# Peano’s Conception of a Single Infinite Cardinality

Claudio Ternullo, Isabella Fascitiello

###### Abstract

While Peano’s negative attitude towards infinitesimals, in particular,
geometric infinitesimals, is widely documented, his conception of a single
infinite cardinality and, more generally, his views on the infinite, are a lot
less known. The paper reconstructs the evolution of Peano’s ideas on these
questions, and formulates several hypotheses about their underlying
motivations.

##  1 Introduction

Peano is widely known for his contribution to the creation of modern logical
symbolism and to the axiomatisation of arithmetic, in particular, for the
axioms which still bear his name.111For Peano’s varied contributions to
mathematical logic, see (Kennedy,, 1980), (Borga et al.,, 1985), (Grattan-
Guinness,, 2000), Ch. 5, and the 2021 special issue of Philosophia Scientiae
‘Peano and His School’ edited by P. Cantù and E. Luciano. His proof of the
impossibility of infinitesimal segments in geometry, has also received the due
attention;222Cf. (Peano,, 1892). For a reconstruction of Peano’s views about
the infinitesimals, see (Ehrlich,, 2006), and the more recent (Freguglia,,
2021). a lot less, if at all, explored are, on the contrary, Peano’s views on
the infinite, an area which, precisely when Peano was most active
mathematically, had experienced major and unprecedented breakthroughs,
especially thanks to the work of, just to mention a few names, Cantor,
Dedekind, Schröder, Veronese and other mathematicians.333Cf., again, (Grattan-
Guinness,, 2000) and (Ferreirós,, 2010).

Recently, Peano’s early conception of a single infinite cardinality has been
examined by (Mancosu,, 2016), which, besides showing how the conception
locates itself in the history of the infinite, also casts ‘Peano’s Principle’
as a bona fide cardinality principle putting pressure on the neo-logicist
doctrine of the analyticity of Hume’s Principle.444 Hume’s Principle [HP],
taken by Frege in (Frege,, 1884) as the correct statement of ‘numerical
identity’ and, as a consequence, of the concept of number, is the universal
closure of the following (second-order) principle:
$\\#x:F(x)=\\#x:G(x)\leftrightarrow F\approx G$ which reads as follows: ‘the
number of $F$s is equal to the number of $G$s if and only if $F$ is
equinumerous with $G$ (ie., if $F$s and $G$s can be put in a one-to-one
correspondence)’. The principle is, arguably, equivalent to Cantor’s
Principle, for which see later in the text, section 4.1. Mancosu has called
the formalisation, in second-order logic, of Peano’s conception of a single
infinite cardinality ‘Peano’s Principle’:
$\\#x:F(x)=\\#x:G(x)\leftrightarrow(\neg Fin(F)\wedge\neg
Fin(G))\vee((Fin(F)\wedge Fin(G))\wedge F\approx G)$ ie., ‘the number of $F$s
is equal to the number of $G$s if and only if either $F$ and $G$ are infinite
or, if finite, equinumerous’, and discusses it in connection with the ‘good
company problem’ for HP, see (Mancosu,, 2016), Ch. 4.

(Mancosu,, 2016) also lists the different stages which Peano’s conception of
the infinite went through, and demonstrates that Peano’s ideas became
increasingly closer to, and from a certain point onwards, indistinguishable
from, Cantor’s, but it does not pin down the exact motivations behind the
conception itself, nor does it explain why Peano, ultimately, became a full-
blown supporter of Cantor’s transfinite.555Cf. (Mancosu,, 2016), pp. 165-6, in
particular fn. 21.

The purpose of this paper is to fill in the gap, by providing, in a way as far
as possible based on the extant textual sources, a more articulated account of
the evolution of Peano’s ideas, which might also shed light on the potential
value and significance of Peano’s early conception.

The structure of the paper is as follows. In the next section, we make some
further preliminary considerations. In §3, we examine the progressive
development of Peano’s ideas. In §4, we indicate two main potential
motivations behind Peano’s conception, and then proceed to make some final
considerations in §5.

##  2 Did Peano Have A Definite Conception of the Infinite?

To be sure, from reading Peano’s terse statements on the subject, one wonders
whether the Italian mathematician had any determinate conception of the
infinite at all; at times, it would rather seem that Peano’s ideas are just
insufficiently developed to be seen as advocating any specific view.

On some authors’ judgement, Peano, while brilliant at spotting inaccuracies,
and in improving on already established results, was poorly armed to devise
(or just propose) a new theory, and rarely took enough pain in exposing in
full detail, or in justifying, conceptions he happened to champion (or use).
However, there is (some) evidence that this might not be the case with
reference to the question of the infinite.666See, for instance, (Grattan-
Guinness,, 2000), p. 221, and Agazzi’s introduction to (Borga et al.,, 1985),
p. 7. For a different assessment of the significance of Peano’s work, see
(Rodriguez-Consuegra,, 1991), especially the beginning of Ch. 3.

To begin with, Peano actively engaged in the lively debates on the nascent
theory of sets, and on infinitesimals, and relentlessly discussed aspects of
these topics with Cantor, Russell, Frege, Veronese and other
mathematicians.777Traces of such discussions may be found in, among other
works, the many letters that Peano exchanged with his mathematical
interlocutors, including Cantor (see later, section 3.2). The correspondence
between Peano and other mathematicians may be found in (Peano,, 2008).
Moreover, as already mentioned, Peano forcefully opposed Veronese’s
introduction of the (geometric) infinitesimals, by producing a (purported)
proof of their inconsistency,888See fn. 2 and section 4 of the present paper.
a fact which, on its own, already demonstrates that he was, at least, willing
to go to great lengths to understand questions about the nature of the
infinite, both philosophically and mathematically.

Peano’s ideas gradually became more and more akin to Cantor’s, to the point
that no traces of his earlier conception can be found in later work. What
motivated such a noticeable change of mind? We believe that Peano became
convinced of the intrinsic weakness of the alternatives to Cantor’s
conception, including his own, precisely as a consequence of his engagement in
the foundational debates which were hosted by, among other journals, his
Rivista di Matematica. The correctness of this interpretation is further
corroborated by the content of Peano’s correspondence with Cantor in the year
1895, which, in our view, proved instrumental for Peano’s ‘conversion’ to the
Cantorian approach.

Therefore, especially in the decade 1890-1900, rather than showing laziness,
or unwillingness to pursue the consequences of any specific conception, Peano
looks very active in searching for the right approach to the issue of the
infinite, by frequently interacting and discussing with his colleagues, and by
testing and revising his position in a coherent manner.

This said, even after inspecting much primary and secondary literature, as we
have done in this paper, the full motivations behind Peano’s early conception
and behind the shift to the Cantorian conception, still appear, to some
extent, mysterious, but the interpretative hypotheses we shall formulate in
§4, will provide us with, at least, some clues.

##  3 The Evolution of Peano’s Conception

In this section we follow closely the evolution of Peano’s ideas between 1891
and the turn of the century. For this, we avail ourselves of three fundamental
textual sources:

  1. 1.

Peano’s own articles, in particular, (Peano,, 1891), where his early
conception is first formulated, and (Peano,, 1892) (discussed in section 4.2);

  2. 2.

Peano’s correspondence with Cantor, consisting of six letters from Cantor to
Peano dating to 1895;999Peano’s letters to Cantor are no longer available, as
confirmed, in a private email to the authors (23/02/2022), by Clara Silvia
Roero, who has recently re-edited Peano’s works and correspondence (cf.,
again, (Peano,, 2008)).

  3. 3.

the first three volumes (and sections thereof) of his Formulaire de
mathématiques, published in rapid succession between 1895 and 1901 ((Peano,,
1895), then (Peano,, 1897), (Peano,, 1898), (Peano,, 1899), (Peano,, 1901)),
where both Peano’s early and ‘Cantorian’ conception may be found.

###  3.1 Peano 1891: ‘On the Concept of Number’

In 1891, Peano published his famous article ‘Sul concetto di numero’ (‘On the
Concept of Number’) in the Rivista di matematica, the journal he had founded
the same year.

Two years after Arithmetices principia, nova methodo exposita ((Peano,,
1889)), in which Peano first presented his axioms of the natural numbers, in
this work, Peano, among other things, simplifies his notation, demonstrates
that his famous five postulates of the natural numbers are mutually
independent, and builds a more complete axiomatic system of numbers by also
taking into account relative, rational and real numbers.101010Cf. (Borga et
al.,, 1985), pp. 79-94, and (Rodriguez-Consuegra,, 1991), Ch. 3.2.

(Peano,, 1891) is the first time Peano endeavours to deal explicitly with the
infinite. In §9, he defines a function, ‘$num\ a$’, whose domain consists of
‘classes’ (denoted $a,b,c,...,u,...$), and whose values are the
‘cardinalities’ of these classes; in Peano’s own words, $num\ a$ is ‘the
number of elements of the class $a$’111111[Con num a intenderemo “il numero
degli individui della classe a” (p. 100).] Page numbers of (Peano,, 1891) are
those of the third volume of the Cassina edition of Peano’s works: Opere
scelte, vol. 3, Cremonese, Roma, 1959. The English translations of Peano’s
quotes are all ours..

Now, if $a$ is a finite class, then $num\ a$ is just the (finite) number of
its elements, ie., a natural number $n$. But then, Peano states that $num\ a$
is not always a natural number, since the set of natural numbers does not
include ‘zero’ and ‘infinity’.121212[Data una classe $a$ non sempre $num(a)$ è
un $N$, poiché $N$ non comprende né lo zero, né l’infinito (p. 101).] This is
the first time Peano mentions the possibility that a class $a$ be empty, or
infinite (that is, that $num\ a=\infty$).131313[Il segno $num$ è un segno
d’operazione che ad ogni classe fa corrispondere o un $N$, o lo $0$, o
l’$\infty$ (p. 102).]

We wish to say something more substantial about ‘$\infty$’. Peano seems to
take it to be a bona fide ‘infinite quantity’, which can be manipulated like
any other (finite) quantity, as is clear from the propositions 3 and 4 in §9:

> 3\. If $a$ and $b$ are two non-empty and finite classes having no element in
> common, then the number of elements of the set of the two classes $a$ and
> $b$ is equal to the sum of the number of $a$s and $b$s.141414[Essendo $a$ e
> $b$ due classi non nulle e finite, non aventi alcun individuo comune, allora
> il numero degli individui appartenenti all’insieme delle due classi $a$ e
> $b$ vale la somma dei numeri degli $a$ e dei $b$ (p. 102).]

