Timeline for Are gaps and loopy games interchangeable in the Surreal Numbers?
Current License: CC BY-SA 4.0
8 events
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| Mar 24 at 11:29 | comment | added | Farran Khawaja | @StevenStadnicki The loopy games that are 'number-comparable' are the ones that are only an element of either the left or right options, not both. So for example ∞ = {N|∞} is a valid loopy number in this construction but something like A={A|A} is not valid. | |
| Mar 24 at 1:30 | comment | added | Joel David Hamkins | Also, for nonArchimedean ordered fields, one usually considers not the Dedekind completion (since this is not generally a field, even in the set case), but the Dedekindean completion, which considers only the bridgeable gaps, which can be traversed by arbitrarily small elements of the field. The Dedekindean completion of an ordered field is Dedekindean complete. However, Gödel-Bernays does not prove that the surreal numbers have a Dedekindean completion. | |
| Mar 24 at 1:29 | comment | added | Joel David Hamkins | It seems to me that any discussion of gaps in the surreal line will be taking place not in set theory, but in class theory, but you haven't specified what your class theory context is, whether Gödel-Bernays or Kelley-Morse or full second-order logic. | |
| Mar 23 at 22:58 | history | edited | Gerry Myerson | CC BY-SA 4.0 |
typo
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| Mar 23 at 22:13 | comment | added | Steven Stadnicki | First issue I can think of: how do you restrict your loopy games to those that are 'number-comparable'? And as to why the Dedekind construction is preferred, a large part of it is that the construction is well-known; it's a canonical method of constructing a complete linear order from any suitable linear order, and since $No$ is a linear order it makes sense to look at the 'standard' completion of it. | |
| Mar 23 at 21:41 | history | edited | Farran Khawaja | CC BY-SA 4.0 |
Fixed formatting
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| S Mar 23 at 21:36 | review | First questions | |||
| Mar 23 at 23:22 | |||||
| S Mar 23 at 21:36 | history | asked | Farran Khawaja | CC BY-SA 4.0 |