FOM: Re: Arbitrary Objects

[Gordon Fisher]

Kanovei wrote:

> Sorry, the problem is to interpret the phrase:
> "take an arbitrary element x of X, then ...." .
> The standard solution (X-many acts of picking x\in X)
> is well known, the problem (as I see it) is to semantically
> interpret the phrase so that SOMETHING is taken just once.
>
> Take an arbitrary x in {0,1} -- in standard understanding
> this means take 0 and see what happens, take 1 see what
> happens, stop end.
> But I want to take an arbitrary element, whatever it be, once.
>
> Best
>
> V.K.

Well, let's see.  Are you saying that once you pick an
object from a set, it's no longer arbitrary, because it
is some known single object?  So any act of selecting
a single object from a set makes the object not arbitary,
i.e. any act of choosing a single object is necessarily
an act of cancelling out any arbitariness?

Suppose I throw an "honest" die in an "honest" way, and
it comes up with a 6 on top.  Can I say that before I threw
the die, it was certain that a 6 would come up?
Similarly, if I pick an element in an "honest" way from the
set {0,1}, and it turns out to be 1, can I say afterward that
it was certain that it would be 1, so it can't have been an
arbitrary choice?

Thinking along these lines, we might postulate that it is
possible to not know _before_ a choice is made of an element
in a set which element will be chosen when a selection is made,
but that we necessarily will know afterward what element has
been chosen.  But what if I choose an element from a set, and
don't perform an act of recognizing which element it is from the
set?  Should such an act be prohibited?

In any case, thinking along these lines appears to introduce a
notion of time (note the tenses of the verbs in the previous
paragraph) which is not customarily introduced into axioms
and theorems of a set theory.

Gordon Fisher     gfisher at shentel.net