In 3. above, Peano is stating, in modern set-theoretic notation, that if two
sets $a$ and $b$ are disjoint, then the number of the elements of $a$ and $b$
is equal to the number of the elements of $a\cup b$. Peano, then, notes that
the proposition holds even if one of the two classes, and even both, contain
infinite elements; but now, we have, as a consequence, that:

$x+\infty=\infty+x=\infty$ (1)

and

$\infty+\infty=\infty.$ (2)

Immediately afterwards, he says:

> 4\. If the classes $a$ and $b$ are such that the second is contained in the
> first, and the class $b$ is non-empty, and is not equal to $a$, and if the
> number of $a$s is finite, then the number of $b$s is also finite, and is
> less than the number of $a$s.151515[Se delle classi $a$ e $b$ la seconda è
> contenuta nella prima, e la classe $b$ non è nulla, e non è eguale ad $a$, e
> se il numero degli a è finito, allora anche il numero dei $b$ è finito, ed è
> minore del numero degli $a$ (p. 101).]

Then he notes: ‘this proposition ceases to be valid if $num\
a=\infty$’.161616[Questa proposizione cessa di esistere se $num\ a=\infty$ (p.
101).] Propositions 3 and 4 also demonstrate that ‘$\infty$’ is taken by Peano
to be different from Cantor’s ‘$\omega$’, since, as is known, by Cantor’s
conception, $\omega+n\neq n+\omega$. Indeed, on the grounds of the
arithmetical laws outlined in (1) and (2), one could be tempted to view
‘$\infty$’ as being equivalent to ‘$\aleph_{0}$’. But this would be a hasty
conclusion. In section 4.2, we shall see that Peano’s own intepretation of his
‘infinitary numbers’ does not automatically sanction the equivalence between
his ‘$\infty$’ and ‘$\aleph_{0}$’.

One further comment is in order. Proposition 4 states that infinite classes
could contain infinite proper subclasses. This implies, among other things,
that ‘standard’ part-whole relationships cease to be valid in the infinite, a
fact which, in turn, may have had some bearings on the development of Peano’s
ideas, as we shall see in the next section.

To sum up, (Peano,, 1891)’s conception clearly is non-Cantorian, as it fosters
the existence of just one infinite ‘cardinal number’. So the question arises
of whether and to what extent Peano had assimilated Cantor’s theory at the
time. On this issue, it should be preliminarily observed, on the one hand,
that only one year later, in (Peano,, 1892), Peano exhibits a more than decent
understanding of Cantor’s transfinite. On the other, as we shall see in a
moment, his 1895 correspondence with Cantor, in particular, the questions he
asked his German colleague, reveal that, at the time of the composition of
(Peano,, 1891), Peano still felt the need to understand Cantor’s theory in
more depth.

###  3.2 The Cantor-Peano Correspondence (1895)

Peano’s early conception of the infinite, which we have just commented upon,
is gradually modified, and then definitively replaced, in 1899, by Cantor’s
theory of the transfinite. What, ultimately, led Peano to change his mind
remains an open question. Although the evidence is insufficient, we conjecture
that the correspondence between Peano and Cantor, in the year 1895, proved
instrumental in that respect.

In what follows, we review bits of the discussion Peano entertained with
Cantor that are relevant to our purposes; as already noted, one side of the
correspondence (from Peano to Cantor) is not extant, so Peano’s comments and
answers can only be, very approximately, deduced from the responses of his
German colleague.171717Cantor’s letters to Peano examined in the present work
are all collected in the Meschkowski edition of Cantor’s letters, (Cantor,,
1990), pp. 359ff. (n. 143-147). An overview of the contents of the Cantor-
Peano correspondence is in (Kennedy,, 1980), pp. 87-90.

The letters cover the following subjects:

  1. 1.

Cantor’s articles that Peano intends to publish in his Rivista Matematica;

  2. 2.

Cantor’s distaste with Veronese’s work on the infinite, which posited the
existence of infinitesimals;

  3. 3.

Peano’s request for explanations about Cantor’s elusive definition of
‘cardinal number’ in Cantor’s works.

In particular, (2) and (3) are fundamental, in our view, for the evolution of
Peano’s ideas on the infinite. Let’s examine them in more detail.

As far as (2) is concerned, in the two letters of, respectively, July 27 and
July 28, 1895, the discussion focuses on the conflict between Cantor’s and
Veronese’s theories. On the German mathematician’s view, what Veronese called
‘ordered groups’ were just a plagiarism of Cantor’s ‘simply ordered sets’. But
then, Cantor points out what he thinks is the mistake that Veronese has made
in trying to automatically extend the arithmetic of natural numbers to
infinite numbers. As Cantor observes, in two passages of the July 27 letter:

> But if it is correct that his $\infty_{1}=\omega+^{*}\omega$, his assertion
> that:
>
> $2\cdot\infty_{1}=\infty_{1}\cdot 2$
>
> must be _incorrect_! […] Anyway, his [Veronese’s, our note] ‘infinite
> numbers’ seem tenable to me only if they are identified with some of my
> ‘transfinite order-types’. In this case, however, they lack the law of
> commutativity for addition and multiplication (in general) on which he
> [Veronese, our note] lays so much stress.181818[Jedenfalls scheinen mir
> seine ”unendlichen Zahlen” nur dann haltbar, wenn sie mit gewißen von meinen
> ”transfiniten Ordnungstypen” identificirt werden. In diesem Falle fehlt
> ihnen aber das Gesetz der Commutabilität bei der Addition und Multiplication
> (im Allgemeinen), worauf er solchen Nachdruck legt ((Cantor,, 1990), p.
> 360)]. The English translation is ours.

We do not know the content of Peano’s answer; in any case, on Cantor’s own
impulse (28 July 1895 letter), the 27 July letter was published by Peano in
the Rivista di Matematica.191919In the August issue of the journal. So, it is
imaginable that Cantor had managed to convince Peano of the correctness of his
arguments.

As far as (3) is concerned, in the subsequent letters, the conversation
between the two mathematicians turns to consider Peano’s qualms about Cantor’s
definition of ‘cardinal number’. This can be safely deduced from Cantor’s
September 14, 1895 letter, in which Cantor tries to clarify some notions
contained in the section 5 of (Cantor,, 1895). From what Cantor says, it seems
quite clear that Peano was unsure about Cantor’s notion of ‘finite cardinal
number’ and about how Cantor introduced the induction principle, and asked his
colleague for clarifications.

In the 21 September 1895 letter, Cantor quotes a sentence written by Peano in
his response to Cantor’s previous letter: ‘Where can one find the definition
of finite cardinal numbers?’,202020[Où est ce que l’on trouve la definition
der endlichen Cardinalzahlen? ((Cantor,, 1990), p. 365)]. a clear sign that
Cantor’s first letter was not entirely clarificatory.

Therefore, what is likely is that, as a consequence of the clarifications
Cantor gave Peano, and of Cantor’s rebuttal of Veronese’s theory, Peano
started considering Cantor’s conception of the transfinite as being the only
correct one, and Cantor’s theory of cardinal numbers as pretty much
definitive. If in 1891 he had chosen not to fully adhere to Cantor’s theory -
or simply not to explore it in depth - now he had further material at hand,
including Cantor’s clarificatory statements, which could re-orient his views,
a fact which would soon be reflected, as we will see, in the Formulaire, where
the theory of the transfinite gradually takes primacy.

###  3.3 The Infinite in the Formulaire (1895 to 1901)

We finally turn to survey, very briefly, the modifications of Peano’s
conception that can be found in the volumes (editions) of Peano’s Formulaire
de mathématiques.212121The volumes of the Formulaire all appeared as
supplements to issues of the Rivista di Matematica between 1895 and 1908. For
an overview of the plan and evolution of Peano’s Formulaire, or Formulario (in
latino sine flexione), see the very exhaustive (Cassina,, 1955). As said at
the beginning, these have already been briefly taken into account by Paolo
Mancosu in (Mancosu,, 2016);222222See fn. 5. what follows aims to expand on,
and complement, Mancosu’s account.

The evolution of Peano’s ideas on the infinite in the Formulaire spans a
period of 4 years, and ends in 1899, when all traces of Peano’s early
conception disappear, and Cantor’s transfinite ultimately takes over.

The first volume ((Peano,, 1895)) saw the collaboration of other Italian
mathematicians, such as G. Vailati, F. Castellano, G. Vivanti, R. Bettazzi. Of
particular importance for our purposes are sections V and VI, written,
respectively, by Peano and Giulio Vivanti; the latter, among other things, had
taken part in the debate with Bettazzi about the infinitesimals raging in the
Rivista in the years 1891-1892.232323The debate is accurately reconstructed in
(Ehrlich,, 2006), pp. 75-101.

In section V of (Peano,, 1895), entitled ‘Classes de nombres’, we find again
the mentioning of just one infinite cardinality, ‘$\infty$’, which is now
explicitly defined as one of the possible values of $num\ u$, where $u$ is a
class, as follows:242424In the text of formulas (4)-(6) below, for the sake of
simplicity, we have chosen not to adopt Peano’s original notation, but the
modern one.

$4.\ num\ u=\infty\leftrightarrow num\ u\notin\mathbb{N}_{0}$

Then Peano explains that $num\ u$ can take only three values:

$5.\ num\ u=n\in\mathbb{N}\vee 0\vee\infty$

Moreover, ‘$\infty$’, in definition 6, is, again, characterised as enjoying
commutativity:

$6.\ a\in\mathbb{N}_{0},\quad a+\infty=\infty+a=\infty+\infty=\infty\quad
and\quad a<\infty$

On the other hand, section VI, due to Vivanti, not to Peano, introduces, and
uses, the symbols ‘Nc’ and ‘Ntransf’, denoting, respectively, Cantor’s
cardinals and ordinals, and deals with the corresponding set-theoretic
notions. Therefore, in 1895, Peano did not entertain ideas much different from
those appeared in (Peano,, 1891), ie. he did not subscribe to Cantor’s views,
while (Peano,, 1895)’s section VI, composed by Vivanti, which mentions
Cantor’s cardinals and ordinals, has no immediate connection with the one
composed by Peano. This is consistent with our hypothesis that the
correspondence with Cantor may have led Peano to change his mind, as the text
of (Peano,, 1895) must have been produced well before 1895.

