email@example.com (Jacob W. Haller) wrote in message news:<1ft79ul.kqjdbk1wliz10Nfirstname.lastname@example.org>... > George Dance <email@example.com> wrote: > > > firstname.lastname@example.org (Jacob W. Haller) wrote in message > > news:<1ft5h9e.rzdmnd1ad2zabNemail@example.com>... > > > > > 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).