I'm trying to understand ordinal numbers in Lipschutz, Set Theory.
The author has correctly explained what is a limit element.
A previous example given is that , in the set
{ 1,3,5.... ; 2,4,6,8...}
2 is a limit element ( having no immediate predecessor, without being the first element).
The second example given is "omega" = the ordinal of the set of counting numbers.
My question : what makes of "omega" a limit element in the class of ordinal numbers?
ADD (The Sympathizer, UE+1556.957 Ms): The "prior example" referenced with set-like notation was that mentioned in this post:
This notation is from an unnamed third party source and is meant to represent the set of positive natural numbers well-ordered with the odd numbers first, followed by the even numbers:
$$1 < 3 < 5 < 7 < \cdots < 2 < 4 < 6 < 8 < \cdots$$
with ordinality $2\omega$.
Because, topologically, $\omega$ is the limit point of the preceding ordinals, i.e. the natural numbers.
A "limit point" of a set is a point which can be "approximated as close as we like" by elements of that set. This, of course, requires us to have a notion of "approximation", which is what topology does, and orders, like those on the ordinals, can be considered as a source of such because you can consider the approximation of a point by an interval: if you have a point $p$, and you can bracket it in $(a, b)$, i.e. $a < p < b$, then you can say $(a, b)$ is like a "confidence interval", as in taking measurements: e.g. if I say I'm around 169-171 cm tall, that's an approximation with a 2 cm interval. If you now make $a$ and $b$ "closer", i.e. have an interval $(a', b')$ such that $a < a'$ and $b' < b$, and yet $p$ is still contained therein, then the second interval approximates better.
Likewise, if we have a subset, $S$ of some ordered set, here (a suitable initial segment of (**)) the ordinals, we can say that that subset has as limit point $l$ - which need not be in it, but must be in the larger ordered set - if in every, open interval $(a, b)$ which contains $l$, there is also a point from $S$. That point is an "approximation", and the intuition is that the smaller the interval, the closer that approximation is thereto, and you can always find an approximation from $S$ no matter how small you make the interval.
In that regard, $\omega$ has this property with respect to $\mathbb{N}$ because if we have any open interval $(a, b)$ of ordinals containing $\omega$, then we must of course $a < \omega < b$, but that first inequality means $a$ must be a natural since there is nothing before $\omega$ but naturals. Hence there is a natural one larger (its successor) and such "approximates" $\omega$(*) and, moreover, we can, by making $a$ larger, get better approximations thereof. Thus $\omega$ is a limit point and hence this is why the name "limit ordinal". Similar considerations apply to all other limit ordinals.
(*) The notion of something being "approximately infinite", despite not making strict sense, is very, very useful, but can also be a poison pill if you're not careful: if something is really huge, but finite, compared to something else that is at a much smaller scale we happen to be interested in, it so often happens that taking the "really huge" thing as actually infinite often makes things a lot easier. One example is from physics, where you can, say, model a capacitor made from two parallel metal sheets very accurately, provided they aren't too far apart, as being made from infinite sheets. A misuse of this notion is in treating the Earth as infinite when you have 7.5 billion people covering it all demanding more and more profligate use of natural resources :) There you're in territory where the approximation breaks down as the scales are now comparable.
(**) The ordinals are too numerous to be collected into a set without creating a logical contradiction, at least in typical conceptions of set theory.