The second volume of the Formulaire consists of three parts, each published
one year apart from the other, following the editorial plan below:

  * •

§1, Logique mathématique (Peano,, 1897);

  * •

§2, Arithmétique (Peano,, 1898)

  * •

§3 (untitled) (Peano,, 1899)

(Peano,, 1898) contains the first manifestation of Cantor’s theory in Peano’s
writings. Peano says:

> Here we define another idea, indicated by the sign $Nc^{\prime}$ similar to
> the previous one, indicated by $num$, but not identical. This definition is
> expressed by the only signs of logic (§1); it is therefore independent of
> the primitive ideas $\mathbb{N}_{0},+,0$. We could start arithmetic here.
> P211 expresses the coincidence of the signs $num$ and $Nc^{\prime}$ when one
> deals with finite classes. But we will have for example: $num\
> \mathbb{N}=num\ \mathbb{R^{+}}$, because the classes $\mathbb{N}$ and
> $\mathbb{R^{+}}$ are both infinite;252525In the original text, Peano uses
> the symbols N and Q, to denote, respectively, the set of the natural and of
> the positive real numbers; in our translation, for the sake of simplicity,
> we have replaced N and Q with the modern $\mathbb{N}$ and $\mathbb{R^{+}}$.
> but $Nc^{\prime}\ \mathbb{N}<Nc^{\prime}\ \mathbb{R^{+}}$, because the power
> of $\mathbb{N}$ is less than the power of $\mathbb{R^{+}}$. Mr. Cantor
> indicated the power of $a$ by $\overline{\overline{a}}$ (See RdM. A. 1895
> p.130), a notation that cannot be adopted in the Formulaire. M. Vivanti in
> $F_{1}$ VII §2 P1 [section VI of (Peano,, 1895), our note] has replaced it
> with $Nc^{\prime}\ a$ “the cardinal number of $a$.”.262626[Nous définissons
> ici une autre idée, indiquée par le signe “$Nc^{\prime}$” semblable à la
> précédente, indiqueée par “num”, mais non identique. Cette définition est
> exprimée par les seules signes de logique (§1); elle est donc indépendante
> des idées primitives $\mathbb{N}_{0},+,0$. On pourrait commencer ici
> l’Arithmétique. La P211 exprime la coincidence des signes $num$ et
> $Nc^{\prime}$ lorsqu’il s’agit de classes finies. Mais on aura par exemple:
> $num\ \mathbb{N}=num\ \mathbb{R^{+}}$, car les classes $\mathbb{N}$ et
> $\mathbb{R^{+}}$ sont toutes les deux infinies; mais $Nc^{\prime}\
> \mathbb{N}<Nc^{\prime}\ \mathbb{R^{+}}$, car la puissance des $\mathbb{N}$
> est plus petite que la puissance des $\mathbb{R^{+}}$. M. Cantor a indiqué
> la puissance de $a$ par $\overline{\overline{a}}$ (Cfr. RdM. a. 1895 p.130),
> notation qu’on ne peut pas adopter dans le Formulaire. M. Vivanti dans
> $F_{1}$ VII §2 P1 l’a remplacée par $Nc^{\prime}\ a$ ”le nombre cardinal des
> $a$”. ((Peano,, 1898), p. 39)].

From the quote above, it seems clear that Peano is beginning to change his
mind, and gradually converting to Cantor’s theory, although he still keeps
mentioning both conceptions: $Nc^{\prime}$, denoting Cantor’s cardinal
numbers, now appears alongside $num$ as cardinality assignment to an infinite
class.

(Peano,, 1899) marks one further transformation of Peano’s conception. In that
work, for the first time, Peano ‘merges’ the concepts of $num$ and
$Nc^{\prime}$ into a unique, and new, concept denoted by ‘$Num$’. $Num$’s are
now nothing but Cantor’s transfinite cardinalities, an ‘expansion’ of Peano’s
former $num$ subtly revealed by the change of notation. From (Peano,, 1899)
onwards, thus, Peano seems to fully adhere to Cantor’s theory.

In the Formulaire’s third volume, (Peano,, 1901), in the $Num$ section, Peano,
finally, explains:

> $Num^{\prime}Cls$ means “the number of a class”. These numbers coincide with
> the [elements of] $\mathbb{N}_{0}$ for the finite classes; G. Cantor calls
> them “cardinal numbers”. In F 1895 the symbol “$Nc$” was introduced to
> represent them [again, a reference to Vivanti’s section VI of (Peano,,
> 1895)].272727[$Num^{\prime}Cls$ signifie “le nombre d’une classe”. Ces
> nombres coïncident avec les $N_{0}$ pour les classes finies; G. Cantor les
> appelle “nombres cardinaux”. Dans F1895 on a introduit le symbole “$Nc$”
> pour les représenter. ((Peano,, 1901), p. 70)]

Thus, already by 1901, and then in the subsequent volumes of the Formulaire
(1903, 1908), any significant distinction between Peano’s conception and
Cantor’s disappears, and Peano’s ‘single-cardinality’ conception of the
infinite is a distant memory.

##  4 Interpretations of Peano’s Conception

In this section, we turn to examine the issue of what may have motivated
Peano’s conception of a single infinite cardinality and present further
textual sources which might help us understand better the evolution of Peano’s
ideas.

We will follow two main interpretative routes: (1) the first will highlight
Peano’s hesitancy to either adopt the Cantorian conception of infinite
cardinalities as opposed to the one based on the Part-Whole Principle; our
underlying presupposition is that, like Galileo before him, Peano was not
fully convinced of the correctness of either method; (2) the second will point
out an ‘incomplete’ understanding, or misuse, of Cantor’s theory of ordinals
and cardinals, which may have misled Peano to believe that his theory could be
‘reduced’ to Cantor’s or, in alternative, that Cantor’s theory of the
transfinite could be bent to suit his own purposes. We examine, respectively,
(1) in §4.1, and (2) in §4.2.

###  4.1 A Galilean View?

In his Dialogues Concerning Two New Sciences (1638), Galileo came to express
skepticism about the possibility of carrying out size comparisons with regard
to infinite ‘collections’ of objects.

In the work’s First Day, Galileo’s spokesperson, Salviati, proposes to compare
the magnitude of a collection of objects with that of another collection by
checking that each object in the former collection is uniquely matched by one
in the latter. His case study consisted in comparing square with natural
numbers (in modern parlance, we would say the set of square numbers
($\mathbb{S}$), and that of natural numbers ($\mathbb{N}$)), and the mapping
function that he is using is the one that associates each natural number with
its square: $n\mapsto n^{2}$:

$0\leftrightarrow 0$ $1\leftrightarrow 1$ $2\leftrightarrow 4$
$3\leftrightarrow 9$ $..\leftrightarrow..$

Salviati and his interlocutor, the learned listener Sagredo, agree that, using
this method, it must be concluded that there are as many numbers as squares
(that is, that $\mathfrak{s}(\mathbb{N})=\mathfrak{s}(\mathbb{S})$, where
$\mathfrak{s}(X)$ denotes the ‘size of $X$’). But now the dire puzzle unfolds
in front of their eyes: it seems to be an ‘established fact’, in any case,
commonsense knowledge, that there are more natural than square numbers, since
$\mathbb{S}$ has lots of gaps ‘in between’, that $\mathbb{N}$ does not have;
this is natural, since $\mathbb{S}$ is a proper part of $\mathbb{N}$ (ie.,
$\mathbb{S}\subset\mathbb{N}$). So, relying on intuitions referring to the
‘density’ of $\mathbb{S}$ in $\mathbb{N}$, Salviati suggests that one should,
in fact, more correctly, conclude that
$\mathfrak{s}(\mathbb{N})>\mathfrak{s}(\mathbb{S})$. This is what has come to
be known as Galileo’s Paradox.

Now, asks Salviati, which of the two horns of the dilemma is correct? Neither,
apparently, as Galileo ultimately points out, since:

> This is one of the difficulties which arise when we attempt, with our finite
> minds, to discuss the infinite, assigning to it those properties which we
> give to the finite and limited; but this I think is wrong, for we cannot
> speak of infinite quantities as being the one greater or less than or equal
> to another. ((Galileo,, 1914), p. 31)

Now, Galileo does not go as far as to say that, but, since one cannot speak of
magnitudes ‘greater than’, ‘less then’, or ‘equal to’ one another in the
infinite, it would, in principle, be fully legitimate and consequential to
think that, to all purposes and intents, there is but one infinite
cardinality. Thus, what we would have at hand, in (Galileo,, 1914), would be
the enunciation of a ‘single-cardinality’ conception of the infinite dictated
by the impossibility to come to terms with a conflict between two ways of
counting in the infinite, one based on ‘bijections’ and one on ‘density’. The
former method is formally enshrined in what would, much later, become the
central pillar of set theory, that is:

Cantor’s Principle (CP). Given two sets $A$ and $B$, if there exists
$f:A\rightarrow B$ which is 1-1 and onto (ie., a bijection between $A$ and
$B$), then $\mathfrak{s}(A)=\mathfrak{s}(B)$.282828CP may be seen as a
‘variant’ of Hume’s Principle mentioned in §1 (see fn. 4).

The latter obeys what was already known in antiquity as Euclid’s Axiom (Common
Notion V of the Elements), that is, the:

Part-Whole Principle (PWP). Given two sets $A$ and $B$, if $A\subset B$, then
$\mathfrak{s}(A)<\mathfrak{s}(B)$.

In our era, the ‘set-theoretic era’, as it were, one tends to think quite
immediately that CP is the correct way to assign cardinalities to infinite
collections, and that, as a consequence, Galileo’s Paradox is no paradox at
all, but rather helps exhibit the hallmark of the infinite itself, as captured
by the notion of:

Dedekind-Infiniteness. A set is said to be infinite if and only if it can be
put in a one-to-one correspondence with a proper part of itself; otherwise, it
is finite.292929The notion was first formulated in (Dedekind,, 1888), V, 64.

But the correctness, and indispensability, of PWP, as is known, was the
prevailing view for many centuries, and still lay almost uncontested in
Galileo’s times.303030For a careful excursus of the history of several
cardinality principles, we refer the reader to (Mancosu,, 2009). Moreover,
oscillations between compliance with CP or PWP could even be found, centuries
later, in avowed supporters of the ‘bijection method’ like Bolzano.313131Cf.
(Mancosu,, 2016), pp. 130-1.

