Thoralf Skolem in *Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre* considers also a relativization of the notion finite. 


Skolem's article is published also in Skolem, T. A.: Selected works in logic, Fenstad, J. E. (ed.) Oslo, Scandinavian University Press, 1970, pp. 137-152. 


As from p. 143 Skolem has:


"Sogar die Begriffe "endlich", "unendlich", "einfach unendliche Reihe" usw. werden nur relativ innerhalb der axiomatischen Mengenlehre. Eine Menge M soll nach der Dedekindschen Definition endlich sein wenn sie keiner echten Teilmenge ihrer selbst ähnlich ist. Die Gültigkeit der Axiome verhindert aber nicht, dass erstens Teile von M definierbar sein können, welche nicht Teilmengen sind, und zweitens auch nicht, dass Zuordnungen definiert werden können, die nich Abbildungen, d.h. "Mengen" von Paaren sind. Es is deshalb sehr wohl möglich, dass inerhalb eines Bereiches B, für den dies Zermeloschen Axiome gültig sind, solche "endliche" Mengen im Dedekindschen Sinne existieren können." 


I unfortunately do not have access to an English version. But here is a [Translation from DeepL](https://www.deepl.com/en/translator#de/en/Sogar%20di%20Begriffe%20%22endlich%22%2C%20%22unendlich%22%2C%20%22einfach%20unendliche%20Reihe%22%20usw.%20werden%20nur%20relativ%20innerhalb%20der%20axiomatischen%20Mengenlehre.%20Eine%20Meinge%20M%20soll%20nach%20der%20Dedekindschen%20Definition%20endlich%20sein%20wenn%20sie%20keiner%20echten%20Teilmenge%20ihrer%20selbst%20%C3%A4hnlich%20ist.%20Die%20G%C3%BCltigkeit%20der%20Axiome%20verhindert%20aber%20nicht%2C%20dasserstens%20Teile%20von%20M%20definierbar%20sein%20k%C3%B6nnen%2C%20welche%20nicht%20Teilmengen%20sind%2C%20und%20zweitens%20auch%20nicht%2C%20dass%20Zuordnungen%20definiert%20werden%20k%C3%B6nnen%2C%20die%20nich%20Abbildungen%2C%20d.h.%20%22Mengen%22%20von%20Paaren%20sind.%20Es%20is%20deshalb%20sehr%20wohl%20m%C3%B6glich%2C%20dass%20inerhalb%20eines%20Bereiches%20B%2C%20f%C3%BCr%20den%20dies%20Zermeloschen%20Axiome%20g%C3%BCltig%20sind%2C%20solche%20%22endliche%22%20Mengen%20im%20Dedekindschen%20Sinne%20existieren%20k%C3%B6nnen).


Can someone explain Skolem's construction and its upshot?



Infinite but Dedekind-finite sets
I think what Skolem is saying amounts, in modern terminology, to the fact that a model of axiomatic set theory can contain a set M that is Dedekind-finite in the sense of the model yet "really" Dedekind-infinite (as seen from outside the model). The reason is that a bijection between M and a proper subset of M (witnessing that M is Dedekind-infinite) might exist in the real world but not be in the model. 


Skolem speaks first ("erstens") of the bijection's range not being in the model and then ("zweitens") of the bijection itself not being in the model. But of course if the bijection were in the model then its range would also be.


Also, where I've talked about things in the real world, Skolem talks about those things being definable ("definierbar"), but I don't think he intends definability in its current sense, which would require specifying a language and a semantics for it. As far as I can tell, when he says that something could be definable, he just means that it could exist. (I won't try to guess whether this conflation of definability and existence represents a philosophical opinion of Skolem's or a general terminological convention of a time before any mathematical analysis of definability.)


