Re: Set Theory: Should You Believe
- From: herbzet@xxxxxxx
- Date: Wed, 19 Jul 2006 10:19:27 -0400
John Jones wrote:
If 'l.i.m.i.t' is the sign which is employed when certain manoevures
need to be made in mathematics, then we need not associate any meaning
with the letter sequence, or word:
I, for one, think this is exactly correct.
if the word 'limit' has any meaning
it would be a meaning we associate with familiar objects and
behaviours.
I'll let you be the judge of that, when we get there. We're not
there yet.
But if mathematics does not deal with familiar objects and
behaviours then its term 'limit' cannot convey meaning. As mathematics
does deal with familiar objects and behaviours then we can safely
assume, if mathematics is consistent in its use of terms, that we can,
as I said before, understand the term in its simple, originary meaning.
Even if the symbols with which the term 'limit' is associated are
different, the intuitive, metaphysical, familiar behaviours or acts
which the term invokes would be similar. That is, the term limit must
always refer to simple behaviours and objects to have meaning.
By saying 'the number which is not a limit of a sequence of numbers is
not found "in a sequence of numbers"', I do not mean to say that
numbers are transferable between sequences. Numerals are transferable
because they are not bound to any application.
Its difficult to imagine how a sequence of numbers can be a sequence of
'numbers'.
Yes; it will turn out, I think, that the notion of "sequence" is
less central to the notion of "limit" then one might, at first,
suppose.
It seems that a different application is made for each
'number', and that the results are gathered and placed in an order that
is largely visual and impressionable, rather than 'mathematical'. In
which case, not least because we are trying to transfer numbers between
applications, it seems that we have numerals, and not numbers.
Yes. Let me go off on a tangent for a moment, and give a small
disquisition on "order". The word "order" as used in this context
(sci.logic, or more generally, in the theory of sets) has little
to do with the familiar, originary use of the term. There is no
before, after, or left, right or above, below or first, last etc.
In set theory the difference between an unordered pair {a,b} and
an ordered pair (a,b) is this: we define the ordered pair (a,b)
as the unordered set {a,{a,b}} (or, if you prefer, {{a,b},a}, or
{a,{b,a}} etc.).
The sole reason for doing this is so we can say: an ordered
pair (a,b) is equal to the ordered pair (c,d) if and only if
a=c and b=d. If you push thru the logic of it, it works out.
I will give the demonstration if you insist, but it's rather
tedious. You might enjoy working it out for yourself, like
a crossword puzzle, or a chess problem.
And a
sequence is pictorial, which is why I said it might be a representative
structure, and strictly not a mathematical device. Regarding your point
about grasping a number, of course I think there is some confusion
where numbers and numerals have the same form.
Yes, it's crucial to keep the distinction in mind.
Also, as you say 'there
is no number to be grasped' unless we make one.
This, of course, is a perennial point of contention. But let's
push on.
Perhaps a different notation will be helpful. Suppose we represent
the sequence of numbers (or numerals) as a_1, a_2, a_3, etc.
We can now specify the sequence by specifying what a_n is to
refer to for each number (or numeral) n, e.g., in this particular
case we can say that a_n = 1 for each n. In the other example
(below) that for each n, a_n = n. This would fix the position of each
element, because each element has a unique nametag, which not only
names the element but gives its place in the sequence.
Yes, if each element has a different name tag then I can distinguish
the start of the sequence from the end; or if not exactly the 'end',
then the identifying act which identifies the 'limit' of the sequence.
Just off the cuff, I think the problem with this idea is that I need to
place the tags in order, if I want to identify the object to which they
are attached. We could just have sequence of numbers as they arise in
addition, 1,2,3,4 seems to identify element and position... but does it
identify an element with its position? After all, the tags are
arbitrary.
OK, in the previous paragraph, where I said "This would fix the
position of each element," I meant each element of the concretely
given sequence of numerals.
There may be a problem here in distinguishing between "the sequence"
as an object concretely given, i.e., numerals, and "the sequence"
as the (presumably existing) object referred to.
Yes, it seems that sequence is more like an act that yields a pictorial
form or order. In which case the act that constructs this order is not
the act of making the numbers that are said to compose it. This might
mean that I have no means of mapping 'number' to 'position'.
