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June 19

Surreal numbers

Where can I find a proof of the statement on the page surreal numbers that every ordered field is a subfield of the Field of surreals? 76.67.73.4 (talk) 18:28, 19 June 2011 (UTC)[reply]

If you follow the references listed in the article you will see that "In the original formulation using von Neumann–Bernays–Gödel set theory, the surreals form a proper class, rather than a set, so the term field is not precisely correct; where this distinction is important, some authors use Field or FIELD to refer to a proper class that has the arithmetic properties of a field. One can obtain a true field by limiting the construction to a Grothendieck universe, yielding a set with the cardinality of some strongly inaccessible cardinal, or by using a form of set theory in which constructions by transfinite recursion stop at some countable ordinal such as epsilon nought". I'd start following some of those links to begin with. — Fly by Night (talk) 18:37, 19 June 2011 (UTC)[reply]

I noticed that footnote; I even linked to it in my original question. None of the links seem very relevant, however. 76.67.73.4 (talk) 20:40, 19 June 2011 (UTC)[reply]

Ah, so you did. My apologies. I assumed that a blue field would link to the field article; which I didn't need to read. Sorry about that. — Fly by Night (talk) 20:42, 19 June 2011 (UTC)[reply]

Please discuss any question of conflict of interest without disclosing personal information. User:Fred Bauder Talk 02:04, 20 June 2011 (UTC)[reply]

Can't you just build an embedding inductively? Start by mapping 1 to 1, extend the map to the arithmetic closure, then the real closure. At every subsequent step, grab an element which is transcendental over the domain, map it to an appropriate element of the surreals, take the arithmetic closure and then the real closure. At limits, take unions. Conway's Simplicity Theorem justifies mapping transcendentals and taking real closures.--Antendren (talk) 02:32, 20 June 2011 (UTC)[reply]