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Topic: An new and very interesting intellectual game!
Replies: 194   Last Post: Jun 25, 2005 8:48 PM

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 Arthur J. O'Dwyer Posts: 91 Registered: 12/13/04
Re: Math posers. Re: An new and very interesting intellectual game!
Posted: Jun 13, 2005 8:18 PM

[fups trimmed]

On Mon, 13 Jun 2005, Minus XVII wrote:
>
> a paradox is just poorly chosen words.
>
> this "paradox" simply doesn't include the time
> after which all the other guys in town have been shaven,
> although babers invariably get other barbers
> to do their hair, I beleive.

Hence the story about the town with two barbers, one with neat hair and
one with unkempt hair. Smart visitors to this town invariably went to the
shop of the barber with the unkempt hair --- because he was obviously the
guy who cut the hair of the neat barber!

> how is this "paradox" any different\
> than the one about sets not containing themselves,

In fact it is isomorphic to Russell's set paradox, which deals with
"the set of all sets not containing themselves as members". Now, before
Russell it was considered quite possible for a set to contain itself ---
e.g., the set of all sets seems very well-defined! As does the set of
all sets-containing-two-items, or the power set of {1,2,3}.

Russell's paradox shows that contrary to intuition, just because we
can name something in mathematical notation doesn't mean it "exists"
in the mathematical sense. And thus there are sets that don't exist!
(Not merely empty sets. The empty set is kind of like zero; sets like
Russell's set are more like numbers-that-don't-exist.)

The paradox is "solved" by denying that certain uncountable collections
are really "sets" at all; thus we talk about the "class of all sets"
instead of the "set of all sets". (And "the class of all classes" is
also meaningless, I think; do we need meta-classes and meta-meta-classes?)

</display of innumeracy>

> or "I am lying, now?"

This one's fundamentally different, as far as I can tell.
I think someone already mentioned Smullyan's Humpty Dumpty's solution
to all these paradoxes, which is similar to yours --- "That's not a
up to a point, IMHO.

-Arthur

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