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Topic: Choice!
Replies: 16   Last Post: Apr 20, 2012 9:30 AM

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 Michael Stemper Posts: 671 Registered: 6/26/08
Re: Choice!
Posted: Apr 20, 2012 9:30 AM

In article <87aa27yfbq.fsf@uta.fi>, Aatu Koskensilta <aatu.koskensilta@uta.fi> writes:

>> Could somebody give a layman-accessible description of how this
>> differs from the standard statement of Choice (if it does)?

>
> Well, let me try. In set theory we can and do introduce various
>operations on sets. These include the powerset operation P that takes a
>set x to the set P(x) of all its subsets, the union operation taking a
>set to its union, and so on. These operations do not correspond to
>functions in set theoretic sense, i.e. they're not sets of ordered pairs
>stipulated to exist by this axiom or that.

[snip lucid explanation]

That was very helpful. I allowed me to better understand the particular
formulation being presented. It also gave me a better understanding of
why Choice is considered problematic by some. [1]

It especially helped me with some concerns that I had with fundamental
things such as union, intersection, subset, and power set. For instance,
"is a subset of" looks so much like a relation that I had been thinking
that it was defined as one. Similarly "powerset of" looked a lot like a
function, and union and intersection looked a lot like binary operations.

But, in each case the issue of domain and range came up. You've addressed
that quite well.

Thanks for the time taken to write that up.

[1] I knew it couldn't be Banach-Tarski that was the hang-up.
--
Michael F. Stemper
#include <Standard_Disclaimer>
2 + 2 = 5, for sufficiently large values of 2