On Unprecise Numbers

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DOI: 10.6084/m9.figshare.29066705

Introduction

This paper, On Unprecise Numbers, was intended as a second offshoot of Gödel Incompleteness and Turing Completeness. My aim was to extend its linguistic viewpoint to numbers. However, the final version is more a complementary than a derivative work.

While writing it, I had to change my mind because I was wrong. My initial idea was to show that the unprecise numbers were the complete numbers, where a complete number is any that requires a complete language to be defined. I was also surprised when I realized that number M is computable.

Syntax and semantics

The equation of Turing completeness is: $$\exists\,{\cal U},\forall{\cal P},\forall{\frak d}:\; \overleftarrow{{\cal U}\langle\,{\frak p}|\vec{\frak d}\,\rangle} = {\cal P}\langle{\frak d}\rangle \;.$$

In the equation, $\langle\,{\frak p}|\vec{\frak d}\,\rangle$ is the computation that applies coded data $\vec{\frak d}$ to program $\frak p$, where $\frak p$ is the Turing machine $\cal P$ coded in the complete language $\cal L$ implemented by the universal Turing machine $\cal U$. For the equation of Turing completeness to work, the string $\langle\,{\frak p}|\vec{\frak d}\,\rangle$ has to be a well-formed formula of language $\cal L$. In other words, the syntax of $\cal L$ has to be decidable, because otherwise the equation of Turing completeness does not work.

Since the well-formed formulas, or sentences, of the complete language $\cal L$ are computations, then the semantics of $\cal L$ expresses computations. And since the halting theorem says that

Therefore, the Corollary to the halting theorem says that

Paradoxes

As a consequence of the Corollary to the halting theorem, there are paradoxes in every complete language. A paradox is a non-halting computation, that is, a well-formed formula, or sentence, of a complete language $\cal L$ the semantics of which we cannot reach because it closes an infinite loop.

The halting theorem can now be stated as follows:

Last paragraph

Let me conclude answering our first question: Why do not we use, in practice, any uncomputable number? Because, being under the law of Post, we cannot conceptualize what is not computable. In figurative language, what is uncomputable is in our conceptual blind spot. While paradoxes show that our conceptual blind spot is not precise, so the unprecise numbers are in its twilight, the blind spot itself provides another argument for the law of Post.

References

Link to the page of my paper on “On Unprecise Numbers” in figshare, and direct link to the pdf file.

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