quantities including infinitely small and infinitely great quantities must be infinite too. (§16).

In all Bolzano’s demonstrations of infinite multitudes, the emphasis is not on their *existence* but on their *infiniteness.* The existence of collections (multitudes) follows from their definition. But it’s necessary to prove they are infinite.

Part-whole principle and one-to-one correspondence {#correspondence}
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Bolzano claims unequivocally to the *part-whole principle.* Not all infinite multitudes are to be regarded as equal to one another with respect to their plurality. If one multitude is contained in another as a mere part of it then the former is smaller than the latter.

The example, “to whom must it not be clear,” is the straight line with two points $a$ and $b$. The length of the straight line from the point $a$ continuing without limit in the direction to the point $b$ is infinite. It may be called greater than the straight line from the point $b$ continuing without limit in the same direction, by the piece $ab$. And the straight line continuing without limit on both sides may be called greater by a quantity which is itself infinite. (§19).

Simultaneously Bolzano is also aware of the existence of the *one-to-one correspondence* 