Re: infinity
- From: rusin@xxxxxxxxxxxxxxxxxxxxx (Dave Rusin)
- Date: 7 Sep 2005 19:45:45 GMT
In article <853boh4h1c.fsf@xxxxxxxxxxxxxx>, David Kastrup <dak@xxxxxxx> wrote:
>Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
>
>> Apparently you think that you can increment a value forever and
>> ever, and it will always be finite, so why do you think you can add
>> 1 element at a time forever and ever to a finite set and get an
>> infinite set? This thinking is inconsistent at the most basic level.
>
>There is no "add" involved. The set of naturals is described by
>static properties, not by a process generating its elements,
Right, although it might help poor Tony to phrase it a little closer to
his way of thinking. We can define "lots of" sets X_n by
X_1 = { 0 }
X_2 = { 0, 1 } (i.e add 1 to the previous largest element and
X_3 = { 0, 1, 2 } throw that in, too.)
etc. Then we can define the set of natural numbers by N = union of all X_n.
So there's Tony's dynamic process of creating each of the integers, i.e.
of the X_n's, and then the swell foop of combining them all into one
set as David suggested. As Tony likes to say, we "increment a value
forever and ever" ("it" here meaning the maximum value of X_n, so that
"it" is a variable which depends on n, which we can think of as "time"
I guess), and indeed "it" is indeed finite at all times n .
Even more importantly, I should add that the other elements of X_n
are also present in X_{n+1} -- they did not "increment".
But none of the X_n's itself is the set of all integers; making N
itself requires the additional step of taking a union of the previous
things (or maybe "waiting until after the end of time", whatever
that means). We're taking the union of infinitely many things; but
none of those things is itself infinite, and none of them contains any
"infinite numbers" either.
Of course David doesn't need to hear me say this, and Tony won't be
helped by hearing me say this, so I don't know why I'm bothering.
Beats grading papers I guess.
dave
.
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