Now, one could speculate that: 1) Peano shared Galileo’s concerns about the
possibility of articulating a theory of distinct infinite cardinalities, and
that: 2) he retreated to a more sober ‘one-cardinality’ conception precisely
because he was hesitant to choose among one of the two (known) methods of
counting in the infinite.

In particular, Peano’s 1891 conception might have reflected dissatisfaction
both with a purely Cantorian and with the Euclidean point of view.

This is (indirectly) proved by a remark made by Peano in a footnote of his
(Peano,, 1891), concerning Rodolfo Bettazzi’s use of the ‘bijection method’ to
characterise the concept of ‘cardinal number’ in (Bettazzi,, 1887).323232The
passage is cited (and translated to English) by (Mancosu,, 2016), p. 164-5.
Apparently, Bettazzi thought that, if two sets $A$ and $B$ could be put in a
one-to-one correspondence, then any correspondence between them would
inevitably be one-to-one. For the sake of our argument, let us state this as:

Bettazzi’s Principle (BP). Given two sets $A$ and $B$, if there exists a
bijective function between $A$ and $B$, then there is no function between $A$
and $B$ which is injective, but not surjective.

Peano correctly objects to Bettazzi that one can biject infinite sets with
their own infinite parts, that is, one can have injections of sets with
themselves which, evidently, are not surjective.

Considerations such as these could have led Peano to doubt that both CP and
PWP could be a satisfactory characterisation of the notion of ‘infinite
cardinality’, and may ultimately have convinced him that all infinite classes
had the same, unique cardinality. This can further be deduced from two facts.
One the one hand, if Peano had accepted PWP as the correct conception of
infinite cardinalities, then he would have had to admit of different orders of
infinity, that, for instance, of a subset of $\mathbb{N}$ and of $\mathbb{N}$
itself. But this he doesn’t do either in (Peano,, 1891) or nowhere else, for
that matter. On the other, as we have already seen in section 3.3, Peano does
not mention the existence of distinct infinite cardinalities as based on the
‘bijection method’, and, as late as (Peano,, 1895), he still maintains that
$num(\mathbb{N})=num(\mathbb{R^{+}})$.

There is still one further way in which Peano’s position could be viewed as
Galilean. If one just takes into account $\mathbb{N}$ and its subsets, one may
be perfectly entitled to conclude that there is but one infinite cardinality
(maybe recognition of this fact already underlay Galileo’s own thinking). Now,
with the exception of later examples he makes of infinite cardinalities where
he also takes into account, e.g., $\mathbb{R}$, Peano’s considerations
(especially those in (Peano,, 1891)) mostly refer to sets and subsets of
natural numbers.

In sum, assuming that our ‘Galilean’ interpretation of Peano’s conception is
correct, one might conclude that either Peano felt that there was a tension
between CP and PWP in the infinite which, ultimately, implied that there were
no distinct infinite cardinalities, or he thought that CP, restricted to
$\mathbb{N}$ (and subsets thereof), that is, to the realm of arithmetic in
which he was especially interested, licensed the view that there is just one
infinite cardinality.

###  4.2 (Geometric) Infinitesimals and Cantorian Set Theory

We have expressed many times the view that Peano may have progressively
embraced Cantor’s transfinite because he finally came to grips with it after a
resolutive exchange of letters with Cantor himself which clarified in full the
details of Cantor’s theory. There are, however, passages of Peano’s earlier
works which do not lend support to this thesis. In particular, we are
referring here to the article on infinitesimals, (Peano,, 1892), where Peano
aims to improve on Cantor’s earlier proof that infinitesimal segments are
inconsistent.333333A detailed analysis of Cantor’s proof (for which see
(Cantor,, 1887)) is carried out in (Ehrlich,, 2006), pp. 27-51. In the
article, Peano shows a good command of Cantorian set theory, but seems to want
to use it, very originally, in a way compatible with his ‘single-cardinality’
conception of the infinite. Let us see how in more detail.

The key notions in that work are those of ‘bounded’ and ‘infinitesimal
segment’. Given the half-line with origin in $o$, a ‘bounded’ segment is a
segment $op$ with origin in $o$ and end in $p$. Peano also denotes a bounded
segment with $u$, or with $\overline{ou}$. An ‘infinitesimal segment’ is a
$\overline{ou}$ lying inside a bounded segment $v$, and denoted $u=v/\infty$.
The latter’s full definition reads as follows:

> We say that the segment $u$ is infinitesimal with respect to $v$ and we
> write $u\ \epsilon\ v/\infty$, if every multiple of $u$ is less than $v$
> […].343434[Dicesi che il segmento $u$ è infinitesimo rispetto al segmento
> $v$, e scriveremo $u\ \epsilon\ v/\infty$, se ogni multiplo di $u$ è minore
> di $v$ […] (p. 113)] Note the use of Peano’s $\epsilon$ symbol, which means:
> ‘is’. Page numbers for (Peano,, 1892) are, again, those of Cassina’s
> edition, for which see fn. 11.

Then Peano proceeds to define the multiple of infinite order of $u$, ‘$\infty
u$’. He says:

> We shall posit:
>
> $\infty u=\bigcup\mathbb{N}u$
>
> that is, we call multiple of $u$ of infinite order the set of points which
> either lie on some segment $u$, $2u$, $3u$, $...$, or the upper bound of the
> multiples of $u$.353535[Porremo  $\infty u=\bigcup\mathbb{N}u$ cioè
> chiamiamo multiplo d’ordine infinito di $u$ l’insieme dei punti che stanno
> sopra qualcuno dei segmenti $u$, $2u$, $3u$, $...$, o il limite superiore
> dei multipli di $u$. (p. 113)] For the sake of simplicity, in the formula,
> we have chosen to replace Peano’s original notation: $\cup^{\prime}Nu$ with
> $\bigcup\mathbb{N}u$.

Now, Peano asserts that, by the definition of infinitesimal, if $u$ is
infinitesimal, also ‘$\infty u$’ must lie inside $v$, and so must be all other
infinitary multiples of $u$. In a crucial passage of (Peano,, 1892), he says:

> We can add $\infty$ to itself, thus obtaining $2\infty u$, and, generally,
> we can form all multiples of $\infty u$; we can multiply $\infty u$ by
> $\infty$, and obtain $\infty^{2}u$ and so on. But all these various
> segments, which one obtains multiplying $u$ by Cantor’s transfinite numbers
> are all equal to one another […].363636[Possiamo sommare $\infty u$ con sé
> stesso, ottenendo così $2\infty u$, ed in generale possiamo formare tutti i
> multipli di $\infty u$; possiamo moltiplicare $\infty u$ per $\infty$, ed
> ottenere $\infty^{2}u$ e così via. Ma tutti questi varii segmenti, che si
> ottengono moltiplicando $u$ pei numeri transfiniti di Cantor sono eguali fra
> loro, […]] (p. 113)

As a consequence, infinitesimal segments, if such things existed, would
blatantly violate the properties of bounded segments, which require that, for
instance, a segment of length $2\infty u$, twice as much as $\infty u$, be
greater than $\infty u$. Hence, concludes Peano, infinitesimal segments are
inconsistent with ‘standard’ representations of the geometric space.

Leaving aside entirely the issue of the correctness of Peano’s argument, what
is most relevant for our purposes is to try and understand Peano’s reference,
in it, to ‘Cantor’s transfinite numbers’. The mentioning of such numbers is
indeed puzzling, as, by Cantor’s own theory, as we know, $\omega+1>\omega$,
hence, $\omega+1\cdot u$ should be greater than $\omega\cdot u$.

Indeed, Giuseppe Veronese, the main target of Peano’s (and Cantor’s) polemic
against geometric infinitesimals, was able to react very quickly to Peano’s
argument. In (Veronese,, 1892), which is meant to be a response to (Peano,,
1892), the Italian mathematician correctly diagnoses what seems to be the
trouble with Peano’s proof:

> But these equalities [$\infty u=2\infty u$ = $\infty^{2}u=...$, our note] do
> not depend on the properties of Mr. Cantor’s transfinite numbers, for which
> hold: $\omega+1>\omega$, $2\omega>\omega$, etc., but precisely on
> considering $\infty u$ as unlimited (our italics).373737(Veronese,, 1892),
> p. 74.

Veronese is, here, drawing attention to the fact that it is the very
definitions of $\infty u$, $2\infty u$, etc., not transfinite arithmetic,
which allow one to deduce that these quantities are all equal. Similar remarks
were made by Giulio Vivanti, who, in his (Vivanti,, 1895), explained why the
argument was doomed to failure. Vivanti argued that Peano viewed $(\infty+1)u$
as the upper bound of $(n+1)u$, but the points of $(\infty+1)u$ must lie
either on one of the finite multiples of $u$, or be $(\infty+1)u$ itself. So,
$(\infty+1)u$ is precisely the same the segment as $\infty u$; hence,
$(\infty+1)u$ and $\infty u$ must be equal. But, if it is true that Cantor’s
$\omega$ is the limit of $n$, $\omega+1$ is no limit number (ordinal), and, as
already pointed out by Veronese, $\omega+1>\omega$, and, consequently,
$(\omega+1)u>\omega u$.383838(Vivanti,, 1895), p. 69.

So, Peano’s reliance upon ‘Cantor’s transfinite numbers’ in his proof does not
serve well his purpose of showing that geometric infinitesimals are
inconsistent. In order to rescue the force of the argument, (Freguglia,, 2021)
conjectures that the reason why Peano holds the correctness of the equation:

$\infty u=2\infty u=\infty^{2}u=...$

is the fact that Peano is, intentionally or not, ‘assimilating’ ‘$\infty$’ to
‘$\aleph_{0}$’.393939Cf. (Freguglia,, 2021), p. 152ff. The author says: ‘Peano
explicitly assimilates $\infty$ to $\aleph_{0}$ […]’ and then explains ibid.,
fn. 14: ‘[i]n the sense that it has the same arithmetic behaviour.’ This is
because, Freguglia argues, while, as said, $\omega\neq\omega+1$, on the
contrary, $\aleph_{0}=\aleph_{0}+1$ and, in general, if $\kappa$ is a
transfinite cardinal number, and $n$ a natural number:

$\kappa+n=\kappa\cdot n=\kappa^{n}=\kappa$

Freguglia’s interpretation has some merit, but seems to be missing, at large,
the mark. In particular, it seems to be rather unjustified to force this
interpretation on Peano’s reference to ‘Cantor’s transfinite numbers’; the
numbers Peano is referring to here are, it seems to us, and as understood by
Veronese and Vivanti, Cantor’s transfinite ordinals
($\omega,\omega+1,...,\omega+\omega$, …).

