Re: Constructibility of X -> X^2 bijection

David C. Ullrich wrote:

On Tue, 31 Jul 2007 08:52:58 -0400, "Stephen J. Herschkorn"
<sjherschko@xxxxxxxxxxxx> wrote:



Herman Jurjus wrote:



Herman Rubin wrote:
[snip]



It is trivial to order the power set of a well-ordered set

Would you care to explain what you have in mind, here?

I am pretty sure that Herman R. refers to the following: Given a well-ordered set X, order F = 2^X as follows. For f, g in F, let x be least such that f(x) != g(x). Then f < g iff f(x) < g(x).

Without referring to 2^X, this becomes: Let A and B be subsets of X. A < B iff and A contains the smallest element in the symmetric difference of A and B. Exercise: Show this is a well-ordering of P(X) without reference to 2^X.



Did you work out a solution to this "exercise" before posting it?



I erred in calling this a well-ordering (as opposed to an ordering), as I acknowledge in a later post. Did you read later posts before responding to this one?

The disproof and correction of an even unintendedly erroneous statement is often a very constructive exercies. For example, Spanier's topology tests always consisted of sequence of statements which were either true or false. The student had to determine and prove which status was correct. (By no means do I deny my error or wish to compare myself to Spanier with these remarks.)

I am curious. Why did you choose to put the word, "exercise," in quotation marks? (My quotation marks in the previous sentence are grammatical. Actually, I am uncertain about the usage of commas here. Should there be only one? Or none?)

--
Stephen J. Herschkorn sjherschko@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey and Manhattan
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