FOM: Re: Arbitrary Objects

[Harvey Friedman]

Reply to Franzen 6:53AM 1/30/02. Franzen had the following exchange with
Silver:

>Charlie Silver says:
>
>  >It's difficult for me to
>  >believe that an answer to what arbitrary objects are is just to say that
>  >'arbitrary' is not an adjective.   If not an adjective, what is it?
>
>  A part of the phrase "an arbitrary object". "Let x be an arbitrary
>real number..." has the same meaning as "Let x be any number...". When
>asked to "press any key", Homer Simpson asked "Where is any key?", and
>the present conundrum is rather similar!
>
>  >Since
>  >mathematicians *use* arbitrary objects constantly, it seems to me there
>  >should be some well-developed and clearly understood account of them.
>
>  Mathematicians use arbitrary objects the way we press any key.

Let me go a bit further along these lines and ask:

*what is an arbitrary bit?*

If "arbitrary" is a predicate here, then there seems to be four
possibilities for its extension: emptyset, {0}, {1}, {0,1}. I think we can
rule out (?) emptyset.  We are left with {0}, {1}, {0,1} as the
possibilities.

For the people supporting the idea that it is a predicate, and for the
people following the people supporting the idea that it is a predicate,
which of the possibilities is it, or is it an open question which of the
possibilities it is? Or somehow the question of what the extension of this
particular predicate is (i.e., "arbitrary") is not meaningful?

Incidentally, by symmetry it seems to me that the extension can't be {0},
because then it would be also {1}, and can't be {1}, because then it would
be also {0}. Hence it would seem that one can "prove" that it is {0,1}. So
one is left with the conclusion (?) that

i) 0 is an arbitrary bit; and
ii) 1 is an arbitrary bit.

How do the supporters and followers of "arbitrary" is a predicate intend to
deal with this?