ONAG: "Surreal Numbers" make a field?

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revslaughter
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ONAG: "Surreal Numbers" make a field?

Hi, I'm sorry to make this my first post - I love xkcd, and I didn't notice that there was a forum for it until today. It just so happens I'm in the middle of a medium-grade panic about Surreal Numbers. Looking around a bit, I noticed that some of you brought up Knuth and Conway in other threads, so it seems likely that some of you might know where I'm coming from.

I'm taking an individual studies course going through Conway's On Numbers and Games, and it's been a trip so far. I've been working on the "On Numbers" part of the book pretty much since the beginning of the semester, and I've run into a brick wall - divison. I can see easily how addition works in Conway's scheme. With some rectangle-drawing I can see how multiplying two numbers makes a number, though I can't exactly prove it yet...It's division that's giving me all sorts of headaches. I tried for an hour last night trying to find 1/2 by making the inverse of 2, and I couldn't begin to get rid of the "inductive" (?) options of the inverse. I can see how this is important in making a field, but must it be so complicated?

It seems that there are two inductions at work in his definition of division, and it's just been...frustrating. Anyone have any tips?
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Xanthir
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Re: ONAG: "Surreal Numbers" make a field?

I'll take a look at my copy of ONAG tonight and see if I can be of any help. Won't be able to post until tomorrow - I still haven't gotten power back after Hurricane Ike.
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jestingrabbit
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Re: ONAG: "Surreal Numbers" make a field?

Doesn't answer your question but wouldn't the surreal numbers have to be a set to be a field?
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Xanthir
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Re: ONAG: "Surreal Numbers" make a field?

The surreals do form a set. Unless there's some crazy definition of "set" involved here that I'm not aware of. I know they don't run into any of the self-reference issues.
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revslaughter
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Re: ONAG: "Surreal Numbers" make a field?

Xanthir wrote:I'll take a look at my copy of ONAG tonight and see if I can be of any help. Won't be able to post until tomorrow - I still haven't gotten power back after Hurricane Ike.

Thank you! Hope you get power back soon.
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jestingrabbit
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Re: ONAG: "Surreal Numbers" make a field?

Xanthir wrote:The surreals do form a set. Unless there's some crazy definition of "set" involved here that I'm not aware of. I know they don't run into any of the self-reference issues.

I'm pretty sure there is self reference going on.

http://sci.tech-archive.net/Archive/sci ... 04588.html

WP also specifically mentions the thing about them not forming a set, but instead a proper class, with a sentence that is like "they're a field but..."
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Re: ONAG: "Surreal Numbers" make a field?

::facepalm:: Blarg. Yes. The surreals are too large to form a set; they form a class instead.

They still form a Field, with the convention being that Field uses a class as it's basis (while a field uses a set).
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Re: ONAG: "Surreal Numbers" make a field?

From wikipedia:
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.
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revslaughter
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Re: ONAG: "Surreal Numbers" make a field?

Hmmm...I'm still rather stuck on how the surreals are closed under multiplication, class or no.

Does anybody have any advice? The proof given in ONAG is really cryptic.
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Com3t
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Re: ONAG: "Surreal Numbers" make a field?

I'm still rather stuck on how the surreals are closed under multiplication, class or no. Does anybody have any advice?

Theorem 8. (i) If x and y are numbers, so is xy. The proof is given in On Numbers and Games, by John H. Conway. What part of the proof is the sticking point?

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Re: ONAG: "Surreal Numbers" make a field?

Com3t wrote:
I'm still rather stuck on how the surreals are closed under multiplication, class or no. Does anybody have any advice?

Theorem 8. (i) If x and y are numbers, so is xy. The proof is given in On Numbers and Games, by John H. Conway. What part of the proof is the sticking point?

When they called the proof in ONAG cryptic, they were talking about the proof in On Numbers and Games.

So, they knew where the proof was.

Also, the previous post was like, 9 years ago.
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