However, following the spirit, not the letter, of Freguglia’s intepretation,
one could say that Peano may have taken his numbers ($\infty u$,
$\infty^{2}u$, …) to be equivalent to $\aleph_{0}$ in the sense that,
although, like Cantor’s ordinals they were linearly ordered by ‘$<$’, and,
thus, differed from each other in ‘magnitude’, they had, nonetheless, the same
cardinality, in particular, they were all countable. Now, the reasoning goes,
since what counts as length of a geometric segment is a measure expressed by a
cardinal number, one must conclude that segments of length $\infty u$,
$\infty^{2}u$ are all equal. But there’s a problem with this interpretation:
Cantor’s ordinals may be uncountable, that is, they may have different
cardinalities, and Peano might have been aware of this.

We wish to propose an alternative interpretation of Peano’s argument, which
aligns more with the content of Vivanti’s and Veronese’s comments above. This
avers that Peano viewed his ‘infinite numbers’ as being all equal to each
other not because he thought that ‘$\infty$’ was equivalent to ‘$\aleph_{0}$’,
but because he thought that there was just one, maximal, unsurpassable,
infinite cardinality, ‘$\infty$’. By this interpretation, when Peano is
referring to ‘Cantor’s numbers’ in the aforementioned passage, he is really
likening his own numbers to Cantor’s ordinals (as supposed by Veronese and
Vivanti), but he is also committing himself to the view that those numbers
must all have the same cardinality, because, by his own conception, there
exists only one, ‘$\infty$’, which cannot be transcended. So, in a sense,
Peano’s reference to ‘Cantor’s numbers’ can be seen as an attempt to make
Cantor’s conception suit (to some extent, even merge it) Peano’s own, ‘single-
cardinality’, conception.

Our interpretation entails that Peano had developed a carefully thought of,
and original, conception of the infinite in the years 1891-1895, which he
believed could be employed in practical mathematical contexts (such as that
discussed in (Peano,, 1892)). By this very interpretation, the fact that he,
subsequently, abandoned this conception does not mean that he thought it to be
faulty, ‘half scarce made up’ in any respect, but only that he thought, as we
have have already said many times, Cantor’s theory to be more general, far-
reaching and, ultimately, fruitful than his own.

##  5 Concluding Remarks and Ways Forward

We have reviewed the development of Peano’s conception of the infinite up to
the 1901 publication of the third volume of his Formulario. We have seen that,
already by the publication of the third section of the second volume in 1899,
Peano’s conception had aligned with the, then already ‘meainstream’, set-
theoretic conception of infinite cardinalities.

In section 3.2, we have conjectured that Cantor may have played a role in
persuading Peano to abandon his earlier view. As far as the latter is
concerned, we have shown that, in his (Peano,, 1892), Peano showed to have
perfectly clear in his mind the consequences of his own conception, the way it
could be properly used in ‘practical’ mathematical contexts, and how it
related to Cantor’s transfinite. Moreover, we have seen that, up to a point,
Peano might have been doubtful about the efficacy of the ‘bijection method’,
as this conflicted with PWP, and we have conjectured that these doubts might
have led him to formulate the conception of a single infinite cardinality.

By way of conclusion, we wish to draw attention to some recent developments,
which could be, if loosely, connected to Peano’s views.

The debate on alternatives to Cantor’s conception of cardinality has, lately,
flourished again, in ways which potentially suggest a re-thinking of the
primacy of CP. The concrete way this is usually done is through re-
legitimising the use of PWP.404040For recent, alternative conceptions of
infinite cardinality based on PWP see Sergeyev’s theory of grossone (cf.
(Sergeyev,, 2017)), and Benci et al.’s theory of numerosities ((Benci et al.,,
2006)).

For another development, one could attempt to narrow down the extent of
Cantor’s transfinite by introducing suitable restrictions on generating
principles for sets, as imagined, for instance, by Vopěnka’s ‘alternative set
theory’ with just two infinite cardinalities, the countable and the
uncountable, Randall Holmes’ ‘pocket set theory’, or Meir Buzaglo’s
cardinality principles discussed in (Buzaglo,, 2002).414141For Vopěnka’s
‘alternative set theory’, see (Vopěnka,, 1979). For ‘Buzaglo’s Principles’,
see (Buzaglo,, 2002), pp. 43-4, 49-50. ‘Pocket set theory’ is outlined in
(Randall Holmes,, 2017), pp. 62ff.

Just to make a concrete example, let us review one of Buzaglo’s ‘principles’.
This is based on defining the ‘$\nless$’ (‘not smaller than’) relationship
between the sizes of sets, in turn the negation of a special ‘$<$’
relationship defined as follows:424242The ‘$|...|$’ sign, in the definitions
below, should be taken to denote a generic concept of ‘size of a set’, not the
set-theoretic one.

######  Definition 1 (Smaller Than)

$|A|$ is smaller than $|B|$ if and only if: (i) there is a function $f$ from
$A$ to $B$; (ii) there is a $b\in B$, such that $b\notin ran(f)$; (iii) there
is no function $g$ ‘greater than’ $f$, that is, such that $ran(f)\subset
ran(g)$ and $b\in ran(g)$.

If one or more conditions (i)-(iii) do not hold, then one says that ‘$A$ is
not smaller than $B$’ (in symbols, $|A|\nless|B|$). Now we have the full
statement of what we will call ‘Buzaglo’s Principle’:

Buzaglo’s Principle (BuP). $|A|=|B|\leftrightarrow|A|\nless|B|$ and
$|B|\nless|A|$.

There are trivial cases of a set’s not being smaller than another one: for
instance, $\mathbb{R}$ is not smaller than $\mathbb{N}$, as, clearly, one can
find an $f:\mathbb{R}\rightarrow\mathbb{N}$, such that there is no $n\notin
ran(f)$. But the fact that $|\mathbb{N}|\nless|\mathbb{R}|$ is less trivial
and helps to show that, by BuP:

$|\mathbb{N}|=|\mathbb{R}|$

In fact, it can be shown that, for any uncountable set $A$:

$|\mathbb{N}|=|A|$

so, BuP is one further example of a ‘one-cardinality’ theory, which assigns
the same cardinality to all infinite sets.

Although the motivations behind BuP are different from those behind Peano’s
early conception of infinite cardinalities, BuP shows that there are, in
principle, ways to sanction the adoption of just one infinite cardinality,
which may be as legitimate as those underlying the ‘bijection method’ which
sanctions the existence of Cantor’s transfinite. Of course, whether, and in
what sense, BuP (as much as Peano’s ‘single-cardinality’ conception, for that
matter) is an interesting and mathematically fruitful principle is another
question, which is besides the scope of this paper.

## References

  * Benci et al., (2006) Benci, V., Di Nasso, M., and Forti, M. (2006).  An Aristotelian notion of size.  Annals of Pure and Applied Logic, 143(43-53).
  * Bettazzi, (1887) Bettazzi, R. (1887).  Sul Concetto di Numero.  Periodico di matematica per l’insegnamento secondario, 2:97–113; 129–43.
  * Borga et al., (1985) Borga, M., Freguglia, P., and Palladino, D. (1985).  I contributi fondazionali della scuola di Peano.  Franco Angeli, Milano.
  * Buzaglo, (2002) Buzaglo, M. (2002).  The Logic of Concept Expansion.  Cambridge University Press, Cambridge.
  * Cantor, (1887) Cantor, G. (1887).  Mitteilungen zur Lehre vom Transfiniten, i-ii.  Zeitschrift für Philosophie und philosophische Kritik, I, vol. 91, p. 81-125; II, vol. 92, p. 240-65.
  * Cantor, (1895) Cantor, G. (1895).  Beiträge zur Begründung der transfiniten Mengenlehre, 1.  Mathematische Annalen, 46:481–512.
  * Cantor, (1990) Cantor, G. (1990).  Briefe.  Springer, Berlin.
  * Cassina, (1955) Cassina, U. (1955).  Storia ed analisi del “Formulario completo” di Peano. Nota I.  Bollettino dell’Unione Matematica Italiana, 10(2):244–265.
  * Dedekind, (1888) Dedekind, R. (1888).  Was sind und was sollen die Zahlen? Vieweg, Braubschweig.
  * Ehrlich, (2006) Ehrlich, P. (2006).  The Rise of Non-Archimedean Mathematics and the Roots of a Misconception I: the Emergence of Non-Archimedean Systems of Magnitudes.  Archive for History of Exact Sciences, 60(1):1–121.
  * Ferreirós, (2010) Ferreirós, J. (2010).  Labyrinth of Thought. A History of Set Theory and its Role in Modern Mathematics.  Birkhäuser, Basel.
  * Frege, (1884) Frege, G. (1884).  The Foundations of Arithmetic.  Pearson-Longman, New York.
  * Freguglia, (2021) Freguglia, P. (2021).  Peano and the debate on the infinitesimals.  Philosophia Scientiae, 25(1):145–156.
  * Galileo, (1914) Galileo, G. (1914).  Dialogues concerning two new sciences.  Dover Publications, London.
  * Grattan-Guinness, (2000) Grattan-Guinness, I. (2000).  The Search for Mathematical Roots, 1870-1940. Logics, set theories and the foundations of mathematics from Cantor through Russell to Gödel.  Princeton University Press, Princeton.
  * Kennedy, (1980) Kennedy, H. (1980).  Life and Works of Giuseppe Peano.  Reidel, Dordrecht.
  * Mancosu, (2009) Mancosu, P. (2009).  Measuring the Size of Infinite Collections of Natural Numbers: was Cantor’s Theory of Infinite Number Inevitable?  Review of Symbolic Logic, 2(4).
  * Mancosu, (2016) Mancosu, P. (2016).  Abstraction and Infinity.  Oxford University Press, Oxford.
  * Peano, (1889) Peano, G. (1889).  Arithmetices principia, nova methodo exposita.  Bocca, Torino.
  * Peano, (1891) Peano, G. (1891).  Sul concetto di numero.  Rivista di matematica, 1:87–102; 265–67.
  * Peano, (1892) Peano, G. (1892).  Dimostrazione dell’impossibilità di segmenti infinitesimi costanti.  Rivista di matematica, 2:58–62.
  * Peano, (1895) Peano, G. (1895).  Formulaire de mathématiques, volume 1.  Bocca, Torino.
  * Peano, (1897) Peano, G. (1897).  Formulaire de mathématiques, volume 2, sezione 1.  Bocca, Torino.
  * Peano, (1898) Peano, G. (1898).  Formulaire de mathématiques, volume 2, sezione 2.  Bocca, Torino.
  * Peano, (1899) Peano, G. (1899).  Formulaire de mathématiques, volume 2, sezione 3.  Bocca, Torino.
  * Peano, (1901) Peano, G. (1901).  Formulaire de mathématiques, volume 3.  Carré et Naud, Paris.
  * Peano, (2008) Peano, G. (2008).  Opera omnia (cd-rom).  Dipartimento di Matematica, Università di Torino.
  * Randall Holmes, (2017) Randall Holmes, M. (2017).  Alternative Axiomatic Set Theories.  Stanford Encyclopedia of Philosophy.
  * Rodriguez-Consuegra, (1991) Rodriguez-Consuegra, F. (1991).  The mathematical philosophy of Bertrand Russell: origins and development.  Birkhäuser, Basel.
  * Sergeyev, (2017) Sergeyev, Y. (2017).  Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems.  EMS Surveys in Mathematical Sciences, 4:219–320.
  * Veronese, (1892) Veronese, G. (1892).  Osservazioni su una dimostrazione contro il segmento infinitesimo attuale.  Rendiconti del circolo matematico di Palermo, VI:1–4.
  * Vivanti, (1895) Vivanti, G. (1895).  Review of G. Peano’s ‘Dimostrazione dell’impossibilità di segmenti infinitesimi costanti’.  Jahrbuch über die Fortschritte der Mathematik, 24:68–69.
  * Vopěnka, (1979) Vopěnka, P. (1979).  Mathematics in the Alternative Set Theory.  Teubner, Leipzig.

