Coding is hard
में बचाया:
| में प्रकाशित: | arXiv.org (Sep 6, 2024), p. n/a |
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| मुख्य लेखक: | |
| प्रकाशित: |
Cornell University Library, arXiv.org
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| विषय: | |
| ऑनलाइन पहुंच: | Citation/Abstract Full text outside of ProQuest |
| टैग: |
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MARC
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| 001 | 3102584699 | ||
| 003 | UK-CbPIL | ||
| 022 | |a 2331-8422 | ||
| 035 | |a 3102584699 | ||
| 045 | 0 | |b d20240906 | |
| 100 | 1 | |a Sanders, Sam | |
| 245 | 1 | |a Coding is hard | |
| 260 | |b Cornell University Library, arXiv.org |c Sep 6, 2024 | ||
| 513 | |a Working Paper | ||
| 520 | 3 | |a A central topic in mathematical logic is the classification of theorems from mathematics in hierarchies according to their logical strength. Ideally, the place of a theorem in a hierarchy does not depend on the representation (aka coding) used. In this paper, we show that the standard representation of compact metric spaces in second-order arithmetic has a profound effect. To this end, we study basic theorems for such spaces like a continuous function has a supremum and a countable set has measure zero. We show that these and similar third-order statements imply at least Feferman's highly non-constructive projection principle, and even full second-order arithmetic or countable choice in some cases. When formulated with representations (aka codes), the associated second-order theorems are provable in rather weak fragments of second-order arithmetic. Thus, we arrive at the slogan that coding compact metric spaces in the language of second-order arithmetic can be as hard as second-order arithmetic or countable choice. We believe every mathematician should be aware of this slogan, as central foundational topics in mathematics make use of the standard second-order representation of compact metric spaces. In the process of collecting evidence for the above slogan, we establish a number of equivalences involving Feferman's projection principle and countable choice. We also study generalisations to fourth-order arithmetic and beyond with similar-but-stronger results. | |
| 653 | |a Mathematical logic | ||
| 653 | |a Arithmetic coding | ||
| 653 | |a Theorems | ||
| 653 | |a Metric space | ||
| 653 | |a Arithmetic | ||
| 653 | |a Hierarchies | ||
| 653 | |a Representations | ||
| 773 | 0 | |t arXiv.org |g (Sep 6, 2024), p. n/a | |
| 786 | 0 | |d ProQuest |t Engineering Database | |
| 856 | 4 | 1 | |3 Citation/Abstract |u https://www.proquest.com/docview/3102584699/abstract/embedded/7BTGNMKEMPT1V9Z2?source=fedsrch |
| 856 | 4 | 0 | |3 Full text outside of ProQuest |u http://arxiv.org/abs/2409.04562 |