What is the difference between "infinite" and "transfinite"? I have never properly got my head round exactly what the difference is between "transfinite" and "infinite".
Wikipedia says: transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite.
Huh?
For example, the set of all natural numbers $\mathbb N$ is "infinite" in cardinality, in fact "countably infinite" -- but its cardinal $\aleph_0$ and the ordinal $\omega$ which is the "order type" of $\mathbb N$ are defined as being "transfinite".
The rest of that wikipedia article on "transfinite number" does not help too much, except to explain that Cantor coined the term "transfinite" as a sweetener, so to speak, so as to make the medicine of his work go down easier.
But apart from historical reasons to do with battling with hidebound modes of thought (and I am familiar with Cantor's difficulties with Kronecker and those of his way of thinking), does there exist a concrete definition that one can go to that says: "this is what transfinite means: ... and this is what infinite means: ... and the difference between the two is ..."?
 A: No, there is no such definition.  The term "transfinite" is just not used at all as a technical term in modern mathematics.  It is used in a couple fixed phrases: "transfinite induction" and "transfinite recursion", which refer to induction or recursion that is indexed by a general well-ordered set (or more generally, a set with a well-founded relation) rather than just ordinary induction on the natural numbers.  But the term "transfinite" on its own has no standard precise meaning, and is rarely used outside these two phrases.  To the extent that it is used in other contexts, it is generally connotes something similar to those phrases: something involving well-ordered sets (typically, ones that are longer than the natural numbers).
A: Infinite simply means "not finite", both in the colloquial sense and in the technical sense (where we first define the term "finite"). There is no technical definition that I am aware of for "transfinite".
Nevertheless, I can attest to my personal use. Transfinite is good when there is a notion of order, so "transfinite ordinal", or when you want to talk about non-standard real numbers which are larger than all the standard natural numbers (in the context of non-standard analysis, that is), then "transfinite" is clearer than "infinite".
The reason being, especially in the non-standard analysis case, that "infinite number" is sort of awkward and can make people think about $\infty$ or infinite cardinals somehow, which may be giving the wrong impression. But "transfinite number" sends, to me, a somewhat clearer message that there is a particular context in which the term takes place.
A: When Cantor first outlined his theory of transfinite numbers, he wanted to stress that there are indeed distinct numbers beyond the finite numbers. He was clear that there are numbers that measure infinite size (infinite cardinal numbers) as well as numbers that measure infinite (well) orderings (infinite ordinal numbers). Cantor did not define these numbers out of intellectual curiosity, but because they provided new proof techniques, especially in the subject that we now call set-theoretic topology. For example, if a set is thought of as comprising branches (sequences) of a tree with a root, and if a branch is called "isolated" if there is a node of the branches beyond which there are no other branches, then by iteratively removing isolated branches from a tree any finite number of times, we see that a set comprises a countable set of branches and a remainder set (which could be empty). In the case of a set of real numbers with all of its limit points (a closed set), Cantor showed that the remainder set is a set of limit points of the same size as the set of real numbers (called a "perfect" set). The technique can be generalised to sets where branches transfinite sequences and (dropping the use of trees) to metric spaces and certain topological spaces. For further reading on Cantor's mathematics I would recommend the classic books by J. Dauben and M. Hallett, and for a readable take on what would now be called descriptive set theory, F. Hausdorff's Set Theory (from the 1930s).
A: I give another interpretation on the differences between "infinite" and "transfinite". Note that the following propositions involve no Axiom of Choice.
Since the word "infinite" means "not finite", we begin from the definition of "finite". Since we can prove
Lemma 1 For any set $S$, there is a bijection from $S$ into $n$ for some natural number $n$ if and only if there is an injection from $S$ into $n$ for some natural number $n$. (Although the left implication is not very easy, the proposition is indeed true.)
we define "finite" and "infinite" as follows.
Definition 2 Suppose $S$ is a set.
(1) $S$ is finite, if there there is an injection from $S$ into $n$ for some natural number $n$;
(2) $S$ is infinite, otherwise, i.e., there is no injection from $S$ into $n$ for any natural number $n$.
Since the word "transfinite" means "beyond finite" (note that the prefix "trans" means "beyond") which is equivalent to "bigger than or equal to all the finite natural numbers", we define "transfinite" as follows.
Definition 3 Suppose $S$ is a set. $S$ is transfinite, if there is an injection from $n$ into $S$ for any natural number $n$.
Note that someone may define "transfinite" the same as "Dedekind infinite" which is an abuse of words in my opinion.
With some effort, we can prove the following theorem.
Theorem 4 Suppose $S$ is a set. Then $S$ is infinite if and only if $S$ is transfinite.
This means "infinite" and "transfinite" are the same in comparing the size of sets. But are "infinite" and "transfinite" the same in other cases? Let's first consider the standard $\leq$ relation in textbooks about set theory.
Since we can prove
Lemma 5 For any natural number $n$ and ordinal $\alpha$, we have
(1) there is some injection from $\alpha$ into $n$ if and only if $\alpha\leq n$;
(2) there is some injection from $n$ into $\alpha$ if and only if $n\leq \alpha$.
we could define "infinite" and "transfinite" on ordinals as follows.
Definition 6 Suppose $\alpha$ is an ordinal.
(1) $\alpha$ is infinite if $\neg(\alpha\leq n)$ for any natural number $n$;
(2) $\alpha$ is transfinite if $n\leq\alpha$ for any natural number $n$.
Clearly $\alpha$ is infinite if and only if $\alpha$ is transfinite. But note that it is based on the fact that $\leq$ is trichotomous, i.e., for any ordinals $\alpha,\beta$ either $\alpha\leq\beta$ or $\beta\leq\alpha$. So generally for another relation $R$ which is not trichotomous, it's clear that "$\alpha$ is infinite with respect to $R$" is not equivalent to "$\alpha$ is transfinite with respect to $R$". Now the difference between "infinite" and "transfinite" comes out.
SUMMARY The words "infinite" and "transfinite" are the same in comparing the size of sets, while not the same in comparing some other relations which are not trichotomous.
