Hyperreal differentiation with an idempotent ultrafilter
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910708540637184 |
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| author | Alexander, Samuel Allen Dawson, Bryan |
| author_facet | Alexander, Samuel Allen Dawson, Bryan |
| contents | In the hyperreals constructed using a free ultrafilter on R, where [f] is the hyperreal represented by f:R->R, it is tempting to define a derivative operator by [f]'=[f'], but unfortunately this is not generally well-defined. We show that if the ultrafilter in question is idempotent and contains (0,epsilon) for arbitrarily small real epsilon then the desired derivative operator is well-defined for all f such that [f'] exists. We also introduce a hyperreal variation of the derivative from finite calculus, and show that it has surprising relationships to the standard derivative. We give an alternate proof, and strengthened version of, Hindman's theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14689 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hyperreal differentiation with an idempotent ultrafilter Alexander, Samuel Allen Dawson, Bryan Logic 26E35, 26A24 In the hyperreals constructed using a free ultrafilter on R, where [f] is the hyperreal represented by f:R->R, it is tempting to define a derivative operator by [f]'=[f'], but unfortunately this is not generally well-defined. We show that if the ultrafilter in question is idempotent and contains (0,epsilon) for arbitrarily small real epsilon then the desired derivative operator is well-defined for all f such that [f'] exists. We also introduce a hyperreal variation of the derivative from finite calculus, and show that it has surprising relationships to the standard derivative. We give an alternate proof, and strengthened version of, Hindman's theorem. |
| title | Hyperreal differentiation with an idempotent ultrafilter |
| topic | Logic 26E35, 26A24 |
| url | https://arxiv.org/abs/2411.14689 |