# Has Bolzano's opinion about the irrelevance of an infinite bijection ever found supporters?

From the only reason that two sets, $A$ and $B$, are corresponding to each other by the fact that for every part $a$ being in $A$ there is a part $b$ being in $B$ such that all pairs $(a + b)$ formed in this way contain every thing contained in $A$ or $B$ and each thing only once – only from this reason it is, as we see, not yet allowed to conclude that these two sets, if they are infinite, with respect to their number of things (disregarding the differences of their parts) are equal to each other. [...] Two finite sets, if they are of a kind such that we can find to every thing $a$ of the first set a thing $b$ of the second set and no thing is remaining and no thing appears in two or more pairs, have always the same number of things. It appears as if this property should be maintained for infinite sets too. It appears so, but on closer inspection we see that it is not at all necessary because the reason lies in the finiteness and therefore vanishes if the sets instead of being finite are infinite. [...] The conclusion becomes invalid as soon as the set of things in $A$ is infinite because, by the definition of an infinite set, we never encounter a last thing in $A$ – how many things we may have counted, there are always others to be counted. And although there is no lack of things in the set $B$ too which can be paired with the things of $A$, the reason becomes invalid to conclude that the multitudes of both sets are equal. [Bernard Bolzano: "Paradoxien des Unendlichen", Reclam, Leipzig (1851) 31-33]

• Definition of set equality via bijection contradicted Euclid's part-whole axiom, and was not accepted until Cantor built a theory upon it rich enough to make the part-whole intuitions moot. Bolzano wasn't first, the history is described by Mancosu in MEASURING THE SIZE OF INFINITE COLLECTIONS. It explores historical and modern alternatives to Cantor's treatment of infinity. May 14, 2017 at 23:07