OK, here's the deal: let's take the sequence (of numerals!) 0/1, 1/2,
2/3, 3/4 etc., or better, take the sequence (of numerals) a_n = n/(n+1)
for each n.
1) Each element of the sequence (of numerals) refers, presumably, to
some number. That is, each numeral of the sequence identifies,
hopefully, some number.
2) We don't have to think of the numbers so identified as having any
particular order, or sequentiality. They just sit there, wherever
"there" is. They are the numbers picked out, elected out, identified
by the given sequence of numerals.
3) In this case, the sequence of numerals is non-redundant: each
numeral a_i identifies a different number. Redundancy is
allowed; in the example where a_n = 1 for each n, only one
number, "1", is identified by each term a_i of the sequence of
numerals. So in that case, there is only one number identified,
over and over again, and there can be no question of "a sequence
of numbers".
4) None of the numbers so indicated by the sequence of numerals
a_n = n/n+1 is equal to one: the numerator is smaller than
the denominator for each term of the sequence of numerals.
No numeral indicating, identifying the number "1" occurs
in the sequence.
5) Although the number one is not indicated by any numeral a_i
in the sequence of numerals, it is the limit of that sequence
of numerals because:
6) If we go out far enough in the sequence of numerals, that
is, if we choose a large enough i, then a_i, and every term
succeeding a_i, identifies a number near the number one, and
the number one is unique in that respect. By "near the number
one" I mean: let p be a positive number; in particular, a small
positive number. Really small. Like 3/1,000,000,000. Let "near 1"
be anywhere in the interval 1-p to 1+p. No matter how small
we choose p, no matter how small is the interval 1-p to 1+p,
we can find an i such that the number identified by a_i is in
that interval, and every succeeding term of the sequence identifies
a number that falls within that interval, and "1" is the unique
number that has this property.
7) Not all sequences have a limit, thus defined. E.g. the sequence
a_n = n has no limit. Of those sequences that have a limit, some
have the limit as an element of the sequence, some don't. I mean,
of course, that of the bunch of numbers identified, redundantly
or not, by some sequence of numerals, some of those bunches contain
their limit, some don't.
8) Please note that I haven't used the term "infinity" in any
of the above. At least, I haven't explicitly invoked the
concept. Neither have I said, or implied, that "infinity"
is the limit of any sequence.
Do we agree that the limit of this sequence is a member
of this sequence?
Not if my examination above, top, was correct.
I suggest that the new notation meets the objection of indeterminacy of
position.
Yes, this needs some more exploration. See also my response above.
While we're on the subject, I might as well mention the closely
related concept of a "limit point". A sequence of numerals a_n
has a limit point x if there is a number, referred to by an element
of the sequence of numerals, near x but other than x. Again, by
"near x" I mean in the interval 1-p to 1+p, for any and every
(small) positive number p.
The difference between the limit of a sequence (if it has a limit)
and a limit point of a sequence, (if it has one) is:
a) The numbers referred to by the numerals of a sequence that are
near x have to be different from x for x to be a limit point,
whereas for a limit L of a sequence, it is allowed for L to count
as a number near L.
b) A sequence can have only one limit, but it can have more than
one limit point.
c) The limit of a sequence, if it has one, may or may not also
be a limit point.
d) If a sequence has a limit, it has at most one limit point,
which would be the limit itself, if it is also a limit point.
Here's an example of a sequence with two limit points: Let
a_(2n) = n/(n + 1) and a_(2n+1) = (n/(n + 1)) + 1. The first
few members of this sequence are:
a_(2*0) = a_0 = 0/1
a_((2*0)+1) = a_1 = 0/1 + 1
a_(2*1) = a_2 = 1/2
a_((2*1)+1) = a_3 = 1/2 + 1
a_(2*2) = a_4 = 2/3
a_((2*2)+1) = a_5 = 2/3 + 1
a_(2*3) = a_6 = 3/4
a_((2*3)+1 = a_7 = 3/4 + 1
.... and so on.
This sequence has no limit (though it is bounded), but it has
two limit points.
The limit points of this sequence are 1 and 2, because there's
a number indicated by a numeral of this sequence in every
interval 1-p to 1+p (and different from 1)and interval 2-p to 2+p
(and different from 2) however small we make (positive) p.
--
hz
.
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