# Well founded induction attributed to Noether

What I know as well founded induction, namely the rule $$\big(\forall y.(\forall z.z\lt y\Rightarrow\phi z)\Rightarrow\phi y\big)\Longrightarrow\big(\forall x.\phi x\big),$$ whose validity is the definition of well-foundedness of the relation $\lt$, I have recently heard called Noetherian induction. In fact Wikipedia includes this attribution.

With Excluded Middle and Dependent Choice, $\lt$ is well founded iff it has no infinite descending sequence $$x_0 \gt x_1 \gt x_2 \gt x_3 \gt \cdots$$

This suggests at least an analogy with the ascending chain condition that any sequence $$I_0 \subset I_1 \subset I_2 \subset \cdots$$ of ideals in a Noetherian ring is eventually constant.

Does the attribution of this idea to Emmy Noether amount to more than this analogy? Did she identify the principle at the top as one of logic? Or is this simply a case of the common phenomenon that names of concepts in Mathematics get transferred to more general or simply analogous situations without very much sound historical basis?

Thanks in particular to Joel Hamkins and Antoine Chambert-Loir (ACL) for their comments. Whilst it would be nice to fill in the small gap between Noether and Bourbaki, we have a plausible explanation of the connection between Emmy Noether the algebraist and this logical idea.

Generally speaking, I am a great admirer of Emmy Noether as a conceptual mathematician and rarely have much to say in favour of set theory. However, on this occasion, it would appear that the latter has a much better historical claim to this idea than algebraists do.

The work of Dimitry Mirimanoff that Joel mentions is in the paper Les antinomies de Russell et de Burali-Forti: et le problème fondamental de la théorie des ensembles, which appeared in Enseignement mathématique in 1917 but seems not to be available online. As Joel says, it defines the rank of the $\in$ relation of a set and uses induction in the form of infinite descent.

However, infinite descent in algebra goes back to Euclid: Book 7, Proposition 31 proves that any composite number is measured by some prime number, saying in the proof that "if the prime number is not found, an infinite series of numbers will measure the number $A$, each of which is less than the other: which is impossible in numbers".

The well founded relation $x\lt y$ in Euclid is that $x$ is a proper divisor of $y$. Algebraists from Kummer to Noether generalised divisibility to inclusion of ideals, $(y)\subset(x)$, which accounts for the reversal of the order. Noether generalised Euclid's arguments about prime factorisation from numbers to ideals.

Another thing that bugs me, both as a conceptual and constructive mathematician, is that the principle of well founded induction that I stated at the top is not the same thing as infinite descent. Quite apart from priority, it seems to me that one should take account of how ideas are formulated when naming or attributing them. I don't know who first wrote down the principle above, but Gentzen seems plausible.

In fact, the place where I recently heard well founded induction attributed to Noether was neither in the context of algebra nor logic: it was at a workshop on induction in automatic theorem provers.

I am not sure which rule Joel is labelling "well founded induction" in his further comments below. The rule at the top that I gave that name is an appropriate style to use as a proof rule in a constructive setting, whereas using infinite descent as a formulation of induction depends on an ambient logic with Excluded Middle and Dependent Choice.

One can also generalise the principle at the top by restricting the class of $\phi$s to which it is applicable. For example if the domain has an order with directed joins and $\phi$ is required to be closed under them then we have Scott induction. This proves properties of fixed points, in particular in the semantics of recursion in programming languages. I imagine that this idiom might be adapted to finitary constructive algebra too.

• I think you have correctly identified the reason for the terminology. In algebraic geometry it's an incredible common proof technique to do induction over the poset of closed subvarieties of an algebraic variety (or rather closed subschemes of a noetherian scheme), common enough to warrant its own name. Searching Google scholar, it seems a majority of the uses of the phrase are within algebraic geometry. Nov 24, 2013 at 22:02
• You can get a more direct analogy using $\leq$ instead of $<$: assuming excluded middle (which in particular allows us to use $\leq$ to express any property of $<$) and dependent choice, $\leq$ is well-founded iff every infinite descending sequence $x_0 \geq x_1 \geq x_2 \geq x_3 \geq \cdots$ is eventually constant. (Or one could use $\subsetneq$ with the ideals to also get a more perfect analogy.) Nov 24, 2013 at 22:13
• As a constructivist, I insist that any well founded relation is necessarily irreflexive, whilst I write $\subset$ and not $\subseteq$ for (reflexive) inclusion of sets etc. Nov 24, 2013 at 22:19
• Anyway, the point of the question was whether Noether (whom I always thought of as an algebraist) identified this as a general principle of logic. Nov 24, 2013 at 22:21
• Regarding your remarks about well-foundedness being not the same thing as infinite descent, I agree, but isn't well-foundedness the more fundamental notion? After all, this always implies the impossibility of infinite descent, but one needs DC to prove the other implication. Meanwhile, the well-founded induction principle is just a reformulation of saying that the relation $\lt$ is well-founded: saying that every inductive set is the whole set is the same as saying that every set with no minimal elements is empty, or in other words, that every nonempty set has a minimal element. Nov 27, 2013 at 0:37

