Re: Infinite Binary Strings: A Question

leonstreet wrote:
On Wed, 24 Sep 2008 14:00:06 GMT, "*** T. Winter" <***.Winter@xxxxxx>
wrote:

In article <jsoid4hq16fcts1dmpu8mud65ci66o9fbo@xxxxxxx> leon street writes:
On Tue, 23 Sep 2008 12:57:00 GMT, "*** T. Winter" <***.Winter@xxxxxx>

*** says:
When you consider finite sets only, counting the element
of a set is nothing more, nor less, than creating a bijection between that
set and a set of the form {1, 2, ..., n}. So it is intuitive to me that
counting is seen as the implementability of a bijection.

Not much to argue with here, except that "implement" is an odd verb to
apply to "bijection".

On the contrary, the implementing of a bijection between finite
collections is a form of counting.

Where is the contradiction?

Huh? Don't understand who is claiming that what is a contradiction,
but never mind...

Let us describe it. We have a collection
of objects, A, on the one hand, and a collection of objects B, on the
other. We have to take an object from the collection A and discard it, and
at the same step take an object from the B collection and discard it from
the B collection. When we say that we discard an object from the A
collection, we do not mean that we physically remove it (we may not have a
/physical/ collection to begin with), but that it should be regarded as
separated from that collection, so that, in the process we are about to
embark on of successively removing objects from the A collection, it should
not be 'counted' again. We are to distinguish between a (growing) discarded
pile and a (depleting) to-be-discarded pile in a stepwise exchange. And we
perform this process for each collection A and B in parallel with each
other. A 1:1 correlation is 'implemented' when the two collections are
exhausted at the same step.

This seems an awfully long-winded description. What is it supposed to
be? A _definition_ of LS's term "1:1 correlation"? Or of an
"implemented 1:1 correlation"??

All it says is that there is an "implemented 1:1 correlation" between
two sets if the sets (a) are finite and (b) have the same number of
elements. Actually it's equivalent to the following definition of
"size":

Two sets have the same "size" if counting each of them stops at the
same number-name.

It would be perfectly consistent with ("standard") mathematics to use
the word "size" only in this way. It would also simplify vastly the
process of responding to large amounts of crank-waffle.

<snip longwindedness>

Firstly, the fact that two finite collections are equal in number
is nothing to do with bijection, but is due to the fact that they have the
same count.

Ah, now you are just wrong, unless you have your own private meaning
for "nothing to do with".

Two finite collections are equal in number, in the (ordinary) sense of
having the same count *precisely* if and only if there exists a
bijection between them. In mathematics if A is true precisely iff B is
true, we do not say that "A has nothing to do with B".

Of course, you may want to "feel" that the ticking off against a
sequence of counting numbers is "really" counting, and the existence
or otherwise of a bijection with an initial set of the natural numbers
is somehow "derived", but this is rather like wanting to argue whether
4 x 7 = 28 *because* 28 / 4 = 7, or whether 28 / 4 = 7 *because* 4 x
7 = 28. It simply isn't anything mathematical.

Bijection may be a consequence, but it is not what this
equality is about.

"Is about" being a reference once again to your non-mathematical
emotions about this?

But let's go to the infinite case. Do I accept, given
two sets, such as Q and R, so that Q is a subset of R but is not in
bijection with R, that on the face of it there appear to be more Rs than
Qs? Yes, I suppose I do.

What do you mean by "more"? Can you really not understand that you
cannot take ordinary words with ordinary meanings and somehow
magically interpret them in an entirely new context? Everyone knows
what "more oranges than apples" means in the context of greengrocer's.
But in the context of (e.g.) infinite sets of integers or reals, the
word has no intrinsic meaning. It is equally valid to say any of the
following as a *definition* of what we mean by "more":

(a) Set A has more members than set B iff B is a proper subset of A
(b) Set A has more members than set B iff B can be put in 1-1
correspondence with a proper subset of A
(c) Set A has more members than set B iff counting A stops at a number-
name and counting B stops at a number name, and the number name for A
is after that for B in the number-name sequence.
(d) Set A has more members than set B iff B can be put in 1-1
correspondence with a proper subset of A and B cannot be put in 1-1
correspondence with A.
(e) More


But even the acceptance of the prima facie case
would be accompanied with much suspicion. I would want to know just how you
can have more than an infinite number ot things.