###### Contents

  1. 1 Introduction
  2. 2 Did Peano Have A Definite Conception of the Infinite?
  3. 3 The Evolution of Peano’s Conception
    1. 3.1 Peano 1891: ‘On the Concept of Number’
    2. 3.2 The Cantor-Peano Correspondence (1895)
    3. 3.3 The Infinite in the Formulaire (1895 to 1901)
  4. 4 Interpretations of Peano’s Conception
    1. 4.1 A Galilean View?
    2. 4.2 (Geometric) Infinitesimals and Cantorian Set Theory
  5. 5 Concluding Remarks and Ways Forward


# Competing Uniaxial Anisotropies in Epitaxial Fe Thin Films Grown on
InAs(001)

James M. Etheridge  Joseph Dill  School of Physics and Astronomy, University
of Minnesota, Minneapolis, Minnesota 55455  Connor P. Dempsey  Mihir
Pendharkar  Electrical and Computer Engineering, University of California
Santa Barbara, Santa Barbara, California 93106, USA  Javier Garcia-Barriocanal
Guichuan Yu  School of Physics and Astronomy, University of Minnesota,
Minneapolis, Minnesota 55455  Vlad S. Pribiag  Paul A. Crowell
<EMAIL_ADDRESS>School of Physics and Astronomy, University of Minnesota,
Minneapolis, Minnesota 55455  Chris J. Palmstrøm  Electrical and Computer
Engineering, University of California Santa Barbara, Santa Barbara, California
93106, USA  Materials Department, University of California Santa Barbara,
Santa Barbara, California 93106, USA

###### Abstract

We report on the interplay of two uniaxial magnetic anisotropies in epitaxial
Fe thin films of varying thickness grown on InAs(001) as observed in
ferromagnetic resonance experiments. One anisotropy originates from the
Fe/InAs interface while the other originates from in-plane shear strain
resulting from the anisotropic relaxation of the Fe film. X-ray diffraction
was used to measure the in-plane lattice constants of the Fe films, confirming
the correlation between the onset of film relaxation and the corresponding
shear strain inferred from ferromagnetic resonance data. These results are
relevant for ongoing efforts to develop spintronic and quantum devices
utilizing large spin-orbit coupling in III-V semiconductors.

Ferromagnetic thin films grown epitaxially on semiconductor substrates have
been of interest in the field of spintronics for several decades. They serve
as model systems for spin injection and spin detection, which would be
critical components of a spin transistor.[1, 2, 3] For example, the system of
Fe on GaAs(001) has been extensively studied due to the near lattice match of
GaAs and Fe as well as the ability to form Schottky tunnel barriers.[3, 4, 5,
6, 7, 8] However, the introduction of semiconductors with larger spin orbit
coupling (SOC) is necessary for the realization of a fast and efficient spin
field-effect transistor. [9, 10, 11] Recently, there has also been growing
interest in developing quantum devices that integrate magnetic elements with
large-SOC semiconductors. [12, 13, 14, 15, 16, 17] The two most commonly used
III-V semiconductors with large SOC are InAs and InSb, whose lattice mismatch
with Fe are 5.4% and 11.5%, respectively. This is quite large compared to the
1.4% mismatch between Fe and GaAs. Despite the possible challenges that a
large lattice mismatch may present, ferromagnets (FM) on III-V semiconductors
with strong SOC must be investigated further given their potential for
applications.

When epitaxial ferromagnets with a bulk cubic symmetry are grown on III-V
surfaces, there are two primary sources of in-plane uniaxial magnetic
anisotropy. The first arises from the anisotropic interfacial bonding between
the substrate and FM film due to the breaking of the fourfold symmetry at the
III-V surface. Independent of the III-V surface reconstruction, the easy axis
of this anisotropy is along the [110] direction.[18, 19, 20, 21, 22] The
second uniaxial anisotropy is due to a magnetoelastic energy originating from
anisotropic in-plane strain.[4, 23] This results in a magnetoelastic energy
with a minimum along the [1$\bar{1}$0] direction. The end result is a
thickness-dependent competition between these two uniaxial terms as well as
the FM bulk anisotropy. In this Letter we demonstrate the FM thickness
dependence (1.4 to 39.0 nm) of the competition between the two uniaxial
anisotropies as well as the cubic anisotropy of bulk Fe, whose easy axes are
along the <100> direction of Fe/InAs(001) heterostructures. To probe these
anisotropies room temperature broadband ferromagnetic resonance (FMR)
measurements were performed. In addition to the FMR experiments, reciprocal
space mapping (RSM) was performed using x-ray diffraction (XRD) to determine
lattice constants and subsequent lattice strain[24].

The Fe films were grown by molecular beam epitaxy (MBE). The InAs(001)
substrates were cleaned for one hour in an ultrahigh vacuum (UHV) chamber with
a base pressure $\sim$ $1\times 10^{-10}$ mbar using atomic hydrogen. The
temperature of the substrate was 475 ∘C, as read by a thermocouple close to
the sample, and the hydrogen cracker was 1700 ∘C. After cleaning, the
substrates were loaded into a VG V80 UHV growth chamber with a base pressure
$\sim$ $5\times 10^{-11}$ mbar, where elemental Fe was deposited on the
substrate from an effusion cell. During deposition the substrate was
maintained at room temperature except for the 4.6 nm film, where the sample
was heated to 75∘C as measured by a thermocouple. After the deposition of Fe
was completed, the samples were either capped with approximately 6 nm of Ti in
the same VG V80 chamber as the Fe, or in vacuo transferred to an inter-
connected UHV chamber where a 10 nm Au cap was grown at room temperature. The
lower growth temperatures used in this work were chosen to avoid diffusion of
Fe into InAs and to minimize any interfacial reactions.

Reflection high energy electron diffraction (RHEED) performed during growth
confirmed the film was oriented as expected as well as suggesting a single
crystalline phase. Post-growth, grazing incidence x-ray diffraction (GIXRD)
was used to determine the precise Fe thickness for each sample. Vibrating
sample magnetometry yielded hysteresis loops for each sample at multiple
orientations, enabling the determination of the saturation magnetization and
coercivity for each sample. Using the Stoner-Wolfarth model, theoretical
hysteresis loops were generated with results qualitatively agreeing with the
experimental results. Wide-angle XRD was performed in concert with the RSM,
with both pointing to the same results, which are discussed in detail below.

Figure 1: a) The inset shows the geometry of the sample with respect to the
applied magnetic field. a)-c) Azimuthal angular dependence of resonance fields
with corresponding fits to Eq. 2 shown in the solid curves.

Broadband FMR measurements were performed at room temperature using a coplanar
waveguide (CPW) in a simple transmission geometry with modulation of the
applied magnetic field.[25] The applied field was swept through resonance, and
the signal was fit to a Lorentzian derivative function. From these fits
$H_{FMR}$, the strength of the applied field at resonance, was extracted. This
procedure was repeated as $\phi_{H}$, the angle of the applied field in the
plane of the film, was varied from 0∘ to 360∘ in increments of 5∘. The data
for several samples are shown in Fig. 1. They show a combination of a two-fold
and four-fold symmetry of varying strengths dependent on the sample thickness.
For the thinnest sample, the two-fold component is strongest with respect to
the four-fold component. The magnetic easy axis of this sample is along the
[110] direction of the crystal. For the intermediate thickness of 4.6 nm, the
easy axis switches to the [1$\bar{1}$0] direction, and there is an increase in
the relative strength of the four-fold component. The 90∘ rotation of the easy
axis signals the relaxation of the Fe film, and this will be discussed in
depth later. In the thicker samples, the magnetic easy axis is along the
[1$\bar{1}$0] direction, indicating that they have also relaxed. When the film
thickness reaches 39.0 nm, the data show a dominant four-fold symmetry that
would be expected from a bulk Fe sample. However, a small two-fold component,
whose easy axis is along the [1$\bar{1}$0] direction, still remains,
suggesting that complete relaxation has not yet occurred.

