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Re: number numeral
Posted:
Apr 11, 2003 11:43 AM
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spog@jwgh.org (Jacob W. Haller) wrote in message news:<1ft79ul.kqjdbk1wliz10N%spog@jwgh.org>...
> George Dance <georgedance@hotmail.com> wrote:
>
> > spog@jwgh.org (Jacob W. Haller) wrote in message
> > news:<1ft5h9e.rzdmnd1ad2zabN%spog@jwgh.org>...
> >
> > > And in general if it's OK to look at supersets, then how can there be a
> > > maximum integer?
> > > -jwgh
> >
> > Good question. I dunno; how?
>
> I don't think there can be. However, you said previously:
>
> | So what's a natural number? Do they exist in space and time? I've
> | argued that they do, as properties of things; they exist in things,
> | not independently. Since (assuming) there is only a finite number of
> | things existing in space and time, there is again only a finite number
> | of natural of numbers existing in space and time.
An excellent point; I missed that completely. Those two statements of
mine certainly sound incoherent, and if they are one of them is false.
Let me put the numbers aside, for a minute, and see if I can come up
with some analogy in which the two statements, in at least one
interpretation, are true.
Let's take 'human' to be any homo sapiens who has ever been alive; ie,
once a human, always a human. The set of humans (H) has a successor
principle - a child of a human is always a human - and no apparent
upper bound. So is there a maximum number of humans (and, therefore,
is the set of humans finite or infinite)? That depends on exactly
what 'maximum number' means.
One thing 'maximum number' could mean is that, "at any specific time,
there is always a finite number of humans' - ie, this many, and no
more. We know that is true by induction (I'll have to write out a
'proof' in natural language, but I'm sure you can easily translate
it): 2 humans would be a finite number; if they had a child, the
result (2+1) is a finite number; for any number of humans m, if m is a
finite number, (m+1) is a finite number; therefore H is finite.
At all times, for any number of humans n, it is true that
~(Ex)(x=(n+1)).
The other thing 'maximum number' could mean is that, "there is a
largest possible number of humans' - ie, there's a number n such that
there cannot be any greater number. And that is absurd, due to the
way we've defined 'human' - H never loses members (as dying does not
make one not a human), while it constantly gains members (as new
children are born). So for any number n of humans, it is always
possible that there can a larger number; therefore, it is false that
~<>(Ex)(x=(n+1)).
On this reading, H is finite and unbounded. But that's not the only
possible reading. Suppose someone else defined a 'human' as any homo
sapiens who has been alive, is alive, or will be alive. Now, because
of the lack of limits on the succession principle, H is as large as
the largest possible number of humans, and we know there isn't one:
because <>(Ex)(x=(n+1)) is true, (Ex)(x=(n+1)) is also true, for any
value of n. By this latter reading, H is infinite.
But in what sense does a human who 'will be alive,' but has never been
alive, exist? Obviously he does not exist in space and time, and
never has. (While he can be 'named,' as in parents naming their
planned offspring, that's mere wishful thinking, not an assertion of
existence). One way is to postulate the existence of something like a
'soul', defined as 'a human (ie, an essential characterstic of a
human) that exists either in or out of space-time', and a separate
reality (such as 'Heaven'), outside of space and time, in which those
souls exist.
I hope the analogies between H and the set of naturals, N, are
obvious, so I won't belabour them.
> However, I would question the idea that a finite number of objects can
> only have a finite number of properties. Indeed, by taking arbitrary
> number of supersets of a set containing only one object we can make a
> set larger than the number of elementary particles in the universe
> (assuming that such a concept makes sense).
> -jwgh
Back to my lighters. Indeed, there are a lot of properties of the
set; it's a set of blue things as well, eg. In this case, we're
looking at only one property of a set; its cardinality. But here the
possibilities are infinite: I can certainly think that, "If there
were one more lighter on my desk, there'd be three, if two, there'd be
four..." and go on forever; and if my successors did the same,
eventually we'd reach a number larger than the particles in the
universe (assuming the latter is a finite set; but that's another
digression).
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