Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)

From: Chas Brown (cbrown_at_cbrownsystems.com)
Date: 11/19/04 

Date: 19 Nov 2004 02:53:57 -0800

examachine@gmail.com (Eray Ozkural exa) wrote in message news:<320e992a.0411180152.30a39fa7@posting.google.com>...
> Josh Purinton <usenet-noreply.a.jp@xoxy.net> wrote in message news:<izUmd.110236$R05.98736@attbi_s53>...
> > In article <320e992a.0411171738.31a07c75@posting.google.com>,
> > Eray Ozkural exa <examachine@gmail.com> wrote:
> > >The question is whether Cantor's diagonal proof is like that.
> >
> > This is Cantor's argument as I understand it:
> >
> > 1. A binary expansion of a real number in [0,1) is given by a function
> > E: Nat-->{0,1}; the jth digit of that expansion is E(j)
> >
> > 2. Let S be any sequence of binary expansions of reals in [0,1), and let
> > S_i be the ith element of S. Then S_i(j) is the jth digit of the
> > ith element of the list.
> >
> > 3. Let G: Nat-->{0,1} be the function defined by:
> > G(i) = 1 if S_i(i) = 0
> > G(i) = 0 otherwise.
> >
> > 4. G is well-defined, and, by construction, G differs from each
> > expansion in S by at least one digit. Thus any sequence of binary
> > expansions of the reals in [0,1) is necessarily incomplete.
>
> What does it mean to say that something differs from something else by
> construction when things talked of are supposedly infinite objects?

When you make a statement like the one above, it makes me wonder: if
we define, for all naturals i, f(i) = i; and, for all naturals j, g(j)
= j+1, are you comfortable determining for yourself the truth or
falsity of the statement: "There exists no natural number k such that
f(k) = g(k)"?

Do you find yourself getting hung up in _this_ case by the thought
that f and g are "supposedly infinite objects"? If so, just skip the
rest, and focus on that!

Otherwise, can you see that it makes sense to say, as a kind of
shorthand, that f is "not the same as" g; or f differs from g? The
above statement is clearly false if we say "There exists no natural
number k such that f(k) = f(k)", so we could also say "f is not
identical to g", or that f is not equal to g.

Do you agree that, given the assumptions, for each pair of naturals
i,j, "S_i(j)" is meaningful description of a _particular_ natural
number (either 1 or 0), i.e. that S is well-defined?

Do you agree that, given S, then for each natural number j, "G(j)" is
a meaningful description of a _particular_ natural number (either 1 or
0), i.e. that G is well-defined?

Then what "G differs from each expansion in S by at least one digit"
means is "for each natural number i, G differs from S_i".

And, in turn, what "G differs from S_i" means is "there exists a
natural number j such that G(j) is not equal to S_i(j)".

For each natural number i, doesn't G(i) = 1 - S_i(i), by definition
(i.e., by its "construction" from the given entity, S)?

And, for each natural number i, "1 - S_i(i) = S_i(i)" cannot be a true
statement, given the properties we assumed for S, right? Does 1 - 1 =
1? Does 1 - 0 = 0? Are there any _other_ possibilities to consider?

Then it follows that, for each natural number i, "G(i) = S_i(i)"
cannot be a true statement, right?

And this is exactly the same as saying, for each natural number i,
G(i) is _not_ equal to S_i(i), right?

So, therefore, for each natural number i, there exists a natural
number j (namely, j = i) such that G(j) is not equal to S_i(j), right?

And this satisfies our definition, "G differs from S_i", right?

Therefore it makes sense to say that for each natural number i, G
differs from S_i, right?

Therefore, from our definition, it makes sense to say that G differs
from each expansion in S by least one digit, right?

That's what it means (the _definition_) in this context to say that
"... something differs from something else by construction... ".

Do you see how it all just follows logically from some set of
assumptions, ultimately based on clear definitions?

It has nothing to do with "supposedly infinite objects". There's
really no mystery here; this type of thing in proofs is so utterly
basic and commonplace that it is rarely laid out, because otherwise
most proofs would be about a million lines long, and practically
impossible to read.