To quantitatively describe the magnetic anisotropy of this series of
heterostructures, a model of the free energy density was introduced with
Zeeman, two uniaxial, cubic, and shape anisotropy contributions,

$\begin{split}F=&-\mathbf{M}\cdot\mathbf{H}+K_{u}\,\sin^{2}\phi+K_{u2}\,\sin^{2}(\phi+\pi/2)\\\
&+K_{1}(\alpha_{1}^{2}\alpha_{2}^{2}+\alpha_{2}^{2}\alpha_{3}^{2}+\alpha_{3}^{2}\alpha_{1}^{2})+2\pi
M_{eff}^{2}\,\cos^{2}\theta,\end{split}$ (1)

where $\mathbf{M}$ is the magnetization vector, $\mathbf{H}$ is the applied
magnetic field vector, $K_{u}$ is the uniaxial anisotropy constant originating
from the Fe/InAs interface, $K_{u2}$ is the uniaxial anisotropy constant
originating from anisotropic strain, $K_{1}$ is the cubic anisotropy constant,
$\alpha$’s are the directional cosines of the magnetization, and $M_{eff}$ is
the effective magnetization. The concept of effective magnetization comes from
the combination of demagnetizing energy and perpendicular magnetic anisotropy.
This free energy density can be used to determine the condition for FMR, first
shown by Suhl,[26] with the result being,

$\omega=\frac{\gamma}{M_{s}\sin{\theta}}\sqrt{F_{\theta\theta}F_{\phi\phi}-\left(F_{\theta\phi}\right)^{2},}$
(2)

where $F$ is the free energy density, $\omega$ is the frequency, $\gamma$ is
the gyromagnetic ratio, $M_{s}$ is the saturation magnetization, $\phi$ is the
azimuthal angle of the magnetization with respect to the [110] direction, and
$\theta$ is the polar angle of the magnetization with respect to the [001]
direction. The subscripts denote derivatives with respect to $\theta$ and
$\phi$.

We made the ansatz that $K_{u}$ is a 3D effective surface energy density, so
it must be inversely proportional to film thickness.[23, 20, 21] Thus,

$K_{u}=\frac{K_{u}^{int}}{t},$ (3)

where $t$ is the thickness of the film and $K_{u}^{int}$ is the purely surface
portion of the uniaxial anisotropy. It is assumed that $\mathbf{M}$ and
$\mathbf{H}$ are in the plane of the film. A previously reported value of 1714
$\textrm{emu}/\textrm{cm}^{3}$ was used for the magnetization.[27] At
resonance, the applied magnetic field is large enough, greater than 800 Oe, to
assume that the magnetization is in the same direction as the applied field.
This assumption was verified for all films through hysteresis loops, including
measurements with the applied field along the magnetic hard axis. Eq. 2 was
manipulated to express $H_{FMR}$ as a function of $\phi_{H}$ in order to fit
the experimental data of $H_{FMR}$ versus $\phi_{H}$. From these fits,
$K_{u}$, $K_{u2}$, $K_{1}$, and $M_{eff}$ were extracted.

For the thinnest sample, 1.4 nm, the film is pseudomorphic, so $K_{u2}$ can be
set to zero. This determination was made in light of the following results:
for thicker samples, greater than 3.2 nm, the easy axis is rotated 90∘, and an
Fe peak appears in the x-ray diffraction data. A pseudomorphic sample has an
Fe peak on top of the InAs peak making it very difficult to separate the two.
The remaining free parameters were determined through fitting. With $K_{u}$
extracted and the thicknesses determined through x-ray reflectivity (XRR),
$K_{u}^{int}$ was determined to be $2\times
10^{-2}\textrm{erg}/\textrm{cm}^{2}$. Previous works on Fe/GaAs
heterostructures have seen this surface energy term $\sim
10^{-1}\textrm{erg}/\textrm{cm}^{2}$.[20, 21] With all thicknesses known and
$K_{u}^{int}$ extracted, $K_{u}$ was constrained for subsequent fits. The data
from the 3.2 nm film are more ambiguous than that of the 1.4 nm film. The
magnetic easy axis of the 3.2 nm sample is along the [110] direction of the
crystal, just as it is in the 1.4 nm sample. In the x-ray diffraction data,
seen in panel c) and d) of Fig. 4, there is a shoulder of the InAs peak which
is most likely the emergence of an Fe peak. However, $K_{u2}$ must still be
small compared to $K_{u1}$ as there is not a 90∘ rotation of the easy axis.
Thus, $K_{u2}$ again was set to zero for its corresponding fit. None of the
other samples were pseudomorphic and the x-ray data were readily analysable,
so $K_{u2}$ was not constrained for their fits.

Figure 2: The inset shows the structure of the InAs(001) including the
direction of the As dimer rows. Anisotropy constants are plotted as a function
of Fe thickness.

The values of $K_{u}$, $K_{1}$, and $K_{u2}$ resulting from this procedure are
shown in Fig. 2. By the above ansatz, the magnitude of $K_{u}$ is inversely
proportional to thickness as defined in Eq. 3. The magnitude of $K_{1}$
increases significantly with thickness, saturating for thicknesses above 7.1
nm to a value of $5.2\times 10^{5}\textrm{erg}/\textrm{cm}^{3}$. This is
slightly larger than the literature value for bulk Fe at room temperature,
$4.8\times 10^{5}\textrm{erg}/\textrm{cm}^{3}$.[27] $K_{u2}$ is zero for the
two thinnest samples and then also decreases with thickness. The appearance of
a non-zero $K_{u2}$ signals that the Fe has relaxed, albeit not completely,
resulting in an in-plane shear strain of the Fe lattice. From the definition
of magnetoelastic energy,[28, 29] $K_{u2}$ must take the following form,

$K_{u2}=B_{2}\epsilon_{6},$ (4)

where $B_{2}$ is the magnetoelastic coefficient of Fe associated with shear
strain, and $\epsilon_{6}$ is the in-plane shear strain defined as
$\epsilon_{[110]}-\epsilon_{[1\bar{1}0]}$. The extracted values of
$\epsilon_{6}$, shown in Fig. 3, are consistent with values from previous
Fe/GaAs heterostructures, which are on the order of 0.1%.[4, 5] For the set of
samples studied, the critical thickness at which shear strain is introduced to
the system occurs between 3.2 and 4.6 nm. After the relaxation of the film and
subsequent onset of shear strain, the shear strain decreases with thickness,
but does not reach zero. This indicates that the film is not completely
relaxed even at a thickness of 39.0 nm.

Figure 3: The inset shows a parallelepiped and its in-plane diagonals. Shear
strain is plotted as a function of Fe thickness with values determined from
fitting FMR data to Eq. 2.

We attempted to determine $\epsilon_{6}$ directly through XRD and compare to
the value extracted from the FMR data. To do this the lattice constants of
each sample where determined through reciprocal space mapping with several
results shown in Fig. 4. Data taken for two different geometrical
configurations offset by a 90∘ rotation of the sample about the [001]
direction. This was done so that maps around both the (202) and (022)
substrate peaks could be produced. From the RSM’s along with the FMR results,
it is determined that the thinnest sample is pseduomorphic to the substrate.
The data of the second thinnest sample has a shoulder on the InAs peak, most
likely corresponding to an Fe peak. Due to low signal to noise lattice
constants could not be extracted. Despite this, the FMR data for this sample
points to a small and possibly negligible shear strain. The appearance of a
definite offset Fe peak in the thicker samples shows that the Fe layer has
relaxed. To determine the amount of relaxation, the RSM’s, oriented by the
substrate peak, were fit to Gaussian curves and the effective Miller indices
were extracted. With the effective Miller indices of the (101) and (011) Fe
peaks known, the lattice constants of each orientation were calculated. The
map centered on the (202) peak gives the a and c lattice constants while the
one centered on the (022) peak gives b and c. Lattice parameter values are
shown in Table 1. The c parameter determined by wide angle XRD in a coupled
geometry was consistent with the results of the RSM’s, confirming the validity
of the procedure.

Figure 4: a) c) e) g) Reciprocal space maps around the (202) peak of InAs(001). b) d) f) h) Reciprocal space maps around the (022) peak of InAs(001). The Au peak seen in panels e)-h) comes from the Au capping layer. This peak is not seen in a)-d) as the 1.4 and 3.2 nm samples were capped with Ti. The units are Miller indices of InAs, with a0 = 6.06 Å. The red line in f) corresponds to the projection of the RSM in Fig. 5. Table 1: Summary of the lattice parameters determined from RSM’s shown in Fig. 4. Fe Thickness (nm) | a (Å) | b (Å) | c (Å) | aInAs/2 (Å)
---|---|---|---|---
4.6 | 2.908 | 2.908 | 2.868 | 3.03
7.1 | 2.906 | 2.907 | 2.840 | 3.03
10.0 | 2.902 | 2.901 | 2.836 | 3.03
39.0 | 2.892 | 2.892 | 2.839 | 3.03
Figure 5: An example of Gaussian fitting, used to extract the effective Miller
indices, of one of the Fe peak projections of the 4.6 nm sample shown as the
red line in panel f).

Like the FMR data, the RSM’s show that films thicker than 3.2 nm are relaxed.
The data from the 3.2 nm sample is ambiguous in regards to if it pseudomorphic
or not. However, even if the sample has relaxed, little to no shear strain is
introduced to the lattice as inferred from the FMR data. For films 4.6 nm
thick and greater, the lattice develops a tetragonal distortion, with equal a
and b lattice parameters larger than the out-of-plane c parameter. As the
thickness of the film increases the in-plane lattice constants relax towards
the bulk value of 2.866 Å, while maintaining the tetragonal distortion. It is
inferred from the FMR data that the lattice develops decreasing amounts of in-
plane shear strain with increasing thickness as expected. This strain distorts
the cubic lattice into a parallelepiped. All angles defining the
parallelepiped should be extremely close to but not exactly 90∘, resulting in
a base that is nearly a square, but is in actuality a parallelogram whose
diagonals differ in length. The shear strain, $\epsilon_{6}$, which is defined
as the difference of the two diagonal strains, will be nonzero. To determine
the in-plane shear strain, both in-plane lattice parameters and the resulting
diagonals must be determined from a single XRD scan, before rotating the
sample 90∘. Unfortunately, from the XRD geometry used to acquire the data,
this was not possible. A determination of the shear strain using a constant
volume assumption along with the lattice parameters shown in Table 1 was
attempted. However, the resulting in-plane shear strains are orders of
magnitude larger than those expected from the FMR data and thus non-physical.
It is believed that the constant volume assumption is not valid in our
samples. This is not surprising as Poisson’s ratio for Fe is 0.29, and a
material that perfectly adheres to the the constant volume assumption would
have a Poisson’s ratio of exactly 0.5.[30]

It is interesting to note that the 39.0 nm sample is not completely relaxed
with a and b parameters larger than bulk and the c parameter less than bulk.
Although this is surprising, the FMR data also supports the conclusion that
the 39.0 nm sample is not completely relaxed. Fig. 3 clearly shows a small
nonzero shear strain. Although direct evidence of shear strain through in-
plane x-ray diffraction was not investigated, the agreement between current
FMR and x-ray data is strong evidence of its existence.

