On Set Theory: On Sets (-1874)

"The existence of an actual infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited." (Thomas Aquinas,"Summa theologia", 13th century)

"One very important genus of complex ideas that we encounter everywhere are those in which the idea of collection (Inbegriff ) appears. There are many types of the latter [...] I must first determine with more precision the concept I associate with the word collection. I use this word in the same sense as it is used in the common usage and thus understand by a collection of certain things exactly the same as what one would express by the words: a combination (Verbindung) or association (Vereinigung) of these things, a gathering (Zusammensein) of the latter, a whole (Ganzes) in which they occur as parts (Teile). Hence the mere idea of a collection does not allow us to determine in which order and sequence the things that are put together appear or, indeed, whether there is or can be such an order. [...] A collection, it seems to me, is nothing other than something complex (das Zusammengesetztheit hat)." (Bernard Bolzano, "Wissenschaftslehre" ["Theory of Science"], 1837)

"[a set is] an embodiment of the idea or concept which we conceive when we regard the arrangement of its parts as a matter of indifference." (Bernard Bolzano, 1847)

"Already within the domain of those things which do not have any pretension of reality, but only of possibility, there indisputably are sets that are infinite. The set of propositions and truths in themselves is infinite, as one can easily see." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"Even in the realm of things which do not claim actuality, and do not even claim possibility, there exist beyond dispute sets which are infinite. The set of all ‘absolute propositions and truths' is easily seen to be infinite." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"[...] from that circumstance alone we are not allowed to conclude that both sets, if they are infinite, are equal to each other with respect to the multiplicity of their parts (that is, if we abstract from all differences between them); [...] Equality of those multiplicities can only be inferred when some other reason is added, for instance that both sets have absolutely equal grounds of determination, i.e., that their mode of formation is absolutely equal." (Bernard Bolzano, "Paradoxien des Unedlichen", 1851)

"The existence of infinite sets, at least with non-actual members, is something which I now regard as sufficiently proved and defended; as also, that the set of all absolute truths is an infinite set." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)