See Bernard Bolzano (1781–1848)'s Paradoxes of the Infinite (1851):
Bolzano's proof for the existence of an infinite set is well known among mathematicians, and there are references to it even in Cantor 1883/84, Dedekind 1888, and Russell 1903. Bolzano proves by mathematical induction that there are infinitely many “truths in themselves”, i.e., true propositions.
An extract:
The set of all ‘absolute propositions and truths’ is easily seen to be infinite. [...] if we fix our attention upon any truth taken at random, say the proposition that there exist such things as truths, and label it $A$, we find that the proposition conveyed by the words ‘$A$ is true’ is distinct from the proposition $A$ itself, since it has the complete ‘proposition $A$’ for its own subject. Now by the same law which enables us to derive from the proposition $A$ another and different one, which we shall call $B$, we are further enable to derive a third proposition $C$ from $B$, and so forth without end. [...] The reader does not need to be remained of the similarity borne by this series together with its principle of construction to the series of numbers [...] The similarity consists in the fact that to each member of the former there corresponds a member of the latter; in the fact that howsoever large an integer be chosen, there exists a set of so many among the above propositions; and finally in the fact that we can always continue the construction of such
propositions [page 85].
See: Richard Dedekind, Was sind und was sollen die Zahlen (1888), Preface to the 2nd edition:
The property which I have employed as the definition of the infinite system had been pointed out before the appearance of my paper by G.Cantor (Ein Beitrag zur Mannigfaltikeitslehre, Crelle's Journal, Vol.84, 1878), as also by Balzano (Paradoxien des Unendlichen, §20, 185). But neither of these authors made
the attempt to use this property for the definition of the infinite and upon this foundation to establish with rigorous logic the science of numbers, and just in this consists the content of my wearisome labor which in all its essentials I had completed several years before the appearance of Cantor's memoir and at a time when the work of Bolzano was unknown to me even by name.
and:
§66. Theorem. There exist infinite systems. Proof [footnote: "A similar consideration is found in §13 of the Paradoxien des Unendlichen by Bolzano (Leipzig, 1851).]
Can be interesting to compare Dedekind's argument with "late" Russell's refutation, taking into account that Dedekind is considered a father (with Frege) of logicism.
In his proof, Dedekind consider "the realm of thoughts" as something objective; this view is the same of Frege.
Russell in Introduction to Mathematical Philosophy (1918) refute the argument in brief:
An object is not identical with the idea of the object, but there is (at least in the realm of being) an idea of any object. [...] The main error consists in assuming that there is an idea of every object. It is, of course, exceedingly difficult to decide what is meant by an “idea”; but let us assume that we know.
Here Russell speaks of "ideas in the mind", while for Frege and Dedekind thoughts have objective reality, not reducible to the psychological nature of "ideas".
Now it is plain that this [the infinite iteration of new ideas] is not the case in the sense that all these ideas have actual empirical existence in people’s
minds. [...] If the argument is to be upheld, the “ideas” intended must be
Platonic ideas laid up in heaven, for certainly they are not on earth. But then it at once becomes doubtful whether there are such ideas. If we are to know that there are, it must be on the basis of some logical theory, proving that it is necessary to a thing that there should be an idea of it. We certainly cannot
obtain this result empirically, or apply it, as Dedekind does, to “meine Gedankenwelt” — the world of my thoughts.
