Hilbert 2nd problem

a history of the mathematics preceding and relevant to Hilbert’s statement of his 2nd problem, initiating his program for the foundations of mathematics -- see Hilbert problems

By about 1820, mathematicians had developed deductively a large part of analysis using the real numbers and their properties as a starting point.

During the 50 years that followed, in a program that came to be known as the Arithmetization of analysis, Bolzano, Cauchy, Weierstrass, Dedekind, Cantor, and others succeeded in “reducing” analysis to the arithmetic of natural numbers $\mathbb{N}$.

Dedekind himself expressed this as follows:

... every theorem of algebra and higher analysis, no matter how remote, can be expressed as a theorem about natural numbers, -- a declaration I have heard repeatedly from the lips of Dirichlet.

In the final three decades of the 19th century, efforts were underway to axiomatize the whole of mathematics.

It thus became clear that (with the aid of a certain amount of set theoretic and logical apparatus) the entire body of traditional pure mathematics could be constructed rigorously starting from the theory of natural numbers.

These efforts proceeded piecemeal and depended greatly on concurrent developments in logic. The major contributors were these:

• Cantor and Frege in set theory
• Dedekind and Peano in arithmetic
• Hilbert in geometry
• many others in abstract algebras (groups, rings, and fields)

In a 1900 lecture to the International Congress of Mathematicians in Paris, David Hilbert presented a list of open problems in mathematics. The 2nd of these problems is known variously as the compatibility of the arithmetical axioms and the consistency of arithmetic.

The article views the 30-year period from 1872 to 1900 as historical background to Hilbert’s program for the foundations of mathematics. There are other, different and equally interesting views of this same period:

• as a continuation and, indeed, culmination of the previous half-century (1822-1872) during which “mathematicians restored and surpassed the standards of rigour” that had long been established, but then neglected, the whole 80-year period called “the formalisation of mathematics.”
• as the first half of the decades-long effort (1872-193X) “from the days of Cantor and Dedekind in the 1870s, through Russell in the 1900s, to the work of Godel in the 1930s” that resulted in the solid establishment of “the modern discipline of foundations.”

However viewed, this 30-year period, from the construction of the real numbers to the Hilbert Problems address, saw “mathematicians of the first rank” engaged with these questions:

• the character of the infinite
• the relationship between logic and arithmetic
• the status of geometry
• the nature of mathematics itself

For a history of the subsequent development of Hilbert’s program for the foundations of mathematics, which was initiated by his 2nd problem, see the article Hilbert program.

Non-mathematical issues

As is the case for other, especially older programs and periods of mathematics, the history of Hilbert’s program was complicated by non-mathematical issues. Some authors were slow to publish their results; others published only selectively, leaving some important results to be published by students and successors. The works of still others, though published, were partially or completely ignored.

As a first example, consider the work of Galileo. His concerns about the “paradoxical” property of infinite sets are often mentioned in published discussions of the potentially infinite and the actually infinite. Yet, even today, doubts are expressed about whether or not Galileo had influence either on Cantor, the mathematician whose name is most often and most closely associated with the notion of infinite sets, or on any other mathematician.

Further, consider Gauss’ well-known comment about actual infinities in the work of Cantor and the latter’s response:

Gauss: I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a facon de parler, the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without restriction.

Cantor: I answered [Gauss] thoroughly, and on this point did not accept the authority of Gauss, which I respect so highly in all other areas ...

There is some doubt that even today, 130 years after these words were written, that we really understand properly the meaning either of Gauss’ comment or of Cantor’s response.

Again, consider the work of Bolzano. His paper “Paradoxes of the Infinite” contains some remarkable results related to the theory of infinite sets:

• the word “set” appears here for the first time
• examples of 1-1 correspondences between the elements of an infinite set and the elements of a proper subset

Yet Bolzano himself never published these results. The paper itself was not published until 1851, three years after his death, by one of his students. Further, Cantor appears not to have become aware of Bolzano’s paper until 1882, some years after he began his own work on infinite sets, which was motivated by the Arithmetization of analysis. Nor did Cantor mention Bolzano's paper in his own work until 1883

A related historical anomaly is that while Bolzano both knew of and referred to Galileo’s work on the infinite, Cantor did neither.

As a final example, consider that Frege’s work “seems to have been largely ignored by his contemporaries.”

• Three [of six] reviews of the “revolutionary” Begriffsschrift,” including one by no less than Venn, show that their authors were either uninterested in Frege's innovations or had completely misunderstood them.
• The Grundlagen only received a single review, and that one was “a devastatingly hostile” one, by Cantor, whose ideas were [ironically] the closest to Frege's.
• Die Grundgesetze der Arithmetik … except for one review by Peano, was ignored by his contemporaries.

