Quoting the wiki:- a set A is Dedekind-infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. -.

There is a also a categorical definition of Dedekind infinite object, which runs as follows:

-An object $A$ in a category $C$ is Dedekind infinite if there a monomorphism $j:A\rightarrow A$ which is not an iso.-

Now, my question is:

is there some suitable category $C$ over some ground "category of sets" $E$ (for instance a topos with a NNO) and a Dedekind infinite object $A$ in $C$, whose image is finite (in Dedekind's sense, or even in the standard sense of being in a bijiection with a finite number) in $E$?

Basically, I am after some gadget that "looks" (Dedekind) infinite inside of $C$ but not "in reality", ie in the ground category $E$.

Why I am interested: assuming that such animals exists somewhere in the vast world of mathematics, they would possibly be good candidates for mirroring Cantor's paradise inside the finite world.


It seems reasonable to think that a topos $C$ defined over (i.e., equipped with a geometric morphism to) the category $E$ of sets counts as a "suitable category $C$ over some ground "category of sets" $E$." Then presumably the word "image" in the question would refer to the forward-part of the geometric morphism, i.e., the "global sections" functor Hom$(1,-)$ from $C$ to $E$. With this interpretation, you get the following example (and lots more like it). Take $C$ to be the topos of $G$-sets where $G$ is your favorite non-trivial group. Take $A$ to be the disjoint union of infinitely many copies, say indexed by the natural numbers, of the regular action of $G$ on itself. This is Dedekind-infinite because you can shift the $n$-th copy to the $(n+1)$-st. But its set of global sections (i.e., $G$-fixed points) is not only finite but empty.

This sort of example is unlikely to be what you wanted. So could you please clarify how I should interpret "suitable category," "over," and "image"?

  • $\begingroup$ My question was admittedly quite sloppy, so your comment is most welcome, it helps focusing. My problem with G-sets is that someone living inside would know immediately that A is infinite (G-set has the obvious NNO, someone would realize that A is at least countable). I guess what I am after is a category C which forgets in the subcat of finite sets of E, and an object A which cannot be recognized as infinite inside. However, I do not insist that C be a topos itself, so your answer can be salvaged by carving out some subcat of G-set, containing the finite G-sets and your dedekind-finite A $\endgroup$ – Mirco A. Mannucci May 26 '11 at 21:09

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