What If Turing Had Preceded Gödel?
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929384369160192 |
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| author | Oberhoff, Sebastian |
| author_facet | Oberhoff, Sebastian |
| contents | The overarching theme of the following pages is that mathematical logic -- centered around the incompleteness theorems -- is first and foremost an investigation of $\textit{computation}$, not arithmetic. Guided by this intuition we will show the following.
* First, we'll all but eliminate the need for Gödel numbers.
* Next, we'll introduce a novel notational device for representable functions and walk through a condensed demonstration that Peano Arithmetic can represent every computable function. It has achieved Turing completeness.
* Continuing, we'll derive the Diagonal Lemma and First Incompleteness Theorem using significantly simplified proofs.
* Approaching the Second Incompleteness Theorem, we'll be able to use some self-referential trickery to avoid much of the technical morass surrounding it; arriving at three separate versions.
* Extending the analogy between the First Incompleteness Theorem and the Unsolvability of the Halting Problem produces an equivalent of the Nondeterministic Time Hierarchy Theorem from the field of computational complexity.
* Lastly, we'll briefly peer into the realm of the uncomputable by connecting our ideas to oracles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08494 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | What If Turing Had Preceded Gödel? Oberhoff, Sebastian Computational Complexity Logic in Computer Science 03F40 F.4.1 The overarching theme of the following pages is that mathematical logic -- centered around the incompleteness theorems -- is first and foremost an investigation of $\textit{computation}$, not arithmetic. Guided by this intuition we will show the following. * First, we'll all but eliminate the need for Gödel numbers. * Next, we'll introduce a novel notational device for representable functions and walk through a condensed demonstration that Peano Arithmetic can represent every computable function. It has achieved Turing completeness. * Continuing, we'll derive the Diagonal Lemma and First Incompleteness Theorem using significantly simplified proofs. * Approaching the Second Incompleteness Theorem, we'll be able to use some self-referential trickery to avoid much of the technical morass surrounding it; arriving at three separate versions. * Extending the analogy between the First Incompleteness Theorem and the Unsolvability of the Halting Problem produces an equivalent of the Nondeterministic Time Hierarchy Theorem from the field of computational complexity. * Lastly, we'll briefly peer into the realm of the uncomputable by connecting our ideas to oracles. |
| title | What If Turing Had Preceded Gödel? |
| topic | Computational Complexity Logic in Computer Science 03F40 F.4.1 |
| url | https://arxiv.org/abs/2406.08494 |