It was not until Russell acknowledged Frege’s work as the trailblazing foundation for the Principia that the greatness of his accomplishment was recognized. Russell himself contrasted the greatness of Frege’s contributions with the limited nature of his influence among his contemporaries as follows:

In spite of the epoch-making nature of [Frege's] discoveries, he remained wholly without recognition until I drew attention to him in 1903.

As a consequence of these and other non-mathematical issues, some mathematical results in the period under examination were achieved multiple times, albeit in slightly different forms or using somewhat different methods, by different authors. Even without the effects of such issues, the mathematics of the past (both long- and recent-past) is still replete with achievements that are said to be “roughly” or “more or less” or “just about” what we know today. About De Morgan’s work on mathematical induction, for example, two types of claims have been made:

1. that he put a process that had been used without clarity on a rigorous basis
2. that he introduced and defined the term “mathematical induction” itself

Yet, another source, citing the contents of De Morgan’s own published papers, has refuted both of these claims.

Finally, a note on some quasi-mathematical matters that are purposely not discussed in this article. Without doubt, positions in the philosophy of mathematics known as Logicism, Formalism, and Intuitionism, along with important methodological and epistemological considerations, grew out of the mathematical practice of the late 19th and early 20th centuries. Further, these philosophical positions were of great interest to some mathematicians and certainly influenced the mathematical problems on which they chose to work. Yet a discussion of either the past origins of or the current nature and status of these philosophical positions would not significantly aid our understanding of the mathematics that resulted from the work of those mathematicians.

Much of the existing literature [of the period surrounding the Hilbert Problems address] has been philosophically motivated and preoccupied with the exegesis of individual thinkers, notably Frege and Russell, who are widely (and rightly) viewed as founding giants of analytical philosophy. But the wider mathematical context has in the process often been lost from sight.

Hence, a discussion of these philosophical positions is not included in this article.

Introduction of infinite sets

In mathematics, uses of infinity and the infinite (and great concerns about those uses) are as old as Grecian urns. Greek mathematicians followed Aristotle in dividing such uses into two major types, one called “potential infinity”, the other called “actual infinity.”

With respect to magnitudes:

a potential infinity was something endlessly extendible, and yet forever finite;

an actual infinity was something such as the number of points on a line.

Similarly, with respect to sets:

a potentially infinite set was, for example, a finite collection of numbers that can be enlarged as much as one wished

an actually infinite set was, for example, the complete collection of all such natural numbers

Ancient Greek mathematicians developed rigourous methods for using potential infinities. However, with the apparent exception of Archimedes noted below, they avoided using actual infinities. Important early examples of uses of infinity and the infinite include these:

• Euclid skirted the notion of the actually infinitely large in proving that the primes are potentially infinite. This is how he stated his theorem:

Prime numbers are more than any assigned magnitude of prime numbers.

• Archimedes, however, appears to have investigated actually infinite numbers of objects:

... certain objects, infinite in number, are "equal in magnitude" to others [implying] that not all such objects, infinite in number, are so equal. ... [thus] infinitely many objects [of] definite, and different magnitudes … are manipulated in a concrete way, apparently by something rather like a one-one correspondence...

Oresme, an early (12th century) mathematician, examined infinite sets using a method prescient of Cantor’s method of one-to-one correspondence. Oresme demonstrated that two actually infinite sets (the set of odd natural numbers and the set of all natural numbers) could be “different” and “unequal” and yet “equinumerous” with one another. He concluded that notions of equal, greater, and less do not apply to the infinite.

Mathematical induction, as a technique for proving the truth of propositions for an infinite (indefinitely large) number of values, was used for hundreds of years before any rigorous formulation of the method was made

Galileo produced the standard one-to-one correspondence between the positive integers and their squares, reminiscent of Oresme’s work. He termed this a “paradox” that results “unavoidably” from the property of infinite sets and concluded, alike with Oresme, that infinite sets are incomparable.

... the totality of all numbers is infinite, and that the number of squares is infinite.; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and, finally, the attributes "equal", "greater", and "less" are not applicable to the infinite, but only to finite quantities.

As recently as 1831, Gauss himself argued against the actually infinite:

I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a facon de parler, the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without restriction.

For the most part, however, mathematicians of the 19th and 20th centuries developed and readily took up methods for using actual infinities that were as rigorous as those the Greeks developed for potential infinities. Certainly Bolzano had no concerns about the “paradoxical” property of infinite sets. Indeed, his theories of mathematical infinity anticipated Cantor's theory of infinite sets. His contribution to the understanding of the nature of the infinite was threefold:

1. he defined the idea of a set

I call a set a collection where the order of its parts is irrelevant and where nothing essential is changed if only the order is changed.

2. he argued that the infinite set does exist

if the integers are a set, then arbitrarily large subsets of integers are subsets of the set of integers, which must itself be actually infinite

3. he gave examples to show that, unlike for finite sets, the elements of an infinite set could be put in 1-1 correspondence with elements of one of its proper subsets.

