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Recently, I read about MU puzzle: https://en.wikipedia.org/wiki/MU_puzzle

It is said about MU puzzle that:

It can be interpreted as an analogy for a formal system — an encapsulation of mathematical and logical concepts using symbols. The MI string is akin to a single axiom, and the four transformation rules are akin to rules of inference.

So, isn't there any basic rule of interference that would make it possible to transform MI to MU?

https://en.wikipedia.org/wiki/Rule_of_inference

What would it make to a system of logic, if we would accept a simple rule:

  1. xI → xU (Replace I after M with U)

or any other rule that makes the puzzle solvable?

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The puzzle is unsolvable with the four stated trasnformation rules (that acts as inference rules of the formal system).

See the Wiki page linked and the proof of its unsolvability.

Obviously, following your proposal, if we add a new transformation rule we can solve the problem, but now the problem is linked to a different formal system.

See Formal system :

A formal system is used to infer theorems from axioms according to a set of rules. These rules used to carry out the inference of theorems from axioms are known as the logical calculus of the formal system.

Thus, the $\text {MU}$ puzzle is expressed with reference to the very simple formal system with only one axiom : the string $\text {MI}$, and the four original rules of inference (called "transformation rules").

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  • $\begingroup$ I quess, the main amazement of mine is that what other useful information about formal systems MU puzzle gives besides that some formal systems, based on their own axioms, can solve certain theorems that some doesn't? Does MU puzzle make apparent that there will always be some theorems that the formal system cannot solve whatever axioms and rules it obeys? $\endgroup$ – MarkokraM Apr 29 at 8:34

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