In a recent Quanta article about the 9th Dedekind number, Dedekind is credited with the first formal definition of infinity.
Is this an accurate attribution?
Folklore where I’m from dictates that Georg Cantor is the father of modern infinity, but they were alive and working concurrently and I’m not familiar enough with either mathematician’s body of work to pin down when he first proposed a formal definition of infinity. Any assistance is appreciated.
Dedekind actually in effect gave two different definitions of infinity.
Namely, first, as is well known, a set is Dedekind infinite if it is equinumerous with a proper subset of itself.
But second, Dedekind proved the famous categoricity result for the natural numbers, showing that the natural-number structure of the successor operation $\langle\mathbb{N},0,S\rangle$ is characterized up to isomorphism by the properties (1) zero is not a sucessor; (2) the successor operation is one-to-one; and (3) every number is generated from zero by successor, in the sense that if $X$ is a set with $0\in X$ and $n\in X\implies Sn\in X$, then every number is in $X$. This (second-order) theory is now known as Dedekind arithmetic, and Dedekind showed how to undertake definition by recursion and thereby to develop all the usual number theory and arithmetic in this structure. Peano famously provided a hugely successful and quite elegant development of the theory on the basis of Dedekind's axioms, and these days the theory is often attributed to Peano, even though Peano credits Dedekind.
The point is that the categoricity result defines what it means to be finite — a set is (numerically) finite if it is equinumerous with the predecessors of a natural number. A set is infinite, accordingly, if it is not finite.
It seems to me that Dedekind probably believed that his two definitions were equivalent, and indeed they are provably equivalent in ZFC. Nevertheless, we now know that this requires the axiom of choice (countable choice suffices), and it is consistent with ZF that there are numerically infinite sets that are Dedekind finite. It is the second definition (numerically infinite) that is usually taken as the right choice in contemporary mathematics.
Finally, let me mention my essay, Equinumerosity and the definition of finiteness, in which I discuss Aristotle, Galileo, Frege, Dedekind, Cantor, Tarski and others on the meaning of finiteness. Further discussion in my essay Potential versus actual infinity, which is freely available, and which brings in Archimedes, as well as ultrafinitism and the contemporary modal analysis of potentialism.
The article is probably referring to Dedekind's Was sind und was sollen die Zahlen of 1888, in which point 64 is Dedekind's definition of infinite. This of course is after Cantor had been investigating various infinities for several years, though I'm not sure where Cantor made a formal definition.
The idea that infinite sets, if they existed, would admit 1-1 correspondences with their proper parts was well understood already in the 17th century, and were the reason such sets were rejected as contradictory. Dedekind's insight was to turn a contradiction into a definition. Cantor's work certainly prepared the ground for Dedekind turning the tables on the old contradiction.
In the preface of the second edition of Was sind und was sollen die Zahlen Dedekind mentions another definition of `finite': a set $S$ is called finite if there is a map $\varphi$ from $S$ to itself such that $S$ itself is the only non-empty set that is closed under $\varphi$.
He investigated this definition in Zweite definition (1889. 3. 9) des Endlichen und Unendlichen. In a note at the end Emmy Noether shows that this notion corresponds to what Joel calls (numerically) finite and that $\varphi$ is a cyclic permutation of $S$.
Aristotle gave two definitions - possibly three - and probably not 'formally' as you would like it. Nevertheless, they have definitions.
The infinite is described as something without end, ie it is not finite which is what the word itself implies and presumably where the word can be traced back to. And he distinguished two different cases - the potential infinite and the actual infinite. I suspect it is the latter that fed into Cantor's semi-mystical notion of the absolute infinite.
He also distinguished this from the All - of which I would write more if I could remember what he wrote. But it's been sometime since I read his Physics/Metaphysics.