The actual infinite is said to have entered algebra in the 1850s in Dedekind’s work with quotient constructions for modular arithmetic:

[T]he whole system of infinitely many functions of a variable congruent to each other modulo p behaves here like a single concrete number in number theory.… The system of infinitely many incongruent classes—infinitely many, since the degree may grow indefinitely—corresponds to the series of whole numbers in number theory.

The five-year period 1868–1872 has been called "the birth of set-theoretic mathematics." A salient milestone was 1871, when Dedekind introduced “an essentially set-theoretic viewpoint … using set operations and … structure-preserving mappings … and terminology that Cantor was later (1880) to use in his own work.

By 1872, procedures involving infinite sets were employed in constructions of irrational real numbers developed during the Arithmetization of analysis by Weierstrass, Dedekind, and Cantor. “Thus analysis [had been reduced] not simply to the theory of natural numbers, but to the theory of natural numbers together with the theory of infinite sets.”

The constructions of Cantor and Dedekind especially relied implicitly on set theory and, further, “involve the assumption of a Power Set principle.”

The realization that (apparently) all the material needed for analysis could be constructed out of the natural numbers using set-theoretic means led to these new questions:

• What further could be said about set-theoretic procedures and assumptions of logic, both of which underlay these accounts of the real numbers?
• Do we have to take the natural numbers themselves as simply given, or can anything further be said about those numbers, perhaps by reducing them to something even more fundamental?

In the 1870s, however, the notions of set and class themselves appeared straightforward. Their highly problematic aspects did not become apparent until the various paradoxes of set theory and the theory of transfinite sets.

Early development of mathematical logic

The history of logic has been described, “with some slight degree of oversimplification,” as having three stages: (1) Greek logic, (2) Scholastic logic, and (3) mathematical logic. From ancient times through the first half of the 19th century, the state of logic was as follows:

• logic was understood to be “the laws of thought”
• the Aristotelian syllogism was the ultimate form of all reasoning
• the logic that a mathematician used did not affect the mathematics that she did

During the second half of the 19th century, the development of mathematical logic began with these two steps:

1. the algebraization of syllogistic logic
2. the development of the predicate calculus

The algebra of logic

The beginning of mathematical logic has been dated from the years in which Boole and De Morgan published their works on the algebraization of Aristotelian logic.

Whereas in [Greek and Scholastic logic] theorems were derived from ordinary language, [mathematical logic] proceeds in a contrary manner—it first constructs a purely formal system, and only later does it look for an interpretation in everyday speech.

Thus, with the advent of mathematical logic, the logic of the syllogism came to be treated as one interpretation of a calculus of logic.

Syllogistic logic

Aristotle’s system of syllogistic logic is closely linked to the grammatical structure of natural language.

A syllogism is a logical argument consisting (usually) of three statements, one of which (the conclusion) is inferred from the other two (the premises).

Here is an Example Syllogism:

All Greeks are Sapiens

All Sapiens are Mortal

therefore

All Greeks are Mortal

Each statement of this syllogism has two parts: a Noun Phrase and a Verb Phrase

the Noun Phrase consists of a Quantifier (All, Some, No, or Not All) and a Common Noun

the Verb Phrase consists of a Copula Verb (are) and a Common Noun

We can think of the Common Nouns in the statements of a syllogism as expressing properties of things or as referring to classes of things that have those properties.

In each syllogism, there is always one Common Noun that occurs in both premises, but not in the conclusion. This Common Noun, which links the two premises of the syllogism, is called the middle term of the syllogism. In the Example Syllogism above, the middle term is the Common Noun “Sapiens”.

A syllogism is valid if the conclusion follows logically from the premises, no matter what Common Nouns are used in its statements; otherwise, the syllogism is invalid. If the syllogism is valid and the premises are true, then the conclusion is true.

The Example Syllogism is valid. Its validity has nothing to do with the particular Common Nouns that are used. If the Common Nouns in the Example Syllogism were replaced by different Common Nouns, the result would still be a valid syllogism. It is the form of the Example Syllogism that makes it valid, not the Common Nouns used in its statements. Replacing the Common Nouns in the Example Syllogism by symbols for classes makes this clear:

All A are B

All B are C

therefore

All A are C

Each statement of a syllogism is one of 4 types, as follows:

A .. All A are B

I ... Some A are B

E... No A are B (= All A are not B)

O... Not All A are B (= Some A are not B)

(In ordinary language, the forms E and O have the alternate forms shown. Note that by introducing an additional Copula Verb are not and using the equivalent forms, we can eliminate the Quantifiers No and Not All, reducing them to just All and Some.)

The statements of the Example Syllogism are all of type A and, therefore, the Example Syllogism itself is said to be of type AAA. There are 19 such valid Aristotelian syllogistic types.

