Talk:Infinite expression

glossary of mathematical jargon
This may be better off as an entry in such a glossary than a separate page. Tkuvho (talk) 20:28, 11 December 2010 (UTC)

How sound is the definition (first sentence)?
The capital-sigma form  $$\sum_{n=0}^\infty 1 / n! $$   is seen as an  'infinite expression'. But how about the expression 2 e +1  (realizing that the  e  is just shorthand writing for   1 + 1/1 + 1/2 + 1/6 + 1/24 + ··· )  ? Please comment on this.

The third sentence of the article, with two short comments added within brackets, reads as:      Examples of well-defined infinite expressions include infinite sums [linked to Series (mathematics) ], whether expressed        using summation notation or as an infinite series [defined as: infinite expression with pluses in between the arguments], .... With twice 'series' and twice 'infinite expression', this sentence is twofold circulating. --Hesselp (talk) 13:42, 3 May 2017 (UTC)


 * There is no definition of "infinite expression" in this article. And moreover, I never saw a definition in the literature. Some relevant quotes from Wikiproject Mathematics talk:
 * "an infinite tree labeled with symbols of various types" — I am able to turn this hint to a definition; I agree that our "Infinite expression" article gives only informal explanation (for non-mathematicians), not a definition (Boris Tsirelson 19:15, 12 May 2017);
 * It's tempting to try to define infinite expressions as infinite sequences of symbols (as non-infinite expressions are usually defined as finite sequences of symbols) but that turns out to be problematic, for instance because things that people would recognize as infinite expressions can have more than one end. That's why I hinted at using parse trees instead, in the quote you found. But I don't know of any source that actually formalizes them that way or any other... (David Eppstein 05:40, 13 May 2017);
 * Carl could know more... I only remember "Borel codes" (for Borel sets); but they are not widely known, and worse, they are usually defined just for the case, not as a special case of general "infinite expression". (Boris Tsirelson 06:08, 13 May 2017).
 * Boris Tsirelson (talk) 05:57, 14 May 2017 (UTC)

This article seems to confuse two different concepts: It is the second concept that the infinite parse tree idea is intended to capture. But I think we should more clearly distinguish these two concepts and, if they are both adequately supported by the mathematical literature, consider separating them into two different articles. —David Eppstein (talk) 06:17, 14 May 2017 (UTC)
 * Expressions that are written using only a finite sequence of symbols (no ellipses), but that represent processes involving infinitely many arithmetic operations (or that involve taking limits of finite approximations to these infinite processes. This includes, for instance, summation notation, but I think not integration.
 * Expressions that cannot be written down in full because their full expression would require infinitely many symbols. This includes sums and products written out in expanded form, infinite continued fractions, Viète's formula, etc.


 * Why not treat summation notation as just abbreviation for the corresponding parse tree with infinitely many children under this vertex (infinite branching)?
 * On the other hand, a parse tree may be well-founded (no infinite descent) or not. Borel codes are well-founded, this is why they are evaluated without any limiting procedure. They are infinite because of infinite branching; but infinite union or intersection does not need a limit. And they are far not simple; their hierarchy, parallel to the Borel hierarchy, involves all countable ordinals.
 * The problem is that all this seems to be OriginalResearch, until/unless someone finds a source... The first ref in the article (Helmer) is quite unsatisfactory. Boris Tsirelson (talk) 06:53, 14 May 2017 (UTC)
 * I try search for "infinitary formula" and "infinitary term"... with little success.
 * Borel codes are described (quite concisely) in book: A. S. Kechris, "Classical Descriptive Set Theory", Springer-Verlag, 1995 (see Sect. 35.B, pages 283-284). Also, in Sect. 3b of my lectures. Boris Tsirelson (talk) 09:30, 14 May 2017 (UTC)

Infinite or infinitary?
According to Wiktionary, "infinite" means "greater than any positive quantity or magnitude", while "infinitary" means "of or pertaining to expressions of infinite length". Really? Should this article be moved to "infinitary expression"? By the way, "infinite expression" exists outside mathematics, see infinite-expression.com. Boris Tsirelson (talk) 19:46, 14 May 2017 (UTC)