Boyer's History of Mathematics includes a brief account of Bolzano's contributions in the chapter on Gauss and Cauchy.

Building on Galileo's "paradox" on the one-to-one correspondence between positive integers and perfect squares, Bolzano was able to show that similar correspondences between infinite sets and a proper subset are commonplace, including examples of sets of real numbers. Boyer then states :

Bolzano seems even to have recognized, by about 1840, that the infinity of real numbers is of a type different from the infinity of the integers, being nondenumberable.

Bolzano's contributions to our understanding of infinite sets are featured in his posthumously published work of 1850, Paradoxien des Unendlichen. Online copies of this text are available in the original German, but I do not read German. I am unable to locate any explanation as to how Bolzano reached this conclusion.

Did Bolzano conclude that the infinity of real numbers is of a different type from the infinity of integers, and if so, how did he reach this conclusion?


2 Answers 2


Did Bolzano conclude that the infinity of real numbers is of a different type from the infinity of integers ?

Yes and no ...


[page 569] The comparison of infinite sets and the creation of an ‘order’ between them was one of Bolzano’s main concerns. In this respect, Bolzano identifies another of the paradoxes that constitutes ‘one of the most noteworthy characteristics of infinite sets’:

When two sets are infinite, they can stand in such a relation to one another that (1) it is possible to couple each member of the first set with some member of the second in such wise that, on the one hand, no member of either set fails to occur in one of the couples; and on the other hand, not one of them occurs in two or more of the couples.

However, Bolzano notes that these sets may simultaneously present another type of relationship, which is the decisive difference between Bolzano’s and Cantor’s criteria to compare two infinite sets.

(2) one of the two sets can comprise the other as a mere part of itself, in such a wise that the multiplicities to which they are reduced, when we regard all their members as interchangeable individuals, can stand in the more varied relationships to one another (Bolzano, Paradoxes, 1851, p. 96)

For Bolzano, the obvious criterion for comparing two sets ought to be based on the second type of relationship, i.e., on the part-whole relationship between sets.

[page 570] Note Bolzano’s crucial idea that two abstract sets may present a one-to-one relationship, but this does not imply that they are equinumerous, because one of them may be a subset of the other. The possibility of establishing a bijection seems to be a (paradoxical) characteristic of infinite sets, since they may nevertheless have a different number of elements. This position clearly separates the works of Bolzano and Cantor. For Cantor (as well as for Dedekind), the part-whole relationship defines infinite sets and the ‘size’ (and order) of infinite sets is based on cardinality, a measure of the ‘number of elements’ of a set. If it is possible to establish a one-to-one correspondence between the members of two sets, the sets have the same cardinality, and are hence equivalent.

The paradoxical situation that arises from the comparison of infinite sets is a central concern in Bolzano’s work, and can be observed in the following paragraph:

As I am far from denying, an air of paradox clings to these assertions; but its sole origin is to be sought in the circumstance that the above and oft-mentioned relation between two sets, as specified in terms of couples, really does suffice, in the case of finite sets, to establish their perfect equimultiplicity in members. […] The illusion is therefore created that this ought to hold when the sets are no longer finite, but infinite instead (Bolzano, Paradoxes, 1851, p. 98).

[Regarding Galileo's paradox] Bolzano did not accept to extend the bijection criterion so as to establish the equinumerosity between two infinite sets. Nevertheless, as with finite sets, the part-whole relationship did allow him to assert that one of the two infinite sets is larger than the other.

[page 571] Bolzano attempts to define the arithmetic of infinity, by analyzing the possibility of introducing a certain ‘operability’ with infinity. However, as there existed no concept that would allow him to ‘quantify’ infinite sets, such as Cantor’s cardinality, Bolzano approached the problem of arithmetization of infinity in terms that are reminiscent of the theory of proportions:

I confess that the mere idea of a calculation with the infinite has the appearance of contradicting itself. For to try to calculate anything means, after all, to attempt a determination of it in terms of number […] But this scruple vanishes when we reflect that a correctly conducted calculation with the infinite is not a numerical determination of what is therein not numerically determinable […] but only aims at determining the ratio [or relationship] between one infinite and another… (Bolzano, Paradoxes, 1851, p. 107)

Two observations stem from this paragraph. Firstly, when Bolzano talks about establishing the ratio between one infinity and another, he accepts that different infinities exist; this position is essentially different from his predecessors’, in particular from Galileo’s. [...] Secondly, Bolzano abandons the idea of equinumerous sets being defined in terms of a one-to-one relationship, a notion that would be fundamental to Cantor, in order to state the problem of arithmetization of infinity in terms of a theory of proportions.

[page 573] When Bolzano analyzes infinite sets of real numbers, he establishes a ‘metric’ comparison criterion by associating real numbers to points on the straight line. [...] The sets of points are full of geometric meaning for Bolzano. The perception of geometrical characteristics of the regions associated with a given set is a determining factor for establishing the order of magnitude that Bolzano attributes to such a set. The following paragraph illustrates this:

We must attribute to a bilaterally indeterminated straight line an infinite length and a set of points infinite many times as great as the set of point in the unite straight line (Bolzano, Paradoxes, 1851, p. 151).

  • $\begingroup$ Thank-you for your extensive answer. Waldegg's views appear to be correct here. Her analysis highlights a degree of sloppiness in Boyer's account of Bolzano's work. His use of the word "nondenumerable" appears to be entirely out of place and he makes no mention of Bozano's use of bijections as identifying different types of infinity in relation to the whole-part relation as applied to the examples he gives. $\endgroup$
    – nwr
    Jun 27, 2016 at 15:17
  • $\begingroup$ It's worth noting that other "paradoxes" that were more or less known at the time were Aristotle's wheel paradox and the fact radial projection from the common center point can be used to establish a one-to-one correspondence between different sized circles having the same centers. $\endgroup$ Jun 27, 2016 at 21:27
  • $\begingroup$ How do you interpret this exactly? "...in such a wise that the multiplicities to which they are reduced, when we regard all their members as interchangeable individuals, can stand in the more varied relationships to one another." I am at a loss for finding any rigorous mathematical meaning there. $\endgroup$
    – j0equ1nn
    Jun 28, 2016 at 13:38

Let me add one paragraph to the answer of Mauro Allegranza (which is completely correct: Bolzano did not accept the bijection as a tool to prove equicardinality. He claimed that a short line has less points than a longer line. A bijection would make all lines appear as equal. But there are different infinities, for instance there are twice as many focal points of ellipses than centers.)

Bolzano mentiones on p. 102f of Paradoxien des Unendlichen that the number of points of a line, may it be even very short, is infinitely larger than the infinitely many points of an infinite (e.g., geometrical) sequence taken from (or projected into) that line because between two points of the sequence there are infinitely many points of the line. So while the word "nondenumerable" is in fact out of place, the infinite can have different sizes. Bolzano explains this by the example of the circle which has one center (or periphery) but an infinity of diameters. The ratio is infinite. (J. BERG (ed.): Bernard Bolzano, Wissenschaftslehre §§ 1-45, Friedrich Frommann Verlag, Stuttgart (1985), p. 31)


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