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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You are asking the right question. Your intuition that something is missing is accurate. Gödel’s theorem essentially predicts it. Transitive operational level logic is not fundamental.
But are you prepared for the actual answer?
There is in fact an escape from Gödel’s theorem. I’ve spent 20 years peering into the matter myself, though not in search of getting behind the theorem. That was a only consequence of the discovery.
The 3 fundamental laws of logic are not independent axioms that are merely related in a transitive operational sense. In fact, they make no logical sense as a transitive relationship. Their relationship is coherent on a more fundamental level. And that level makes transitive operations like Gödel’s theorem possible.
Each of the 3 laws can only be defined in the presence of all 3 laws. So each law is a distinct expression of all 3.
Take the law of non-contradiction for example:
A thing cannot be A and -A at the same time or in the same sense.
Formulate it however you like, the result is the same. Formulation requires Identity, Distinction, and Exclusion.
The same structure applies to all laws. They are not individual axioms, but mutually dependent statements.
This fundamental level of logical structure makes every other level of logical and mathematical systems possible. It is the level of holism.
Furthermore, the 3 fundamental laws can be mathematically described and is fully coherent as a holism. I’ve included the python script below.
This hollistic structure is not caught in or bound by the transitive axiomatic logical loop that Gödel’s theorem describes. It is grounded in the objectivity of 3 necessary perspectives as expressed in the distinct laws.
In other words, the logical holism contains the minimum number of variables/witnesses required for objective verity.
I asked if you really wanted the answer to your valid and solid intuition, because the implications are staggering. It is not an answer many people will be comfortable with. We have encountered this 3 in 1 anomoly before in the Trinitarian theology stubbornly articulated by Jesus of Nazareth.
We couldn’t make transitive coherent sense of the relationship structure He claimed to embody and didn’t discover the mathematical category until 1900 years later.
Well, now we can make sense of it as a holism. Implications be damned. Python script below:
import numpy as np
Define the shared Relational Matrix
A = np.array([
[0, 1, 1, 1],
[1, 0, -1, -1],
[1, -1, 0, -1],
[1, -1, -1, 0]
], dtype=float)
Extract the isolated peripheral submatrix (Center removed)
A_isolated = A[1:, 1:]
print(f"1:1 Structural Isomorphism Verified: {np.array_equal(A, A.T)}“)
print(f"Full System Spectrum: {np.sort(np.linalg.eigvalsh(A))}”)
print(f"Isolated Submatrix Spectrum: {np.sort(np.linalg.eigvalsh(A_isolated))}“)
print(f"Full System Determinant: {round(np.linalg.det(A))}”)
print(f"Isolated Submatrix Determinant: {round(np.linalg.det(A_isolated))}")
Consciously or not, you are operating under the long held view that the 3 fundamental laws of logic are three separate, independent axioms. That then puts them into a transitive framework subject to Gödel’s incompleteness.
But this longheld assumption is false.
The laws of Identity, Non-Contradiction, and Excluded Middle are not independent axioms. They are co-inherent and mutually dependent. They are simultaneously and equally necessary components of each other, of each law’s distinctness.
This logical structure is the actual ground of all coherent thought, whether in mathematics or natural language. This hollistic level is fundamental to (and distict from) the transitive level where Incompleteness operates.
Because you’re treating logic as three disconnected rules, you’re forcing it into the very category that creates these incompleteness problems.That’s why you think natural language has no primitive recursive axiomatization. You’re caught in a transitive loop.
Once you see the laws as ontologically unified rather than independent axioms, the whole framework changes. Suddenly, the liars paradox is seen as the more obvious invalid transitive loop because it violates the law of non-contradiction.