> countably infinite
Countably infinite means you can count to any particular element of a set in finite time, even though there is no end to the pattern of elements in the set.
You cannot count to infinity plus one in finite time.
[1] https://en.wikipedia.org/wiki/Transfinite_number
[2] https://en.wikipedia.org/wiki/Ordinal_number#Ordinals_extend...
For example, uncountable sets can exhibit the property you mention: you count to x via the (very) finite sequence {x} for any element x of the set of real numbers.
Conversely, countable sets (even if you restrict such enumerations to a certain ordering) don’t in general have this property: ordering the integers by [x < y iff x < y (mod 2) or x < y if x = y (mod 2)] (that is, taking two isomorphic copies of the integers, identified with the even and odd integers respectively, juxtaposed side by side), the sequence {0,2,4,6, …} fails to arrive at 1 no matter how far you count along. So there’s an element we can’t count to in ‘finite time’.
The property you seem to be alluding to, that ‘the set admits a total order such that every element can be reached in finitely many steps starting at the first element’ actually corresponds to the order type of the natural numbers — often called ω. It’s countable, but there are many, many (in fact, uncountably many) more sets with this cardinality that don’t have that property.
The proper definition of ‘countably infinite’ is simply that the set can be placed in bijection with the natural numbers. This has nothing (directly) to do with ordering.
Does it not imply that there exists an ordering which reaches each value in finite time? That's how I'd straightforwardly read GP's comment. A given countable set under its 'natural' ordering might not have order type ω, but there will always exist some alternative ordering with type ω.
(And indeed, the existence of such an ordering implies that the set is countable.)
So it actually does work as a definition, if you phrase it properly.
I guess it's just equivalent to the observation that ω has the property that every initial segment (i.e. every subset of the form {0, 1, ..., n}) is finite.
So you could say: a set S is countable if it admits an ordering that makes it 'covered by all finite initial segments'.
...but this boils down to saying the set admits an ordering that makes it isomorphic to some n, or ω... which just means it's countable in the usual sense.
The natural numbers are of course countably infinite, but so are even numbers, prime numbers, fractions, algebraic numbers and lots of other kinds of numbers that people want to do arithmetic with. "Uncountably infinite" sets can't be put into a one to one correspondence with the naturals, no matter how you produce to produce such a mapping, it's always possible to produce a number in the set which wasn't part of it. Real numbers being the prototypical example of that, but the power set (the set of all the subsets) of the naturals is, also.
People do arithmetic with all kinds of things that "aren't numbers". Math isn't restricted to working with numbers.