We have shown through FMR and XRD that the magnetic anisotropy of Fe/InAs(001)
heterostructures is heavily dependent on sample thickness. Our investigations
show that anisotropic relaxation of the Fe thin films results in shear strain,
producing an additional term of magnetoelastic origin in the free energy
density. This magnetoelastic term was confirmed through a rotation of the
uniaxial easy axis as observed through FMR measurements. Similar phenomena
have been reported in Fe/GaAs(001) structures. This suggests that, despite
Fe/InAs having a significantly larger lattice mismatch than Fe/GaAs, this does
not appear to have a detectable detrimental effect on interfacial magnetic
properties. We would expect to see similar results for other Fe/III-V
semiconductor heterostructures as well, which is encouraging for the
developement of applications relying on the interplay between ferromagnetism
and high-SOC semiconductors.

###### Acknowledgements.

We acknowledge Bill Peria and Brett Heischmidt for their helpful input and
discussions throughout this research. This work was supported by the
Department of Energy under award no. DE-SC0019274. Parts of this work were
carried out in the Characterization Facility, University of Minnesota, which
receives partial support from NSF through the MRSEC program.

## References

  * Datta and Das [1990] S. Datta and B. Das, “Electronic analog of the electro-optic modulator,” Applied Physics Letters 56, 665–667 (1990).
  * Sato and Mizushima [2001] R. Sato and K. Mizushima, “Spin-valve transistor with an Fe/Au/Fe(001) base,” Applied Physics Letters 79, 1157–1159 (2001).
  * Hanbicki  _et al._ [2002] A. T. Hanbicki, B. T. Jonker, G. Itskos, G. Kioseoglou, and A. Petrou, “Efficient electrical spin injection from a magnetic metal/tunnel barrier contact into a semiconductor,” Applied Physics Letters 80, 1240–1242 (2002).
  * Thomas  _et al._ [2003] O. Thomas, Q. Shen, P. Schieffer, N. Tournerie, and B. Lépine, “Interplay between Anisotropic Strain Relaxation and Uniaxial Interface Magnetic Anisotropy in Epitaxial Fe Films on (001) GaAs,” Physical Review Letters 90, 017205 (2003).
  * Lu  _et al._ [2009a] J. Lu, P. Xu, Y. Zhu, H. Meng, L. Chen, W. Wang, X. Zhang, J. Zhao, and G. Pan, “Anisotropic strain relaxation of thin Fe film on c(4×4) reconstructed GaAs (001) surface,” Physica E: Low-dimensional Systems and Nanostructures 42, 150–153 (2009a).
  * Lou  _et al._ [2007] X. Lou, C. Adelmann, S. A. Crooker, E. S. Garlid, J. Zhang, K. S. Reddy, S. D. Flexner, C. J. Palmstrøm, and P. A. Crowell, “Electrical detection of spin transport in lateral ferromagnet-semiconductor devices,” Nature Physics 3, 197–202 (2007).
  * Christie  _et al._ [2015] K. D. Christie, C. C. Geppert, S. J. Patel, Q. O. Hu, C. J. Palmstrøm, and P. A. Crowell, “Knight shift and nuclear spin relaxation in Fe/n -GaAs heterostructures,” Physical Review B - Condensed Matter and Materials Physics 92, 1–8 (2015).
  * LeBeau  _et al._ [2008] J. M. LeBeau, Q. O. Hu, C. J. Palmstrøm, and S. Stemmer, “Atomic structure of postgrowth annealed epitaxial Fe/(001)GaAs interfaces,” Applied Physics Letters 93 (2008), 10.1063/1.2990622.
  * Manchon  _et al._ [2015] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A. Duine, “New perspectives for Rashba spin-orbit coupling,” Nature Materials 14, 871–882 (2015).
  * Koo  _et al._ [2009] H. C. Koo, J. H. Kwon, J. Eom, J. Chang, S. H. Han, and M. Johnson, “Control of Spin Precession in a Spin-Injected Field Effect Transistor,” Science 325, 1515–1518 (2009).
  * Chuang  _et al._ [2015] P. Chuang, S. C. Ho, L. W. Smith, F. Sfigakis, M. Pepper, C. H. Chen, J. C. Fan, J. P. Griffiths, I. Farrer, H. E. Beere, G. A. Jones, D. A. Ritchie, and T. M. Chen, “All-electric all-semiconductor spin field-effect transistors,” Nature Nanotechnology 10, 35–39 (2015).
  * Liu  _et al._ [2020] Y. Liu, A. Luchini, S. Martí-Sánchez, C. Koch, S. Schuwalow, S. A. Khan, T. Stankevic, S. Francoual, J. R. L. Mardegan, J. A. Krieger, V. N. Strocov, J. Stahn, C. A. F. Vaz, M. Ramakrishnan, U. Staub, K. Lefmann, G. Aeppli, J. Arbiol, and P. Krogstrup, “Coherent Epitaxial Semiconductor–Ferromagnetic Insulator InAs/EuS Interfaces: Band Alignment and Magnetic Structure,” ACS Appl. Mater. Interfaces 12, 8780–8787 (2020).
  * Yang  _et al._ [2020] Z. D. Yang, B. Heischmidt, S. Gazibegovic, G. Badawy, D. Car, P. A. Crowell, E. P. A. M. Bakkers, and V. S. Pribiag, “Spin Transport in Ferromagnet-InSb Nanowire Quantum Devices,” Nano Lett. 20, 3232–3239 (2020).
  * Sun  _et al._ [2020] J. Sun, R. S. Deacon, X. Liu, J. Yao, and K. Ishibashi, “Spin filtering in germanium/silicon core/shell nanowires with pseudo-helical gap,” Applied Physics Letters 117, 052403 (2020).
  * Vaitiekėnas  _et al._ [2021] S. Vaitiekėnas, Y. Liu, P. Krogstrup, and C. M. Marcus, “Zero-bias peaks at zero magnetic field in ferromagnetic hybrid nanowires,” Nat. Phys. 17, 43–47 (2021).
  * Yu  _et al._ [2021] M. Yu, S. Moayedpour, S. Yang, D. Dardzinski, C. Wu, V. S. Pribiag, and N. Marom, “Dependence of the electronic structure of the EuS/InAs interface on the bonding configuration,” Phys. Rev. Materials 5, 064606 (2021).
  * Yang, Crowell, and Pribiag [2021] Z. Yang, P. A. Crowell, and V. S. Pribiag, “Spin-helical detection in a semiconductor quantum device with ferromagnetic contacts,”  , 1–7 (2021).
  * Kneedler  _et al._ [1997] E. M. Kneedler, B. T. Jonker, P. M. Thibado, R. J. Wagner, B. V. Shanabrook, and L. J. Whitman, “Influence of substrate surface reconstruction on the growth and magnetic properties of Fe on GaAs(001),” Physical Review B 56, 8163–8168 (1997).
  * Herfort  _et al._ [2004] J. Herfort, W. Braun, A. Trampert, H.-P. Schönherr, and K. Ploog, “Atomically engineered interfaces for spin injection: ultrathin epitaxial Fe films grown on As- and Ga-terminated GaAs(001) substrates,” Applied Surface Science 237, 181–188 (2004).
  * [20] .
  * Brockmann  _et al._ [1999] M. Brockmann, M. Zölfl, S. Miethaner, and G. Bayreuther, “In-plane volume and interface magnetic anisotropies in epitaxial Fe films on GaAs(001),” Journal of Magnetism and Magnetic Materials 198-199, 384–386 (1999).
  * Freeland  _et al._ [2001] J. W. Freeland, I. Coulthard, W. J. Antel, and A. P. J. Stampfl, “Interface bonding for Fe thin films on GaAs surfaces of differing morphology,” Physical Review B 63, 193301 (2001).
  * Lu  _et al._ [2009b] J. Lu, H. J. Meng, J. J. Deng, P. F. Xu, L. Chen, J. H. Zhao, and Q. J. Jia, “Strain and magnetic anisotropy of as-grown and annealed Fe films on c (4×4) reconstructed GaAs (001) surface,” Journal of Applied Physics 106, 1–5 (2009b).
  * [24] G. Yu and J. G. Barriocanal, “RSLab : fast 3-dimensional reciprocal space imaging with laboratory X-ray diffraction, Technology Commercialization, Technology No. 2021-290, University of Minnesota,” https://license.umn.edu/product/fast-3-dimensional-reciprocal-space-imaging-with-laboratory-x-ray-diffraction-rslab#.
  * Peria  _et al._ [2020] W. K. Peria, T. A. Peterson, A. P. Mcfadden, T. Qu, C. Liu, C. J. Palmstrøm, and P. A. Crowell, “Interplay of large two-magnon ferromagnetic resonance linewidths and low Gilbert damping in Heusler thin films,” Physical Review B 101, 1–7 (2020).
  * Suhl [1955] H. Suhl, “Ferromagnetic Resonance in Nickel Ferrite Between One and Two Kilomegacycles,” Physical Review 97, 555–557 (1955).
  * Cullity and Graham [2008] B. D. Cullity and C. D. Graham,  _Introduction to Magnetic Materials_ (John Wiley & Sons, Inc., Hoboken, NJ, USA, 2008).
  * Paes and Mosca [2013] V. Z. Paes and D. H. Mosca, “Effective elastic and magnetoelastic anisotropies for thin films with hexagonal and cubic crystal structures,” Journal of Magnetism and Magnetic Materials 330, 81–87 (2013).
  * Hoselitz [1962] K. Hoselitz, “Modern Magnetic Materials,” Physics Bulletin 13, 107–112 (1962).
  * Ledbetter and Reed [1973] H. M. Ledbetter and R. P. Reed, “Elastic Properties of Metals and Alloys, I. Iron, Nickel, and Iron-Nickel Alloys,” Journal of Physical and Chemical Reference Data 2, 531–618 (1973).