Re: Humble pie.
From: Arturo Magidin (magidin_at_math.berkeley.edu)
Date: 06/25/04
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Date: Fri, 25 Jun 2004 18:53:53 +0000 (UTC)
In article <w%_Cc.30$jB6.6@newsfe6-win>, Andrew <stan370@btinternet.com> wrote:
>There are two reasons for believing that the set of natural numbers can, in
>some sense, be 'complete'. The first depends upon Cantor's proofs of the
>uncountability of certain sets. This is based upon the idea that for any
>supposed complete list of the elements of certain sets (such as the set of
>real numbers), a novel element not already in the list can be demonstrated,
>and after adding that to the list yet another novel element can be shown,
>and so on, and so on (e.g., via the diagonal argument). This is taken as
>proof that the set in question must have a higher order of infinity than the
>set of natural numbers since, even if supplied with a supposed complete
>list, an infinity of elements guaranteed to not already be in the list can
>be demonstrated. If it can not be considered that the list is in fact
>complete, then such demonstrations become meaningless,
No, they do not.
> since it is no great
>achievement to show the existence of novel elements guaranteed to not
>already be in the list if the list is known beforehand to be incomplete.
This is a false dichotomy. You are implicitly assuming that if we do
not assume that the list is complete, then we must assume that the
list is incomplete. But this is not the case.
In fact, the demonstration can proceed perfectly well without either
assumption. One simply shows that given ANY list of real numbers
whatsoever, there are real numbers which are not on the list. From
this, it follows that no list is complete.
>And that is the first reason for believing that the set of natural numbers
>can in some sense be 'complete'.
This seems like a rather novel notion of "complete". You deduce that
the natural numbers are "complete", based on a flawed argument about
other sets.
>The second reason springs directly from the first, in that, according to
>such determinations, the set of natural numbers has an order of infinite
>magnitude less than that of the set of real numbers because the real numbers
>can not be exhaustively paired, one-to-one, with any list of natural
>numbers.
> Conversely, this means that the set of real numbers must be able
>to exhaust the set of natural numbers via one-to-one pairings (otherwise
>one-to-one pairings have no meaning).
You mean: it is possible to pair a subset of the real numbers with the
integers, one-to-one? Your use of 'exhaust' is confusing, because
there is no one-to-one pairing from the set of real numbers to the set
of natural numbers, but that seems to be what you are writing.
> In this sense, the set of natural
>numbers must be considered to have been taken in its entirety, and so
>actually be 'complete' in some sense.
This seems to me to be pure nonsense. Obviously your mileage varies.
>Now, a few simple facts about the set of natural numbers which I hope are
>seen to be fair comment. The set has a least element, i.e., the single
>unit, but it has no greatest element.
Yes.
> Also, there is no natural number
>which, starting from the single unit, can not in principle be discovered by
>repeat applications of a successor function (i.e., by adding single units).
Yes. But these notions have to do with order, not cardinality.
>Since the single unit is finite, and all subsequent discoveries via the
>successor function increase the magnitude by only one finite unit, the
>direct consequence is that no single natural number can ever be infinite in
>extent - i.e., there is no natural number for which !!!. . . is a valid
>formulation (taking "!" as a symbol for a single unit and ". . ." to
>represent a whole non-terminating string of single units).
Yes: each natural number is finite; however, the set of all natural
numbers is infinite.
>In addition to these properties, the set of natural numbers itself has the
>quality of being "infinite", but it has a particular manifestation of
>infinity. Consider the first three of its elements in unary form;-
>
>!
>!!
>!!!
>
>In this way it is plain to see that with each subsequent natural number the
>list of individual units grows in only one direction, i.e., to the right.
>Since each individual natural number must contain only a finite quantity of
>such units, the infinity belonging to the set of natural numbers is revealed
>purely through the inability to ever exhaust the successor function. In
>other words, the set of natural numbers is infinite because the successor
>function can always 'discover' a new next number greater than the last, and
>not because of the quantity of individual units which may be present in any
>single natural number, which must always remain finite.
>
>But this raises an interesting question - what can we say should be the
>consequence if it were possible to have the set of ALL natural numbers in
>the form of a list? Clearly, this must in some way render the successor
>function no longer valid,
No "clearly" about it.
