I'm trying not to be offended, I know how spam filters work and sometimes someone gets wrongly accused and it just happens to be me. I'm just a little frustrated because I carefully craft this question for over an hour, then I get "this looks like spam" with absolutely no suggestion to make it look less like it.

My Notepad++ counts 913 "\w+" instances (so around 913 words), I use 2 links, 2 strongly marked phases ("Theorem 10." and "Here is my problem:"), 3 cursive parts (also short) and 5 lines beginning with > (quotes). Removing links and formatting hasn't helped. I could just shorten the question by leaving parts out and just edit them in again, but that seems cheap and is possibly unwanted.

Please tell me what I can do?

Edit: Original Post at https://pastebin.com/jvt9jPuD and below (first line was heading)

Is the inverse of surreal numbers actually well-defined?

J.Conway wrote in his book "On numbers and games" (1st edition, 1976) on p. 66

It seems to us, however, that mathematics has now reached the stage where formalization within some particular axiomatic set theory is irrelevant, even for foundational studies.

I happen to disagree. In fact, I'm having some fun formalizing mathematics in the Mizar system, which has just one article on Conway games yet. I want to find out if further formalization would be fruitful, i.e. if the operations defined in the book can be formalized at all. The cited article only defines $-x$ for a game $x$, but gives a pretty good idea how to deal with the highly inductive nature of games and I'm certain I could formalize addition and multiplication. What bothers me is the definition of $y=\frac{1}{x}$, given by $$y=\left\{\left.0,\frac{1+(x^R-x)y^L}{x^R},\frac{1+(x^L-x)y^R}{x^L}\right|\frac{1+(x^L-x)y^L}{x^L},\frac{1+(x^R-x)y^R}{x^R}\right\}$$ for $x$ positive and only positive $x^L$ considered, which is needed for defining division. Conway, after giving this definition on p.21, writes himself

Note that expressions involving $y^L$ and $y^R$ appear in the definition of $y$. It is this that requires us to "explain" the definition. The explanation is that we regard these parts of the definition as defining new options for $y$ in terms of old ones.

In a footnote, the rather trivial example of $\frac{1}{3}=\{0,\frac{1}{4},\frac{5}{16},\ldots|\frac{1}{2},\frac{3}{8},\ldots\}$ is given to show "how the definition works".

I can't see why $y$ is well defined in general. For example, given two uncountable cardinals $\alpha<\beta$ I'm having a hard time seeing how $\frac{1}{\beta-\alpha}$ should be computed. The emphasis here lies on "uncountable".

Claus Tøndering gave a seemingly equivalent definition of the inverse here on p.44 (if the definition should not be equivalent, please point out why). He defines $y$ through $y^L$ and $y^R$ as such: $$0\in y^L$$ $$z\in y^L \Rightarrow \tfrac{1+(x^R-x)z}{x^R}\in y^L, \tfrac{1+(x^L-x)z}{x^L}\in y^R$$ $$z\in y^R \Rightarrow \tfrac{1+(x^L-x)z}{x^L}\in y^L, \tfrac{1+(x^R-x)z}{x^R}\in y^R$$

This is still "too" recursive to be formalized. One of my problems is that I can't comprehend the cardinality of $y^L$ and $y^R$. I mean, I could define $y^L_0 = \{0\}, y^R_0 = \{\}$ and for $n\in\mathbb{N}, n>0$ change Tønderings definitions to $z\in y^L_{n-1} \Rightarrow \ldots\in y^L_n$ and so on (or better: $$y^L_n = \left\{\left.\tfrac{1+(x^R-x)z}{x^R}\right|z\in y^L_{n-1}\right\}\cup\left\{\left.\tfrac{1+(x^L-x)z}{x^L}\in y^L \right|z\in y^R_{n-1}\right\}$$ and $y^R$ analogue) and conjecture $$y^L = \{0\}\cup\bigcup_{n\in\mathbb{N}} y^L_n,\quad y^R = \bigcup_{n\in\mathbb{N}} y^R_n$$

Here is my problem: I really doubt I could prove that. First off, I'm having trouble believing the equality holds, that I could miss something by merely having a countable union. Secondly, $y^L$ and $y^R$ are required to be sets and I can image how they accidentally could become classes this way with some $x$ nefarious enough (maybe $x=\beta-\alpha$ is enough already?), because maybe the set generation process never stops at a certain day. I get couldn't information about this topic at all. In papers about surreal numbers either they are just given like here without further doubt or not explicitly given at all. Some papers, like these from Philip Ehrlich, go deeper into cardinality or other theories above my understanding, so if the issue would be resolved there, I wouldn't have noticed.

On the matter of the $y^L_n$ and $y^R_n$ being sets, Conway writes

Theorem 10. We have (i) $xy^L<1<xy^R$ for all $y^L,y^R$.
(ii) $y$ is a number. [(iii) and (iv) left out]
Proof. We observe that the options of $y$ are defined by formulae of the form $$y''=\frac{1+(x'-x)y'}{x'}$$ where $y''$ is an earlier option of $y$, and $x'$ some non-zero-option of $x$. This formula can be written $$1-xy'' = (1-xy')\frac{x'-x}{x'}$$ which shows that $y''$ satisfies (i) if $y'$ does. Plainly $0$ does. Part (ii) now follows, since we cannot have any inequality $y^L\geq y^R$. [...]

As far as my understanding goes, with "$y$ is a number" he means "if $y$ is a game, then it's a number", as this proof (directly following the remark after the definition) does not indicate the sethood of $y^L$ or $y_R$ in my eyes.

  • At least to help you with this specific case it would be probably useful to see the source of the post. Does the system allow you to add it to this question? If not, could you share it somewhere online (like pastebin or something similar)? – Martin Sleziak Dec 7 '17 at 13:11
  • @MartinSleziak should have thought of this myself @_@ – SK19 Dec 7 '17 at 13:13
  • 1
    @MartinSleziak Also added pastebin variant. Surprisingly, I triggered their spam detection, too. – SK19 Dec 7 '17 at 13:14
  • It is strange that I was able to post exactly the same question on Mathematics without any problem. (I have immediately deleted it - I have done this just as a test. I will add a link but it will work only for 10k+ users.) The only constructive suggestion I have is to post a part of the question and add rest in the edits (or other users might help you with editing). However, I am not sure whether this is going to work. – Martin Sleziak Dec 7 '17 at 13:18
  • Maybe we could continue this discussion (about experiments with the post) in chat: chat.stackexchange.com/transcript/9369/2017/12/7 (Just in order not to leave too many comments here.) – Martin Sleziak Dec 7 '17 at 13:18
  • Generally spam filters are more forgiving to "more established users". If you have another question, shorter, with perhaps less math to it, it might be worth trying to post it first, get some good reactions, and then try to post this one. – Asaf Karagila Dec 7 '17 at 13:23

By trimming the question down to half (until example with $\frac{1}{\beta-\alpha}$) I could post it and then edit it to full length without problem. The comments to my question indicated that this is legal behaviour here. I'm still wondering if it is a bad sign that the spam detector can be tricked like that, but as far as I'm concerned, I'm glad the question is out.

  • I do not know how the spam filters work either. One thing I've noticed in the post above is the (compared to the bulk of posts on MathOverflow) excessive use of "I". In general, I don't have a problem with this, but to be blunt, I am more interested in the math than the description of your involvement of it. That last clause is partly wrong, I AM interested in your journey, but the point is the spam filter may not be. Gerhard "Math Has Personality; Emphasize That" Paseman, 2017.12.07. – Gerhard Paseman Dec 7 '17 at 16:13

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