On Strongly-equitable Social Welfare Orders Without the Axiom of Choice
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918161288265728 |
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| author | Serafin, Luke |
| author_facet | Serafin, Luke |
| contents | Social welfare orders seek to combine the disparate preferences of an infinite sequence of generations into a single, societal preference order in some reasonably-equitable way. In [2] Dubey and Laguzzi study a type of social welfare order which they call SEA, for strongly equitable and (finitely) anonymous. They prove that the existence of a SEA order implies the existence of a set of reals which does not have the Baire property, and observe that a nonprincipal ultrafilter over $\mathbb{N}$ can be used to construct a SEA order. Questions arising in their work include whether the existence of a SEA order implies the existence of either a set of real numbers which is not Lebesgue-measurable or of a nonprincipal ultrafilter over $\mathbb{N}$. We answer both these questions, the solution to the second using the techniques of geometric set theory as set out by Larson and Zapletal in [11]. The outcome is that the existence of a SEA order does imply the existence of a set of reals which is not Lebesgue-measurable, and does not imply the existence of a nonprincipal ultrafilter on $\mathbb{N}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_08684 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Strongly-equitable Social Welfare Orders Without the Axiom of Choice Serafin, Luke Logic 03E05, 03E75, 91B15 Social welfare orders seek to combine the disparate preferences of an infinite sequence of generations into a single, societal preference order in some reasonably-equitable way. In [2] Dubey and Laguzzi study a type of social welfare order which they call SEA, for strongly equitable and (finitely) anonymous. They prove that the existence of a SEA order implies the existence of a set of reals which does not have the Baire property, and observe that a nonprincipal ultrafilter over $\mathbb{N}$ can be used to construct a SEA order. Questions arising in their work include whether the existence of a SEA order implies the existence of either a set of real numbers which is not Lebesgue-measurable or of a nonprincipal ultrafilter over $\mathbb{N}$. We answer both these questions, the solution to the second using the techniques of geometric set theory as set out by Larson and Zapletal in [11]. The outcome is that the existence of a SEA order does imply the existence of a set of reals which is not Lebesgue-measurable, and does not imply the existence of a nonprincipal ultrafilter on $\mathbb{N}$. |
| title | On Strongly-equitable Social Welfare Orders Without the Axiom of Choice |
| topic | Logic 03E05, 03E75, 91B15 |
| url | https://arxiv.org/abs/2406.08684 |