms of a isomorphism, such as linear algebra, whether the Kolmogorov complexity would possibly be lower.

I wanted to give a simple example with the color encoding mechanism behind RGB (red, green, blue) on monitor screens or TV screens. The only problem with this example is that I don't know whether Kolmogorov complexity was established in how RGB is usually encoded, so there might be no point to doing whatever I'm saying to begin with.

Anyway, the hypothesis was that by Gödel numbering each term (RGB) in linear algebra, then the very richness of matrix operators and other logical operators, could contribute to a more "concise" format to encode the complexity of different color schemes from RGB, onto a monitor or TV set.

Anyway, thanks for posting, and I don't have much more to say.

Shawn

Also, given some deliberation, in response to what TonesInDeepFreeze said, I'm not entirely sure if what I am attempting on doing (creating an isomorphism between three terms (R,G,B) in a Gödel numbering schematic to linear algebra is possible). By doing so, I would hope to reduce the amount of information by the logical operators utilized in linear algebra to encode the information required to display color on a screen, as a trivial example.

TonesInDeepFreeze

↪Shawn

Perhaps there are other people who understand what you're saying or asking, but I don't.

It would help if you specified in already understood mathematical terms exactly what you mean, step by step.

Starting with your posts today, what do you mean by "Godel numbering [...] in terms of an isomorphism such as linear algebra"?

I know what Godel numbering is. I know what an isomorphism is. And I know what the subject of linear algebra is basically about. But it would be for you to exactly, in already understood mathematical terms, say what you mean by "Godel numbering in terms of an isomorphism such as linear algebra".

Start with "Godel numbering in terms of an isomorphism".

Shawn

Start with "Godel numbering in terms of an isomorphism". — TonesInDeepFreeze

Sure, I can try. Given three terms, R, G, B denoting "red, green, blue." I want to create an isomorphism in the domain of linear algebra with those terms, and by doing so, doing the things only linear algebra allows with those terms... Before I go any further, is what I'm saying possible?

TonesInDeepFreeze

As I said, an isomorphism is a 1-1 structure preserving function.

What is the domain and range of the function you wish to adduce. What is the relation on the domain you wish to preserve?

And you use yet another term, "domain of linear algebra", for which you provide no definition. Is that a term you have read somewhere, or just something you've made up on the fly here?

And you didn't address my question: What do you mean by "Godel numbering in terms of an isomorphism?" I mentioned what a Godel numbering is, and I provided the definition of 'isomorphism'. But what you mean by "a Godel numbering in terms of an isomorphism" is unknown to anyone but yourself.

Shawn

↪TonesInDeepFreeze

In the OP, I already stated what needed to be said about my abilities in the domain of logic. Seemingly you missed that part as you keep on demanding some kind of formalism from me, which unfortunately I can't provide. I clarified later that I had only one interest in Gödel numbering, mainly in utilizing the same logic found in linear algebra, specifically operators with matrices. My assumption being that by doing so, you could concise the amount of information required for the readout of the logic needed to ensure the lowest amount of information needed through the logical operators in matrixes for decoding RGB encoding on the matrix logical operator.

If there's nothing fruitful to do with such logic, then sorry for wasting your time.

TonesInDeepFreeze

↪Shawn

I understood from your first post that you are not well versed in logic. So I have offered information that would help you, have pointed out where your locutions are not understandable, have asked questions whose answers might allow an understanding, and gave suggestions for proceeding methodically with clearly defined terminology, all in a sincere attempt to find out what you have in mind in this subject and thus to possibly guide you to insights on the subject, indeed as you invited questions to relieve the vagueness.

Now your latest post is even more undefined usage. If you're not interested in proceeding step by step with the concepts, then at least it is appropriate for me to remark that I cannot make heads nor tails of such things as "Godel numbering, mainly in utilizing the same logic found in linear algebra, specifically operators with matrices. My assumption being that by doing so, you could concise the amount of information required for the readout of the logic needed to ensure the lowest amount of information needed through the logical operators in matrixes for decoding RGB encoding on the matrix logical operator" even though I do know what Godel numbering, linear algebra and operations with matrices are.

Shawn

↪TonesInDeepFreeze

Yeah, I don't think I'm making much sense. So, I'm dropping this thread. Sorry for wasting your time, et al.

TonesInDeepFreeze

↪Shawn

You could study the subject more to learn how to express your idea in an understandable way. But, of course, that would be a matter of how you spend your own time.

Gödel Numbering in Discrete Systems

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