Peacock's and De Morgan’s contributions

Even before Boole’s and De Morgan's work, important steps were taken towards the development of a calculus of logic. As early as 1830, Peacock suggested that the symbols for algebraic objects need not be understood only as numbers.

'Algebra' … has been termed Universal Arithmetic: but this definition is defective, in as much as it assigns for the general object of the science, what can only be considered as one of its applications.

In his treatise, Peacock distinguished between arithmetical algebra, with laws derived from operations on numbers, and symbolic algebra, which he describes as follows:

the science which treats the combinations of arbitrary signs and symbols by means defined through arbitrary laws…. We may assume any laws for the combination and incorporation of such symbols, so long as our assumptions are independent, and therefore not inconsistent with each other.

In 1847, De Morgan extended Peacock’s vision for a symbolic algebra with the notion that the interpretations not only of symbols for algebraic objects, but also of symbols for algebraic operations were arbitrary.

De Morgan’s contribution to logic was twofold. First, he insisted on the purely formal or, as he put it, “symbolic” nature of algebra, the study of which has as it object “symbols and their laws of combination, giving a symbolic algebra which may hereafter become the grammar of a hundred distinct significant algebras.” Consider his example of a commutative algebra to which he provided five interpretations, among which are the three listed:

Given symbols M, N, +, and one sole relation of combination, namely that M + N is the same as N + M:

1. M and N may be magnitudes, and + the sign of addition of the second to the first
2. M and N may be numbers, and + the sign of multiplying the first to the second
3. M and N may be nations, and + the sign of the consequent having fought a battle with the antecedent

Second, De Morgan clarified the nature of logical validity as “that part of reasoning which depends upon the manner in which inferences are formed…. Whether the premises be true or false, is not a question of logic…. the question of logic is, does the conclusion certainly follow if the premises be true?”

Boole’s algebra of logic

Both Boole and De Morgan were aware of the limitations of syllogistic logic, in particular, that there were inferences known to be valid, but whose validity could not be demonstrated by syllogistic logic. Their intent was to develop “a general method for representing and manipulating all logically valid inferences.” The significant difference in Boole’s approach from De Morgan's was the algebraic methods that Boole adopted.

In 1847, in “a little book that De Morgan himself recognized as epoch-making,” Boole undertook the following:

• the goal: “to express traditional logic more perspicuously using the techniques of algebra” such that deduction becomes calculation
• the program: to develop an algebraic calculus and show that the doctrines of traditional logic can be expressed using this calculus.

In this early work, Boole extended De Morgan’s view about the formal nature of algebra by presenting the view that the essential character of the whole of mathematics is formal, somewhat as follows:

If any topic is presented in such a way that it consists of symbols and precise rules of operation upon these symbols, subject only to the requirement of inner consistency, this topic is part of mathematics.

In 1854, Boole published a second book, the completion of his efforts “to incorporate logic into mathematics by reducing it to a simple algebra, pointing out the analogy between algebraic symbols and those that represent logical forms, and beginning the algebra of logic that came to be called Boolean algebra.”

Boole eventually gave his uninterpreted calculus three interpretations, in terms of classes, of probabilities, and also of propositions. These various interpretations were possible because of analogies among the concepts of a class, an event, and a statement. As a consequence, the "order" relation in a Boolean algebra can be interpreted variously as set-theoretical inclusion, as causal follow-up of events, as logical follow-up of statements. A modified version of the third interpretation of his calculus became modern propositional logic. This latter is today the lowest level of modern logic, but at the time and in effect, it was all of logic, because it was able to be used for Aristotelian syllogistic logic:

• it used symbols for statements rather than numbers
• it defined operations on statements rather than on numbers
• it defined deductions as equations and as the transformation of equations

Boole’s second (1854) book was an effort “to correct and perfect” his first. He introduced the formalisms of his algebra, including these:

• Classes were $x, y, z$.
• There was a universal class $1$ and an empty class $0$.
• Multiplication $x \cdot y$ was intersection, yielding $x \cdot y = y \cdot x$.
• Next, he gave the idempotent law $x \cdot x = x$.
• Addition $x+y$ was aggregation (for $x, y$ disjoint), yielding $x + y = y+x$ and $z(x + y) = z \cdot x + z \cdot y$.
• Also, $x − y = − y + x$ and $z(x − y) = z \cdot x − z \cdot y$.

Boole did not, however, go on from all of this to build an axiomatic foundation for his algebra of logic. Instead, he introduced three theorems (Expansion, Reduction, Elimination) and used them in his “General Method” for analyzing syllogistic arguments.

