READING: I §4 3
Back to reading:
I §4 2
begin quote ita"
CHAPTER I [:] Notions from Set Theory [...] §4. FINITE AND INFINITE SETS. "end quote
These pages assume that the reader has the book and is reading along. topic index contents
Please read these notes from the beginning to here and then the first 3 sentences of paragraph 4 of §4.
PROPER SUBSETS OF INFINITE SETS
First please recall that the meaning of the material implication is , where ~ means "not" and where the symbol means "and". Now recall that all important substitution exercise that was first encountered on the subset page. Replace A above by ~B and replace B by ~A. We get the following:
It is easy to show that a set X is infinite if and only if it may be put into one-one correspondence with a proper subset of itself. To do this, note first that if X is finite then any proper subset has a smaller number of elements, whereas two finite sets in one-one correspondence must have the same number of elements. This proves the "if" part.
The if part is the statement "If X can be put into one-one correspondence with a proper subset of itself, then X is infinite." To use the result presented above, we negate and switch: "If X is finite, then it can't be put into one-one correspondence with a proper subset of itself." This is what Rosenlicht proves.