And it's _not_ the case that this is normally not included in a proof
because mathematicians have "never thought about it"; on the contrary,
as you learn mathematics, you have to learn to do this almost as a
kind of set of mental reflexes, akin to those required when driving a
car.

So there's typically no need to go into such mind-numbing detail. But
the details are always there if you need them - "turn" = "hit the
brake, ease up on the clutch, a little bit of gas, turn the steering
wheel, shift down, ...".

If you really don't understand why Josh's version of Cantor's proof
follows logically after accepting the above painful definition of
"differs from", you (seriously!) should _completely abandon_ studying
Cantor's proof for now, which appears to be only adding to your
confusion, and focus on far simpler proofs.

That way, your objections/confusions about "infinite objects" can be
addressed without worrying about things like "real numbers",
"bijections", "sequences" and other technical terms. These terms
appear in proofs which assume that the issues you seem baffled by have
already been resolved by the reader - it's assumed the reader already
knows how to drive with a "standard" transmission.

> I
> think this was one of the steps of reasoning which Dr. Zenkin
> criticized.
>
> > Does this argument meet your criteria for employing only finitary
> > reasoning?
>
> This is quite nice. To me, it looks fairly finitary and formal

"fairly finitary"? Isn't that a bit like "a little pregnant"?

> when
> you write it like this at the first glance, and this is of course a
> quite common argument. However, Dr. Zenkin thought there was a "fatal
> mistake" in one of the implicit steps. It seemed to Dr. Zenkin that we
> were "jumping to conclusions" about the nature of infinite objects, by
> making assumptions about how we should think about infinite objects.

When "we" make such assumptions, "we" start by _clearly defining_ the
meaning of terms like "infinite object" or "infinitary reasoning".
This has the advantage that we can then prove or disprove whether the
statements we make about them follow from these definitions.

Have you read any logic or set theory?

Of course, if you (or Zenkin) disagree with or fail to make these
definitions, then the onus is on you (or Zenkin) to provide exact
definitions of, e.g., "infinite object". If you (or Zenkin) cannot or
do not do so, then you are talking about something that is not what
anyone else here seems to be talking about - mathematics.

> If this isn't confusing already, you should read his paper.
>

The confusion disappears when one realizes that Zenkin is not doing
mathematics; his paper is mathematical "non"-sense.

As noted above, no-one has the mental fortitude to wade through a
million line proof. So when communicating to each other,
mathematicians learn to use a type of exposition that invites the
reader to "fill in the gaps", of course with the assumption that they
can do so.

When someone is speaking mathematical "non"-sense, there's just no
conceivable way to fill in the gaps. Lack of pertinent definitions is
an example of mathematical "non"-sense.

> Thus, when I read it the second time, I think I spotted a place where
> Zenkin's argument could be applied.

> You've indeed put it in a succinct
> formal form, but formality should not avoid any problems, if there
> were any to begin with. So, if your formalization is correct, which
> seems to be the case, whatever Zenkin criticized must be there.

"Must"? You seem to be putting the cart before the horse. You are
assuming that Zenkin's _logic_ is correct (e.g., refers to something
that exists). It is not.

Why is it that you say (a) you can follow the argument given above,
and see that it is correct; and (b) by your own admission, you cannot
particularly follow Zenkin's argument, let alone come to an opinion
its truth; and yet (c) you continue to claim that there "must" be some
validity to Zenkin's argument, because... I dunno, because he's a
professor with a web page?

Have you considered that maybe what Zenkin criticised _isn't_ there,
because Zenkin's "criticism" is not making mathematical sense? This
would certainly explain why his paper is confusing!

> We
> should send your version to Dr. Zenkin, on which he can comment. I
> cannot really decide.

If you want to do _mathematical_ philosophy, then perhaps you should
set your mind to learning _how_ a mathematician (as opposed to a
philosopher) would make up his or her own mind on this issue.

Not that I wouldn't probably get a chuckle of it anyway.

>
> Regards,

Cheers - Chas

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