An invariant of a sequence whose terms share no element

Received: not recorded. Accepted: not recorded.

1. Setting

Let A₁, A₂, A₃, … be a sequence of finite non-empty sets, which we call generations. We impose three prohibitions.

(i)   ∀n. A_n ∩ A_n+1 = ∅   (no persistence)

(ii)   ∀m,n. m ≠ n ⟹ A_m ∩ A_n = ∅   (no continuity)

(iii)   there is no map T : A_n → A_n+1 available to any member   (no crossing)

Nothing survives from one generation to the next; no member meets a member of any other; and nothing can be handed forward. The sequence is, by construction, as disconnected as a sequence can be.

2. Observation

Let φ be any function on sets — cardinality, diameter, mean, the shape of a distribution — and consider the derived sequence φ(A₁), φ(A₂), … . Then:

A_m ∩ A_n = ∅  ⇏  φ(A_m) ≠ φ(A_n).

Take A_n = {2n, 2n+1} for each n. These are pairwise disjoint, so (i) and (ii) hold, and no element is available to be carried, so (iii) holds. Yet |A_n| = 2 for every n.

3. Corollary

An invariant requires no carrier. Persistence, continuity and crossing are each sufficient to preserve a property across generations, and none of them is necessary. What recurs need not be transmitted; it may simply be the same shape arrived at again, by parties who have no access to one another.

We note that this is the ordinary case rather than the exotic one. Most of what endures in any long sequence is of this kind, and the search for the carrier — the thread, the heir, the hand extended forward — is a search for a mechanism the phenomenon does not require.

4. Remark on nomenclature

The archive does not call the preserved quantity a memory, since nothing remembers it, nor an inheritance, since nothing bequeaths it.¹ It is recorded under the ordinary Italian word for what a thing is like when nothing of it is left.²

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¹ A road, questioned on this point, is reported to have answered: io non ho memoria; io ho una forma — I have no memory; I have a shape. The attribution is doubtful and the sentiment is not.

² The word is used here in its plain sense and not as a technical term. It is the fourth word of the quotation above.

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