This one is pretty slick, though not as neat as the one for integers:
Define f(x) = \frac{1}{2 \lfloor{x}\rfloor - x + 1}. Then 0, f(0), f(f(0)), f(f(f(0)),... is the nonegative rationals. No repeats or the thing would loop !
On my view all formalisms are parasitic on natural language, and therefore one will never surpass natural language by way of a formalism.* The two Incompleteness Theorems apply intuitively to natural language, and therefore they cannot fail to apply to that which is parasitic upon natural language. If this is right then Gödel’s theorems should be entirely uncontroversial. The interesting thing is that he managed to prove them within certain formal systems (because some might assume that the Incompleteness Theorems were themselves unprovable truths within those systems). Beyond that, they caused an additional stir at the time because a group of logicians had actually convinced themselves that they had conjured up a formalism that qualitatively surpassed natural language (as opposed to merely condensing ideas of natural language into logical shorthands and semantic silos).
With that said, mathematics represents a strange middle ground between natural language and a formal system. In particular, it is unclear whether natural language is prior to mathematics or whether mathematics is prior to natural language, and in what way. Put differently, there are important ways in which mathematics is not a formal system at all, and what this means is that there is an important difference between an attempt at a systematic formalization of mathematics, and mathematics itself. There are few other places where it is so easy to equivocate between natural and formal (i.e. between analogical and univocal, for the difference of a formalism lies in the semantic stipulation that artificially generates a way of thinking that lacks shades of grey).
* Incidentally, this is why the Medievals were not interested in formalisms per se.
Incompleteness theorems don’t directly apply to Natural Language (NL), because
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NL doesn’t have a primitive recursive axiomatization: it is impossible to deterministically assign a godel number to each and every valid NL sentence via an effective procedure, due to NL having a fluid, ambiguous and open ended grammar, such that the syntactical correctness of a sentence is vague, subjective and decided contextually on a per-use basis in an an-hoc fashion.
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Even if Natural language was p.r. axiomatized, it would be syntactically inconsistent, e.g. because “This statement is False” is generally considered to be a syntactically valid sentence.
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Semantics in NL isn’t aloof, static and established via acausal platonic rules of metaphysical correspondence to a Tarskian model, but directly determined on a case by case basis in terms of the cut and thrust of speakers interacting with their worlds. Hence semantic completeness doesn’t make sense for NL.
Incompleteness ultimately concerns the unbounded roughness of the fractal known as “formal arithmetic”.
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