Boole's introduction of an Elimination theorem is interesting as an example of his commitment to an algebraic approach to logic. As shown in the Example Syllogism above, the middle term of a syllogism is a Common Noun that occurs in both of the premises. In effect, it links the two other Common Nouns of the syllogism, allowing them to be joined in the conclusion. Observing this, Boole reasoned that syllogistic logic produces a conclusion by eliminating that middle term, so he introduced into his algebra of logic an Elimination theorem, which he borrowed from the ordinary algebraic theory of equations.

The final version of Boole’s method “for analyzing the consequences of propositional premises,” briefly stated, is as follows:

1. convert (or translate) the premise statements of the syllogism into equations,
2. apply a prescribed sequence of algebraic processes to the equations, including application of the three theorems mentioned above, yielding equational conclusions
3. convert the equational conclusions into statements, yielding the desired conclusions of the syllogism.

With this method Boole had replaced the art of reasoning from premise statements to conclusion statements by a routine mechanical algebraic procedure.

Boole showed, with somewhat mixed results, that his algebra provided “an easy algorithm for syllogistic reasoning,” an elementary example of which is as follows:

an Aristotelian syllogism of the AAA type:

(1) $\text{All } A \text{ are } B$ .... $A \cdot B = A$

(2) $\text{All } B \text{ are } C$ .... $B \cdot C = B$

(3) ........................... $A (B \cdot C) = A$ .... substituting in (1) the value of $B$ given by (2)

(4) ........................... $(A \cdot B)C = A$ .... applying associative law for multiplication

(5) $\text{All } A \text{ are } C$ .... $A \cdot C = A$ ........... substituting in (4) the value of $AB$ from (1)

No less than De Morgan himself praised Boole’s work as a remarkable proof that “the symbolic processes of algebra, invented as tools of numerical calculation, [are] competent to express every act of thought, and to furnish the grammar and dictionary of an all-containing system of logic.”

Taken at face value, De Morgan’s praise overstated the adequacy of Boole’s logic of propositions without quantification, in two ways:

1. it was inadequate to express some important statements of mathematics such as the law of mathematical induction, on which De Morgan himself had worked;
2. it was also inadequate to express some statements of ordinary language with a form such as, “If all horses are animals, then all heads of horses are heads of animals.”

In fact, this example was De Morgan’s own, intended “to show the inadequacy of traditional logic” and that, for a logic adequate to express this example, “binary relations are essential.” If, however, we take De Morgan’s comment to be about some yet-to-be-developed logic with quantification, then we can accept that his optimism about Boole's calculus was not misplaced.

Variants and improvements

For three decades after Boole introduced his calculus in 1847, “most researchers interested in formal logic worked on extending and improving [his] system.”

In 1864, Jevons published an alternative system of algebraic logic, retaining Boole’s use of algebraic equations as the basic form of logical statements, but rejecting Boole’s desire to retain “dependence on” the ordinary algebra of numbers. More generally, Jevons replaced the use of classes (associated with quantity) with predicates (associated with quality).

Both Boole and Jevons understood logic to be an expression of “the laws of thought.” Yet Boole had more of an algebraic concept of logic and saw deduction as calculation, while Jevons argued that mathematics proceeds from logic, seeing calculation as deduction..

De Morgan himself extended Boole’s calculus with a law of duality that asserts for every theorem involving addition and multiplication, there is a corresponding theorem in which the words addition and multiplication are interchanged.

Interpreted as a logic of classes, we have this:

If x and y are subsets of a set S, then

• the complement of the union of x and y is the intersection of the complements of x and y
• the complement of the intersection of x and y is the union of the complements of x and y

Interpreted as a logic of propositions, we have this:

If p and q are propositions, then

• not (p or q) equals not p and not q
• not (p and q) equals not p or not q

Though De Morgan had clearly located the inadequacy of syllogistic logic in its inability to express binary relations, he himself lacked “an adequate apparatus for treating the subject.” The title “creator of the theory of relations” has been awarded to C. S. Peirce.

In several papers published between 1870 and 1882, [Peirce] introduced and made precise all the fundamental concepts of the theory of relations and formulated and established its fundamental laws … in a form “much like the calculus of classes developed by G. Boole and W. S. Jevons, but which greatly exceeds it in richness of expression.”

Peirce's work on (binary) relations was continued and extended in a very thorough and systematic way by Schroeder, whose published work of 1895 “is so far the only exhaustive account of the calculus of relations.”

Cantor’s early theory of sets

Set theory is the study sets, their properties, and the operations that can be performed on them. It has been especially concerned with sets that have infinitly many elements.

Broadly defined, the term naive set theory connotes an informal set theory developed in a natural language in which such words as and, or, if ... then, not, for some, and for every are not rigorously defined. The term includes these various versions of set theory:

1. Cantor’s early (pre-1883 Grundlagen) theory of sets
2. Cantor’s later general theory of sets, the basis of the theory of transfinite numbers
3. informally developed set theories (axiomatic or otherwise) developed by Dedekind, Peano, and Frege
4. modern, informally developed versions of an axiomatic set theory, as in Naive Set Theory by Paul Halmos.

