at one end, while one matches the other at the opposite end. Thus, it is neither true that one matches the other without qualification, nor that they fail to match without qualification. Rather, they match at one end and do not match at the other. In that case, the greater sequence is not equal to the smaller, nor are the two finite.
If someone says:
“We match them starting from the end nearest to us. If they are equal, it follows that the greater is equal to the smaller, that the additional portion’s presence is equivalent to its absence, and that a thing is the same whether something else is absent or present. If they are unequal, it follows that parts of what is infinite are unequal.”
The reply is:
Matching them from the finite end, while they differ at that end, is impossible. An impossible supposition may entail an impossible consequence. If the past events counted back from yesterday are supposed to match the past events counted back from today, such a matching is impossible: the one cannot match the other, since the two sequences are unequal. Where there is inequality, a matching that entails equivalence and equality is impossible.
If someone says:
“I conceive the matching in the mind, even though it is impossible in external reality.”
He is told:
You have thus conceived two things in the mind while making one greater than the other at one end and equal to it at the other. It is clear that if you make this supposition, their inequality would not be impossible; rather, inequality would be necessary.
Your argument rests on supposing a matching, from which inequality within what is infinite follows. Both premises are false. If you suppose a valid matching that establishes equality, that supposition is false. But if you suppose it even though it is impossible, then the inequality would not be