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I've been thinking about what "mathematical model" can be used to model every possible thing (including itself).

Examples: a simple neuron network models a function but doesn't model an algorithm. A list of instructions models an algorithm but doesn't model relations between elements...

You might be thinking "maybe there is nothing that can model everything" but in reality "language" does model everything including itself. The issue is that it's not an organized model and it's not clear how to create it from scratch (e.g. if you will send it to aliens that don't have any common knowledge to start with).

So what is some possible formalization of a mathematical model that models every possible thought that can be communicated?

Edit 1:

The structure formalization I'm looking for has to have a few necessary properties:

  1. Hierarchical: the representation of ideas should rely on other ideas. (E.g. an programming function is a set of programming functions, the concept "bottle of water" is sum of two concepts "water" and a "bottle"...)
  2. Uniqueness of elements: When an idea uses in its definition another idea, it must refer to one specific idea, not recreate it each time. For example, when you think of a digit "9" and the digit "8", you notice that both have a small circle at the top, you don't recreate a new concept "circle" every time, instead, you use a fixed concept "circle" for everything. By contrast, a neural network might recreate the same branch for different inputs. So two representations of concepts must be different iff they have a difference.)
RationalFragile
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    Hi and welcome to this community! You should precisely define "everything", "thought" and "X models Y". Also, what do you mean by "in reality "language" does model everything including itself" – nbro Jan 03 '20 at 03:57
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    @nbro Your questions are helpful because I don't know what to explain more. "Everything" means thoughts, concepts, ideas and basically everything that can be expressed with language. For example, you can't explain what "the color red" is, so it doesn't count as a thought for my question. But here are examples of things I would like the same model to be able to represent: two objects have similar color or not (a relation), how to brush teeth (an algorithm), what can be considered a "3" (a classification)... Notice how I used language to make you think of those examples. – RationalFragile Jan 03 '20 at 10:23
  • Category Theory seems promising. There's a great free textbook Seven Sketches in Compositionality. Technically, anything can be a set, but Set Theory is mostly about cardinality and paradoxes, where Category Theory is more about functions and their relations, and more suitable for engineering. – DukeZhou Jan 03 '20 at 19:28
  • @RationalFragile I still have no idea of what you're looking for. What do you mean by "X models Y"? The language is just a means of communication in this case. – nbro Jan 03 '20 at 20:13
  • @DukeZhou I don't know almost anything about category theory, but, AFAIK, it is basically an abstract field. The only application I remember is related to functional programming languages (such as Haskell). – nbro Jan 03 '20 at 20:15
  • @nbro Applied Category Theory wiki. I'm just starting to dip my toe in, but it seems to have the potential to be applied to anything, including the humanities. (Borges wrote about the mathematical underpinnings of narrative, as an example. Been a while since I read his writing on that, but I recall it was rooted in set theory.) AGI would surely require a system for modeling even non-scientific abstraction, such as metaphors and ideas. Also following some mathematicians applying it to game theory. – DukeZhou Jan 03 '20 at 20:21
  • @nbro I'm not looking for something necessarily "applied". I am looking for an abstract way to represent anything. The point is, I want to design a system that can learn anything but if I can't even represent anything, it wouldn't be possible. "X models Y" means that if I never met you and I don't even know what language you speak, I can send you "X" when I want to communicate to you the idea of "Y". For example, I can send you a trained neural network to communicate to you a function. – RationalFragile Jan 03 '20 at 20:44
  • The only thing that comes to my mind now is AIXI. I still think that your question cannot be answered unambiguously. Maybe someone else will be able to answer it or maybe category theory is something you're looking for. – nbro Jan 03 '20 at 20:47
  • It is unclear to me how Category theory or AIXI represent things in a universal way. I have updated the question to mention a few necessary properties of the architecture I'm looking for. To clarify, what I look for is a "data structure" in a sense, but the data itself can represent anything. I can make one that will handle one type of things well, e.g. "action = action then action | action or action". This can be used to represent any complicated action but cannot for example represent a function (like a neural network does). So my goal is to unify all these hierarchies if that makes sense. – RationalFragile Jan 03 '20 at 22:40
  • Doesn't Gödel's incompleteness theorem prove that there is no formal system that can even completely describe itself? – George White Jan 04 '20 at 06:23
  • @GeorgeWhite I'm not quite sure about that. There is no truth value to the "ideas" in the sought after structure. (Being false or true is in itself an idea.) Besides, I am assuming that there will be a finite number of fixed elementary "ideas" such as "neuron is firing or not" but beside those, there will be an infinite number of combinations that each get to be called an "idea". Just like in language, there are no truth value to words but you can combine them to create any infinitely complex book. (You have to rely on the "elementary ideas" while learning the language at first...) – RationalFragile Jan 04 '20 at 10:05

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