This section examines the first of these, namely, Cantor’s early theory of sets.

Bolzano’s contribution

In spite of Cantor’s pre-eminence in the area of set theory, the first to work with sets was Bolzano, as was noted above in #Introduction of infinite sets. It is from him that we have the following early definitions:

... an embodiment of the idea or concept which we conceive when we regard the arrangement of its parts as a matter of indifference (1847)

... an aggregate so conceived that it is indifferent to the arrangement of its parts (1851)

It was also Bolzano who first used the German word Menge for set, a usage that Cantor himself continued in his theory.

Despite this, Bolzano’s understanding of the notion of set was incomplete, especially with respect to the important distinction between the element/set relation and the part/whole relation. Consider, as evidence, his use of the word “parts” (Teile) to refer to the elements of a set in his description given above.”

Further, Bolzano thought absurd consideration of a set with only one element, while he failed entirely to consider the null set. Nevertheless, it was Bolzano who identified sets as “the carriers of the property finite or infinite in mathematics.”

Cantor’s discoveries

Traditional views give to Cantor (not entirely undeservedly) all or most of the credit for having developed set theory:

naive set theory is primarily due to Cantor

the first development of set theory was a naive set theory … created at the end of the 19th century by Georg Cantor.

“For most areas [of mathematics] a long process can usually be traced in which ideas evolve until an ultimate flash of inspiration, often by a number of mathematicians almost simultaneously, produces a discovery of major importance. Set theory however is rather different. It is the creation of one person, Georg Cantor.”

Set theory, as a separate mathematical discipline, was born in late 1873 in the work of Georg Cantor.

To this needs to be added a nuanced caveat, such as these:

Cantor’s work should be considered as a completion of a long historical process

The concept of set is no Athena: school children understand it now; but its development was long drawn out, beginning with the earliest counting and reckoning and extending into the late nineteenth century.

Even today, it is known that early study of naive set theory and early work with naive sets are useful in mathematics education:

• they aid in developing a facility for working more formally with sets
• they aid in understanding the motivation for axiomatic set theory

Cantor’s first ideas on set theory were contained in papers on trigonometric series, but for the most part he developed the set concept and its theory as a consistent basis for his work with infinite sets.

In 1873, he discovered that the linear continuum is not countable, which he treated as an invitation to investigate the “different sizes of infinity” and the domain of the transfinite. The following is a brief account of how his discovery came about:

• Cantor, in correspondence with Dedekind, asked the question whether the infinite sets N of the natural numbers and R of real numbers can be placed in one-to-one correspondence.
• Dedekind, in reply, offered a proof of the following:

the set $\mathbb{A}$ of all algebraic numbers, the set of all real roots of equations of the form an $x_n + a_{n-1} x_{n-1} + a_{n-2} x_{n-2} + . . . + a_1 x + a_0 = 0$, where $a_i$ is an integer, is denumerable (i.e., there is a one-to-one correspondence with $\mathbb{N}$).

• Cantor, a few days later, proved that the assumption that $\mathbb{R}$ is denumerable leads to a contradiction, using the Bolzano-Weierstrass principle of completeness.

Thus, Cantor showed that “there are more elements in $\mathbb{R}$ than in $\mathbb{N}$ or $\mathbb{Q}$ or $\mathbb{A}$,” in this precise sense:

the cardinality of $\mathbb{R}$ is strictly greater than that of $\mathbb{N}$.

A consequence of all this, Cantor noted, was proving anew an old (1844) result of Liouville's, namely, the existence (in every interval) of (uncountably many) transcendental numbers. In effect, there are in any real interval, more transcendental numbers than algebraic numbers.

In 1874, Crelle’s Journal published Cantor’s paper reporting this remarkable result and, in doing so, marked the birth of set theory. Previously, all infinite collections were assumed to be of "the same size." Cantor invoked the concept of a 1-to-1 correspondence to show that “there was more than one kind of infinity.”

Here is a summary of Cantor’s published results involving the early version of his naive set theory:

1. in 1874, a proof that the set of real numbers is not denumerable, i.e. is not in one-to-one correspondence with (is not equipollent to) the set of natural numbers.
2. in 1878, a definition of what it means for two sets M and N to have the same power or cardinal number; namely that they be equipollent.
3. also in 1878, a proof that the set of real numbers and the set of points in n-dimensional Euclidean space have the same power, using a precisely developed notion of a one-to-one correspondence.

Cantor actually achieved this last result -- at the time quite paradoxical -- in 1877, after which he wrote to Dedekind to report it, saying “I see it, but I don't believe it!” There were others who really didn’t believe it! Cantor submitted a paper reporting the result to Crelle's Journal. Kronecker, who had significant influence over what was published in the journal, disliked much of Cantor's set theory and fundamentally disagreed with Cantor's work with infinite sets. The paper was published only after Dedekind intervened on Cantor's behalf.

