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Added
For a treatment in homotopy type theory see
- Dan Frumin, Herman Geuvers, Léon Gondelman, Niels van der Weide, Finite Sets in Homotopy Type Theory, (pdf)
Re-organised slightly. Added a brief introductory section, moved a couple of paragraphs of existing content into it. Removed the two context menus ’foundations’ and ’mathematics’ which I don’t think really fit here (the latter is arguably too general to be useful on any page).
Intend to add a new section in a subsequent edit.
Richard, maybe we can mention (in the “Viewing as schemes” section) that a finite coproduct of affine schemes $Spec R_i$, $i=1,\ldots,n$, is again affine, $Spec (R_1 \times \cdots \times R_n)$. Taking $R_i=\mathbb{Z}$, we can view the finite set $X$ as the (affine) scheme $Spec (\mathbb{Z}^X)$. This agrees with what you wrote, but seems more canonical.
Nice! Great if you can go ahead and make an edit if you have time, as I’ll be tied up until the evening European time!
Maybe keep the explicit description, but add the nice canonical one in addition?
Thanks for the edit!
Correction: not all finite subsets are decidable subsets. Any singleton in the real numbers is finite but generally not decidable unless the real numbers themselves have decidable equality. But finite subsets of sets with decidable equality are decidable subsets, and every finite set has decidable equality.
Owen Coyle
Thanks. I have touched the formatting and moved the item into chronological order (now here).
Also copied it to the author’s entries (at Arnaud Spiwack and Thierry Coquand).
Is $B(S)$ a decidable subset of $2^S$ if $S$ is finite? If so I think one could just replace “smallest subset” with “smallest decidable subset” in that definition.
Merged definition involving decidable subsets into the section “finiteness without infinity”.
Whoever added the section “finiteness predicatively without infinity” was probably thinking about the definition in Agda, because somebody was talking about this topic in the Univalent Agda discord server. And while Agda is predicative in that it doesn’t have propositional resizing, Agda isn’t predicative in that one still has universes $U_i$, and thus a $U_i$-large subobject classifier $Prop_i$, for which every impredicative definition still makes sense and is constructible but does not lie in $U_i$.
Also, the fact that the universes exist means that Agda does have an axiom of infinity, just not the one that we are usually used to via the second-order Peano axioms. Given a universe $U_i$, define that a type $A$ is finite if it satisfies the constructive definition without infinity but with $Prop_i$. Then you could collect all the types in $U_i$ which are finite using the dependent sum type $\sum_{A:U_i} isFinite(A)$. Then since one has $Prop_i$, one also has the impredicative definition of quotient sets through $Prop_i$, and if you quotient $\sum_{A:U_i} isFinite(A)$ by the mere existence of an equivalence, then you gets the natural numbers.
In set theory, imagine if you did not have an axiom of infinity or power sets, but you did have function sets and an internal model $V$ of finitist CZF. Then you could take the set of all subsingletons in $V$ and quotient it by existence of an isomorphism to get a set $\Omega_V$ of truth values that you could use in the definition of finite set. Then take the set of all finite sets in $V$ and quotient it by existence of an isomorphism to get a set $N_V$ of natural numbers.
Edwin Michaelson
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