On the limits of comparing subset sizes within $\mathbb{N}$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916349389832192 |
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| author | Wenmackers, Sylvia |
| author_facet | Wenmackers, Sylvia |
| contents | We review and compare five ways of assigning totally ordered sizes to subsets of the natural numbers: cardinality, infinite lottery logic with mirror cardinalities, natural density, generalised density, and $α$-numerosity. Generalised densities and $α$-numerosities lack uniqueness, which can be traced to intangibles: objects that can be proven to exist in ZFC while no explicit example of them can be given. As a sixth and final formalism, we consider a recent proposal by \citet{Trlifajova:2024}, which we call c-numerosity. It is fully constructive and uniquely determined, but assigns merely partially ordered numerosity values. By relating all six formalisms to each other in terms of the underlying limit operations, we get a better sense of the intrinsic limitations in determining the sizes of subsets of $\mathbb{N}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_03344 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the limits of comparing subset sizes within $\mathbb{N}$ Wenmackers, Sylvia Logic 00A30, 03A05, 11B05, 03H05, 28E15, 03E25, 03F65 We review and compare five ways of assigning totally ordered sizes to subsets of the natural numbers: cardinality, infinite lottery logic with mirror cardinalities, natural density, generalised density, and $α$-numerosity. Generalised densities and $α$-numerosities lack uniqueness, which can be traced to intangibles: objects that can be proven to exist in ZFC while no explicit example of them can be given. As a sixth and final formalism, we consider a recent proposal by \citet{Trlifajova:2024}, which we call c-numerosity. It is fully constructive and uniquely determined, but assigns merely partially ordered numerosity values. By relating all six formalisms to each other in terms of the underlying limit operations, we get a better sense of the intrinsic limitations in determining the sizes of subsets of $\mathbb{N}$. |
| title | On the limits of comparing subset sizes within $\mathbb{N}$ |
| topic | Logic 00A30, 03A05, 11B05, 03H05, 28E15, 03E25, 03F65 |
| url | https://arxiv.org/abs/2408.03344 |