>since this list is supposed to be complete any
>application of the successor function must no longer be able to produce an
>element not already in the list - otherwise the list could in no way be
>considered to be complete.
The list has no final element. You are speaking nonsense again. we
have not "rendered the successor function [in]valid".
> At the same time no single number in the list
>could ever be said to have an infinite quantity of constituent units or it
>would no longer be a member of the set of natural numbers.
>
>The curious thing about this particular conjunction of circumstances is that
>the fundamental manifestation of infinity that the set of natural number
>shows to us is dependent upon the continued validity of the successor
>function, not upon the specific magnitudes of the individual natural numbers
>in the list. If this is then rendered ineffective, what further reason do
>we have for believing that the list must retain its infinite character?
You seem to have jumped into the realm of sophistry and nonsense, and
left mathematics far behind. Let us know if you ever wish to return to
them.
[...]
>There is a way to interrogate the list and find direct evidence for the
>status of the actual situation present. Imagine that I really do have the
>entire set of natural numbers in a list, and since I have all of them the
>successor function, by definition, must become ineffective.
No such definition. The statement "the successor function[...] must
become ineffective" is nonsense.
> Starting from
>the premise that no two finite natural numbers can be summed to produce a
>natural number with an infinite magnitude, it is possible to successively
>sum all natural numbers in a strictly finite list and to only ever have a
>finite sum, no matter what the specific identity of the natural numbers in
>the list.
Yes: a finite sum of natural numbers always yields a natural number,
and hence a finite number.
> However, since this is supposed to be a list of the set of all
>natural numbers, and that particular set has an infinite cardinality, such
>an approach could never be guaranteed to ever terminate and definitively
>resolve the issue.
Huh?
>But there is another way to achieve the effect of having performed however
>many summation operations would be necessary, even if an infinite amount
>were actually needed, without resorting to a doomed iterative process such
>as this. Consider instead that since all natural numbers, viewed in unary
>form, must be formed from a finite list of single units, it is possible to
>consider the entire list containing the set of all natural numbers not as
>individual members, but as one long unary list of single units (this is only
>made possible by virtue of the fact that we are supposed to be in possession
>of the set of all natural numbers in the first place) - which, by virtue of
>the fact that each individual natural number in the set of natural numbers
>must be finite, can not itself escape being of a finite magnitude.
This is the fallacy of composition, the claim that because each
individual item in a collection has a particular property, the entire
set of such items must also have the property. Each natural number is
finite: but the set of all natural numbers is NOT finite; the claim
that "[it] can not [...] escape being of a finite magnitue" is, quite
simply, false.
> This
>follows since viewing the set as a whole in this way is equivalent to having
>performed the summing without needing to actually perform a potentially
>infinite number of separate operations.
This is nonsense yet again.
>To see this consider again the first three natural numbers arranged in a
>list:-
>
>!
>!!
>!!!
>
>If instead of a vertical list like this we take the list horizontally:-
>
>!!!!!!
>
>Then we arrive at a situation identical in effect to the one we would have
>arrived at if we had actually performed two consecutive summations upon the
>vertical list, but without actually having explicitly done anything of the
>sort. We are simply choosing a particular view of an already extant object,
>i.e., the list.
>
>The result of this effective summation operation can not escape being finite
>(since all of its constituents are themselves finite), and hence it can not
>escape the bounds of finite arithmetic either, in which the sum must be
>greater than any of its individual parts.
You can only perform this operation on an initial segment (or a finite
subset) of the entire list, not on the list itself.
>It is also clear that since it is
>composed of nothing but a finite list of individual units, it meets the
>criteria for being a valid natural number and it is also evident that the
>natural number thus represented can not have previously appeared in the
>list. Otherwise the element thus discovered would have the property of
>being both a member of the set of natural numbers and also not a member of
>the set of natural numbers.
No. The "element" you discovered if you try to do this with the entire
set of real numbers is called an "ordinal", and it is the ordinal
omega_0; it is not, however, a natural number.
> It is thus reasonable to add it to the list of
>natural numbers
There is nothing reasonable about it. You are making a categorical
error here.
[.rest deleted.]
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu
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