In 1878, Cantor stated his Continuum Hypothesis, asserting that every infinite set of real numbers is either countable, i.e., it has the same cardinality as $\mathbb{N}$, or has the same cardinality as $\mathbb{R}$. From that point until 1883, these were the only two infinite powers or cardinal numbers.

In all of these early papers, up to his development of the theory of transfinite numbers in 1883, Cantor’s notion of a set was essentially as follows:

a set is a collection of elements that constitute the extension of a (mathematical) concept

with the further important understanding that

the concept involved is defined only for objects of some given (mathematical) domain.

With the proviso noted, we can make these further points about this:

• Cantor’s early notion is the notion of set as it is most often applied in mathematics
• the proviso noted ensures that the paradoxes of set theory simply do not arise

Finally, in an 1882 paper, Cantor made the following point with respect to what were termed “undecidable” concepts:

an algorithm for deciding whether or not the concept determining a set applies to any particular object in the given domain is not needed for the concept to be the basis of a well-defined set.

He gives, as an example, the set of algebraic numbers, which (as mentioned above) he himself had determined was countable. This set, Cantor insisted, is well-defined, even though determining whether or not a particular real number is algebraic “may or may not be possible at a given time with the available techiniques.”

Two presentations of naive set theory

Set theory begins with two fundamental notions, objects and sets of those objects.

Membership is a fundamental binary relation between objects $o$ and sets $A$. If $o$ is a member (or element) of $A$, write $o \in A$.

Set inclusion is a derived binary relation between two sets. If all the members of set $A$ are also members of set $B$, then $A$ is a subset of $B$, denoted $A \subseteq B$. $A$ is called a proper subset of $B$ if and only if $A$ is a subset of $B$, but $B$ is not a subset of $A$.

Set theory features binary operations on sets, such as these:

• Union of the sets $A$ and $B$, denoted $A \cup B$, is the set of all objects that are a member of $A$, or $B$, or both.
• Intersection of the sets $A$ and $B$, denoted $A \cap B$, is the set of all objects that are members of both $A$ and $B$.
• Set difference of $U$ and $A$, denoted $U \setminus A$, is the set of all members of $U$ that are not members of $A$. When $A$ is a subset of $U$, the set difference $U \setminus A$ is also called the complement of $A$ in $U$. In this case, if the choice of $U$ is clear from the context, the notation $A^c$ is sometimes used instead of $U \setminus A$.
• Symmetric difference of sets $A$ and $B$, denoted $A \bigtriangleup B$ or $A \ominus B$, is the set of all objects that are a member of exactly one of $A$ and $B$ (elements which are in one of the sets, but not in both). It is the set difference of the union and the intersection,$(A \cup B) \setminus (A \cap B)$ or $(A \setminus B) \cup (B \setminus A)$.
• Cartesian product of $A$ and $B$, denoted $A \times B$, is the set whose members are all possible ordered pairs $(a,b)$ where $a$ is a member of $A$ and $b$ is a member of $B$.
• Power set of a set $A$ is the set whose members are all possible subsets of $A$.

- - - - -

Beginning with the fundamental notions of set and belongs to or is a member of and assuming that sets have properties usually associated with collections of objects, Paul Halmos, in his 1960 text, developed informally an axiomatic set theory that presented the binary relation of set inclusion and the binary operations noted above, as follows:

1. Axiom of Extension: Two sets are equal if and only if they have the same elements.
2. Axiom of Specification: For every set S and every proposition P, there is a set which contains those elements of S which satisfy P and nothing else.
3. Axiom of Pairs: For any two sets there is a set which contain both of them and nothing else.
4. Axiom of Union: For every collection of sets, there is a set that contains all the elements and only those that belong to at least one set in the collection.
5. Axiom of Powers: For each set A there is a collection of sets that contains all the subsets of the set A and nothing else.
6. Axiom of Infinity: There is a set containing 0 and the successor of each of its elements.
7. Axiom of Choice: The Cartesian product of a non-empty indexed collection of non-empty sets is non-empty.

In addition, there is an axiom stipulating (more or less) that anything intelligent one can do to the elements of a set yields a set:

8. Axiom of substitution: If S(a,b) is a sentence such that for each a in set A the set {b: S(a,b)} can be formed, then there exists a function F with domain A such that

F(a) = {b:S(a,b)} for each a in A.

