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The concept of possibility is vague and cannot be defined syntactically. But that does not show that it is spurious. In fact, it is indispensable.
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— | Tim Williamson, Knowledge and its Limits. |
If a man will begin with certainties, he shall end in doubts; but if he will be content to begin with doubts, he shall end in certainties.
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— | Francis Bacon (thanks @jessicajthames) |
Let A be any set. Consider any pairing-off of members of A
with members of the power set of A that assigns to each distinct
member of A a distinct subset of A. Let S be the set of all members
of A that are not members of the subset assigned to them. S is a
a subset of A. But S is not assigned to any member of A. For suppose
it were assigned to a member, say x, of A. Then x would be a member
of S if and only if it were not a member of S. This is a contradiction.
So any pairing-off of distinct members of A with distinct members
of the power set of A leaves some member of the power set unpaired.
So there is no 1-1 correspondence between A and its power set.
with members of the power set of A that assigns to each distinct
member of A a distinct subset of A. Let S be the set of all members
of A that are not members of the subset assigned to them. S is a
a subset of A. But S is not assigned to any member of A. For suppose
it were assigned to a member, say x, of A. Then x would be a member
of S if and only if it were not a member of S. This is a contradiction.
So any pairing-off of distinct members of A with distinct members
of the power set of A leaves some member of the power set unpaired.
So there is no 1-1 correspondence between A and its power set.
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— | Cantor’s Theorem as explained by Geoffrey Hunter (Metalogic, p. 24) |