Right, here's an exercise. For which of the definitions above (a) -
(d) is it possible to "have more than an infinite number of things"?
(In the sense that there may be a set B which is infinite and a set A
which has more members than B)

More generally, of course it isn't possible to have "more than an
infinite number of things", in the sense that there is no "sort of
number" which is "beyond infinite". The naive (but intelligent) view
before Cantor pointed out you-know-what was that there were finite
sets, which could be compared, and found larger or smaller than each
other, and there were infinite sets, which were in some obvious sense*
larger than finite sets, but "beyond counting", so that it would never
be possible to compare them in a way that made one "larger" than the
other. (* But not in sense (c) above, for example!)

I might consider the
number line, and ask is it really possible to think of all the rational
positions in this line inked in, leaving only these irrational gaps behind?

No. If you try to understand the real line of set theory by imagining
processes like "inking in" you will only ever get more and more
confused.

Suppose we think of the inking in, not as already accomplished, but as a
supertask to be performed. And to be methodical about it, we will not
straight off try to cover all rational positions, but we will mark off,
given a unit interval to start with, the half way point, and all the
halfway points of all subsequent halfway points ad infinitum. Then at any
finite stage of this process, any position we have inked in will be
representable as a finite binary string. But we have granted ourselves
superpowers, and we may suppose ourselves to have completed all inkings in,
at which stage we will, I take it, have covered the entire interval.

Superpowers? Very sixties-ish...

I think the usual meaning of "supertask" is that you imagine (e.g.)
doing something at times noon-1 minute, noon-0.5 mins, noon-0.25 mins,
and so on, so that at noon you might have (for example!) an infinite
number of balls in an urn. But every ball will have a perfectly
ordinary (finite) natural number written on it. Something like that.

No, if you consider the sequence of sets of finite binary fractions
(0.1; 0.01, 0.10, 0.11; ...) you will only get the set of 'binary
fractions': { 0.5, 0.25, 0.75, 0.125, ...} which does not include 5/7,
let alone 1/pi.

infinite binary string, periodic and aperiodic alike, will have its
position on the line. Was there some stopping off point at which we had
only inked in all the rational numbers, but not yet the irrational ones?
Given that we start off with finite inkings-in (finite binary strings), are
there, so to speak, two superjumps here? One to get to the infinite number
of rational positions, and another to finish the job with the irrational
positions? Then it would seem to me that there is only one jump here, and I
would start to think that there is something very fishy about this idea of
a rational number line with gaps in it.

There are no "gaps" in the rational number line, if a "gap" is
anything with nonzero width.

It doesn't seem possible to have a
complete rational line (one with all the rational positions inked in)
without having all the irrational real positions there at the same time.
However we may come to define the irrationals, as limits or partitions or
whatever, the idea of them occupying a definite position on a line (or in a
sequence, one could say, for 'line' here and all along is only figurative),
just as the rationals do, is rather paradoxical.
At this stage I would start to investigate these infinite,
aperiodic strings, and the conclusion I have tentatively come to is that
the reason they won't list is that by there very nature they can't. It's
about the nature of the individual asystematic string, not the whole set of
them. The individual asystematic strings are like the unclassified species
in an incomplete taxonomy, each one a separate species. They do not form a
natural class. They are what's left over when we have done all the
classification we can.

This is, sadly, rather uninformed rambling. The problem is that
emotions like "the reason they won't list is that by [their] very
nature they can't." are not mathematical. Is the reason that in the
integers 3 does not divide into 7 that by its nature it can't? And why
can't it? Perhaps because by its very nature 7 is not a multiple of 3.

I earnestly suggest that you might try doing more reading and less
speculation. It is very hard to respond to posts as long as yours.

Brian Chandler
.

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