It seems to me to be much more than merely an analogy, because (assuming DC) the ascending chain condition is exactly equivalent to asserting that the collection of ideals is well-founded under (reverse) inclusion. Thus, one can make arguments by induction on ideals, where each instance reduces to the instances in larger ideals, if any, and it seems that Noether did this quite well. For example, under the ACC, every nonempty set of ideals must contain a maximal element, and this is simply another way of stating the induction principle: to prove $\phi(I)$ for every ideal, simply prove that if $\phi(J)$ for all $J\supsetneq I$, then $\phi(I)$, since this will rule out a maximal element of the set $\{I\mid \neg\phi(I)\}$, which must therefore be empty. When the ACC holds, therefore, one may assign an ordinal rank to every ideal, in the manner that any well-founded relation supports such ranks, namely, the maximal ideals get rank 0, and penultimate ideals get rank 1 and so on, with the rank of an ideal equal to the supremum of the ranks+1 of the ideals properly containing it.

So my perspective is that the ACC is a quite robust and important instance of well-foundedness, rather than merely analogous to it.

Concerning the history of the terminology, I noticed that the Wikipedia entry on Noetherian induction redirects to the page on well-founded relations, and Wikipedia cites "Bourbaki, N. (1972) Elements of mathematics. Commutative algebra, Addison-Wesley" specifically in connection with this terminology. So perhaps Bourbaki is the origin of the terminology? (See ACL comments below.) Transfinite recursion itself certainly pre-dates Noether, tracing back to Cantor's use of it in the Cantor-Bendixson theorem, which is also the theorem that led Cantor to the ordinals. Meanwhile, the Wikipedia entry on the axiom of foundation asserts that "the concept of well-foundedness and rank of a set were both introduced by Dmitry Mirimanoff (1917)."

Lastly, let me add that one doesn't generally much see this Noetherian terminology used for well-foundedness or well-founded induction in the parts of logic or set theory with which I am familiar, where the use of well-foundedness is pervasive and often a central concern. But I suppose it wouldn't be surprising to find this terminology more commonly used in algebra, because of Noether's successful use of it there.

• Bourbaki's first edition of Algèbre commutative dates from 1968. But Noetherian topological spaces are already defined in Grothendieck-Dieudonné's Éléments de géométrie algébrique, I (1960), and the « principe de récurrence noethérienne » is stated (and proved) in Chapter 0, (2.2.2). Once he had considered spectra of commutative rings as the spaces of interest for algebraic geometry, it was probably natural to Grothendieck to restate and generalize in this way the more classical induction on dimension.
– ACL
Nov 25, 2013 at 9:22
• Update. I had a look at a first version of Bourbaki's Algèbre commutative, archives-bourbaki.ahp-numerique.fr/items/show/567. There (page 1), the « récurrence noethérienne » is stated as a lemma in the theory of ordered sets, and used more or less fluently in the proof (page 10) of existence of primary decomposition. So one should go back and check how this was used in E. Noether's proof of primary decomposition in polynomial rings (or in Lasker's generalization).
– ACL
Nov 25, 2013 at 9:28
• Comment. However, « récurrence noethérienne » is stated in terms of nonexistence of increasing sequences, or on terms of maximal elements, while « well-founded relations » use an opposite ordering. Although this is technically trivial to reverse an ordering, it does not look as a psychologically trivial operation. What would historians say?
– ACL
Nov 25, 2013 at 9:32
• In the 1970 (second?) edition of Bourbaki's Théorie des ensembles, "récurrence noethérienne" is Proposition 7 in III.6.5 (with this terminology). Nov 26, 2013 at 21:42