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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 ..."?

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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).

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  • $\begingroup$ This is sort of what I was expecting (and hoping) to learn. I will bear it in mind the next time I read of "transfinite" concepts in (metaphorically hushed reverent tones. $\endgroup$ – Prime Mover Jun 6 at 19:15
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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.

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  • $\begingroup$ To me "transfinite" strongly connotes concepts related to ordinals, so I find it a poor choice in contexts like nonstandard analysis where it risks contributing to misconceptions people have that hyperreals have some particular connection to infinite ordinals. $\endgroup$ – Eric Wofsey Jun 6 at 16:35
  • $\begingroup$ "Infinite number" (and hence "transfinite number") is really a bit of a solecism anyway -- what is usually meant is "infinite set", "infinite ordinal" (which is sort of the same thing) and "infinite cardinal", and (in the context of analysis, real and complex) "(the) point at infinity". Referring to a "number" as infinite or transfinite smacks to me of miscomprehension and conflation of concepts. Just my 2c worth. $\endgroup$ – Prime Mover Jun 6 at 19:12
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    $\begingroup$ @PrimeMover: How would you refer to a positive non-standard real number which is strictly larger than all the standard reals? "Co-infinitesimal number"? "NSRLTASR"? $\endgroup$ – Asaf Karagila Jun 6 at 19:17
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    $\begingroup$ @PrimeMover: Non-standard analysis is not analysis with "a point at infinity". It might be wise to look certain terms before commenting, trust me, I've made these mistakes repeatedly and often, and I still do. $\endgroup$ – Asaf Karagila Jun 6 at 19:28
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    $\begingroup$ @PrimeMover: Yes, because a formalization of Newton and Leibniz original approach to calculus using infinitesimal is just deplorable. You might want to dial down the offensive parts. I have nothing more to add to this conversation. $\endgroup$ – Asaf Karagila Jun 6 at 20:24

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