# On Numbers and God

Arbtirary thoughts on nearly everything from a modernist poet, structural mathematician and functional programmer.

## Saturday, April 10, 2010

### Interesting notes on Dedekind

I'm reading through Dedekind's The Nature and Meaning of Numbers(as translated by W. Beman), an early treatment of set theory. I find the following convention interesting:

A system [set] $A$ is said to be part of a system $S$ when every element of $A$ is also element of $S$. Since this relation between a system $A$ and a system $S$ will occur continually in what follows, we shall express it briefly by the symbol $A\subset S$. The inverse symbol $S\superset A$, by which the same fact might be expressed, for simplicity and clearness I shall wholly avoid, but for lack of a better word, I shall sometimes say $S$ is whole of $A$ [$S$ contains $A$], by which I mean to express that among elements of $S$ are found all the elements of $A$. Since further every element $s$ of a system $S$by (2) can be regarded as a system, we can hereafter emply the notation $s\subset S$.

(The bold is mine, and the symbol used by Dedekind is not, in fact, $\subset$, but the same symbol is used throughout. The [...] is also my own clarification.)

The question is, of course: is he confusing the two notions $A\subset S$ and $s\in S$, or is he just abusing notation? Considering the context, I doubt the latter, so it would appear he is confusing the two notions. On the other hand, his reasoning seems to be clear throughout, and points (1) and (2) (the text of which I will not force upon you), seem to suggest that he well understands the difference between the idea of "system" and "thing" (as he puts it), that I find the first alternative likewise hard to accept. Although perhaps not, as (2) may provide the source of his confusion. He says "For uniformity of expression it is also advantageous to include the special case where a system $S$ consists of a single (one and only one) element $a$, i.e., the thing $a$ is element of $S$, but every thing different from $a$ is not an element of $S$." This seems to suggest that that the confusion is not elementhood versus subsethood (forgive the abuse of the English language...), but rather, $a$ and $\{a\}$. Either way, it's fascinating, and it seems that this confusion (if that's what it is), does not pop up in the rest of the text.

Another interesting point is that I see the first (that I know of) use of a few common words and notations. A few that come to mind:

• The word identity to mean what is commonly meant in the mathematical community. That is
The simples transformation of [function from] a system is that by which each element of its elements is transformed into itself; it will be called the identical transformation of the system.

• . for composition: "This transformation [the composition] can be denoted briefly by the symbol $\psi .\phi$ or $\psi\phi$." This same paragraph has the first proof I've seen that sets with functions forms a category... albeit, not in those words, and as his set theory is naive, it is not technically correct (as it is not even a consistent system!)

There also appears to be (at least) two flawed proofs.
He shows first that $f(A)\subset f(B)\Rightarrow A\subset B$, and from this concludes that $f(\cap_{i\in I} A_i) = \cap_{i\in I} f(A_i)$. (You can show the first is false by taking some $s\in S\setminus B$, and mapping it into $f(B)$. Then $A=\{s\}$ is a counterexample. Actually, the second is quite correct, assuming the first statement...) Although again, I'm not quite certain: This theorem appears where he is discussing bijections (which he calls "similar transformations"), which might lead one to believe he means only in the case of bijective functions, but in every other theorem in the section he is clear to point out if the function is supposed to be bijective. Further, it appears every function is a priori surjective up until the next section.

While I'm picking apart such a crucial text, I might as well continue complaining: the translation is also infuriating at times as it translates phrases such as dann ist $A$ and dann gibt es as "then is $A$" and "then is there", rather than the more natural "then $A$ is" and "then there is". I do like Miltonic inversion, but this is hardly poetic writing...