Talk:Initial Segment of Ordinals under Lexicographic Order

Does the recent edit explain the connection between the two statements? --Andrew Salmon 23:41, 27 July 2012 (UTC)

Not without considerable background work, which needs linking to in order for it to make sense. Examples:

1. It is taken that "Le" is a "relation on ordered pairs of ordinals", but this is defined nebulously. A relation is defined on ProofWiki as a subset of the cartesian product of two sets. In this context (and on the page Definition:Lexicographic Order, where the subsection needs to be in a separate subpage) those two sets need to be specified as here it is not clear. Presumably it's either $\left({\operatorname{On}, \operatorname{On}}\right)$ or $\operatorname{On} \times \operatorname{On}$. Which or neither? The concepts are confusing at this level of abstraction.
2. When defining a relational structure (i.e. a set with a relation imposed), the notation "$\left({S, \preceq}\right)$" or whatever is used so as to indicate (a) the set and (b) the relation itself. At this stage I am not clear what the relation symbol is, unless the object under discussion is $\left({\operatorname{On} \times \operatorname{On}, \operatorname{Le}}\right)$ (but see above). From the context this is implied, but it is far from straightforward to determine.
3. It is understood that the initial segment of $\left({1, \varnothing}\right)$ is the set of all elements of $\operatorname{On} \times \operatorname{On}$ (but see above) that come before $\left({1, \varnothing}\right)$. It would be useful to indicate what form such elements take. Presumably $\left({\varnothing, x}\right)$ where $x$ is any element of $\operatorname{On}$.
4. The notation $F:\operatorname{On} \to (\operatorname{On} \times \operatorname{On})_{(1,\varnothing)}$ has not been defined anywhere on ProofWiki and needs to be explained.

A more subtle point is that it is not advisable to use the symbols for numbers, e.g. $1$ in this context, without indicating what sort of object it is being used to mean. If it's the ordinal one (apologies for that page not being better organised, this area needs to be tidied up and explained better) then that needs to be stated. Admittedly it is implicit from the context, but (as I'm sure you will have gathered by now from reading around the website) the philosophy of this site (which I am fighting tooth-and-nail to maintain) is that every page ought to be (to as great an extent as possible) understandable independently of any previous knowledge - particularly when what is "expected" to be understood concerns a particular style of notation or exposition --prime mover 08:07, 28 July 2012 (UTC)