An informally developed naive set theory with these axioms is equipped to do the following:

• develop concepts of ordered pair, relation, and function, and to discusses their properties
• discuss numbers, cardinals, ordinals, and their arithmetics,
• discuss different kinds of infinity, in particular, the uncountability of the set of real numbers

Paradoxes and Cantor’s early set theory

A discussion of paradoxes is relevant in two ways to the theory of sets:

• the “paradoxes of the infinite” that had to be overcome before set theory could be developed
• the paradoxes that later arose out of the development of set theory itself

It is interesting to consider that Cantor succeeded in resolving the “paradoxes of the infinite” and providing a coherent account of cardinal number for infinite multiplicities, while Bolzano, though he made great progress with the use and understanding of sets, failed to do so. Two points have been made to account for this.

First, there were notions about which, when applied to infinite sets, Bolzano was confused:

• the cardinal number of the set of points in an interval
• the magnitude of the line interval as a geometric object

Cantor accepted for infinite sets what had long been accepted for finite sets, namely, that “the relation of having the same cardinal number is defined in terms of equipollence.” Thus, since “the interval (0, 1) of real numbers is equipollent to the ‘larger’ interval (0, 2),” then these two sets of points, though different in magnitude, nevertheless have the same cardinal number. Bolzano, along many, rejected this.

Second, Euclid’s Common Notion 5, that the whole is greater than the part, was a barrier to working with infinite sets. Euclid’s principle does indeed apply to geometric magnitude. It may be that Cantor’s clear understanding of the first point above allowed him to see that in the domain of sets, infinite sets are simply a counterexample to Euclid’s principle. Whatever the reason, Bolzano did not see this.

As a final issue, it is worth commenting on the oft-repeated claim that working with naive set theory inexorably leads one to paradoxes. One such claim is the following:

Naïve set theory is intuitive and simple, but unfortunately leads very soon to controversial statements [, because] it relies on an informal understanding of sets as collections of objects, called the elements or members of the set, that is [, it relies] on a predicate indicating that a collection is a set and a relation type symbol to represent set membership.

This claim, however, does not apply to Cantor’s early theory of sets, which is the naive set theory that we have been examining. Certainly an aspect of set theory (naive or otherwise) that can lead to controversy and paradoxes is the use of unrestricted predicates (properties/concepts) to determine sets. Cantor’s early theory of sets, however, determines sets using restricted concepts. It is worth repeating here Cantor’s early notion of set:

a set is a collection of elements that constitute the extension of a (mathematical) concept that is defined only for objects of some given (mathematical) domain.

Sets determined in accordance with such a notion do not give rise to paradoxes.

Primary sources

• Bolzano, B. (1851). Paradoxien des Unendlichen (ed. by F. Pryhonsky), Reclam'; [English translation by D. A. Steele, ‘’Paradoxes of the Infinite’’, London: Routledge & Kegan Paul, 1950].

• Boole, G. (1847, [1951]). The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning, Macmillan, Barclay, & Macmillan, [Reprinted Basil Blackwell].

• Boole, G. (1854, [19158]). An Investigation of The Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, Macmillan, [Reprinte Dover].

• Dedekind, R. (1930-32). Gesammelte Mathematische Werke, Vols. 1-3, R. Fricke et al., eds., Vieweg.

• De Morgan, A. (1847). Formal Logic: or, The Calculus of Inference, Necessary and Probable, Taylor and Walton.

• De Morgan, A. (1849). Trigonometry and Double Algebra, Taylor, Walton & Maberly.

• Hilbert, D. (1900). “Über den Zahlbegriff,” Jahresbericht der Deutschen, Mathematiker-Vereinigung 8, 180–184. (English translation in Ewald, W. (1996). From Kant to Hilbert: A source book in the foundations of mathematics, vol. 2, Oxford University Press.

• Hilbert, D.(1900). "Mathematische Probleme," Nachr. K. Ges. Wiss. Göttingen, Math.-Phys. Klasse (Göttinger Nachrichten) , 3 pp. 253–297 (Reprint: Archiv Math. Physik 3:1 (1901), 44-63; 213-237; also: Gesammelte Abh., dritter Band, Chelsea, 1965, pp. 290-329) Zbl 31.0068.03, URL: https://www.math.uni-bielefeld.de/~kersten/hilbert/rede.html, Accessed: 2015/06/03.

• Hilbert, D. (1902). "Mathematical problems," Bull. Amer. Math. Soc. , 8 pp. 437–479, MR1557926 Zbl 33.0976.07, (Reprint: ‘’Mathematical Developments Arising from Hilbert Problems’’, edited by Felix Brouder, American Mathematical Society, 1976), URL: http://aleph0.clarku.edu/~djoyce/hilbert/problems.html, Accessed: 2015/06/03.

• Jevons, W S. (1890). Pure Logic and Other Minor Works, Robert Adamson and Harriet A. Jevons (eds), Lennox Hill Pub. & Dist. Co. [Reprinted 1971].

• Peacock, G. (1830). Treatise on Algebra.

• Schroeder, E. (1895). Algebra und Logik der Relativ.

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