Re: Is continuum completely filled up?

Andy Smith wrote:
Dave Seaman writes

Actually the iteration as specified gives 3^N after ~N iterations; I
should have said to insert 1 real in each hole on each iteration, but
this doesn't affect the line of argument.

My underlying train of thought was that you define a real via a Dedikind
cut, which implicitly asserts a meaning to infinite sets.

No. Infinite sets are meaningful because we have axioms that tell us
what we can do with them. And, you shouldn't worry about how the reals
are constructed. Simply work with their properties as a complete ordered
field. Spivak lists all the properties of the reals and shows how you
prove things from these properties.

I was trying
to consider a systematic method of creating all the reals from the
bottom up (or maybe top down, depending on your perspective).

- if you have a countably infinite number of bits, then after a
countably infinite number of iterations, you still have as many open
intervals as distinct real numbers?

That line of reasoning holds only as long as there is such a thing as a
"next" point in the set, since an interval lies between a point and its
immediate neighbor. There is not "next" point in the rationals, and
therefore there are no intervals in the sense that you describe.

But at each level of the iteration you have an ordered set of 2^N points
- you have defined them, so you know where they are?

Yes, you can define the set at each iteration. The phrase "know where
they are" doesn't mean anything.

If one wished we
could take e.g. pi or sqrt(2) and successively map that by
proportionately scaling its location into each interval on each
iteration, and we can then have a formula that describes the set of
locations of all points at iteration N ?

Sure, you can add whatever real you like at each step of your
construction. It is up to you.

The iterative construction puts "interval" (absence of specified points)
on an equal footing with number, with a location defined (if you like)
by the real number corresponding to its mid-point.

The intervals may not contain any points that you've listed so far, but
that doesn't mean that they won't contain points that you list later.

But doubtless you are right. As I said before, I don't trust any form of
reasoning about infinities

That is only because you haven't learned the rules. If you only learn
math from engineering or science courses, you won't learn to reason
correctly.

- you just have to consider
the Grandi series +1 -1 +1 -1 ... for which one can construct
apparently solid arguments for the sum being any integer that you wish

The key word is "apparently". The arguments are not correct.

(of course, for series this would not be defined as convergent,

We define the property of a series being convergent. We then prove
theorems about convergent series. We also prove that the series you gave
is not convergent. In particular, we don't define what the "sum" of a
non-convergent series is. So, any discussion which assumes such a series
has a "sum" is incorrect from the start.

but that
rather dusts under the carpet the issue of why the logic is flawed -

On the contrary, it exposes the flaw in the logic quite clearly. You
can't assume that words, like "sum", have meanings. You must carefully
define what they mean.

even if the series is non-convergent, it should be clear why a
particular line of argument, e.g. add successive terms, is invalid,
without circular reasoning (e.g. we can't discuss this because it is a
non-convergent series)).

Once you carefully define all your words, then the flaw in the reasoning
is clear. There is nothing circular about it.

And, even though there are no intervals, there are still gaps in the
rationals.

Well I think one could probably modify the argument in such a way as to
demonstrate that there are still gaps in the rationals, but one doesn't
need to do that because it is easy to show that there exists an
irrational between any two rationals anyway.

What do you mean by the word "gap?

Between any two rationals, there is another rational. Proof: Let x and y
be distinct rationals. Then z = (x+y)/2 is a rational and is between x
and y.

In fact, between any two reals, there are an infinite number of
rationals and an infinite number of irrationals.

Properties of finite sets do not necessarily extend to infinite sets.

I can see that that is true, for sure.

Is there enough information in a countably infinite set of bits to
encode every location on the real line? You would say yes, I guess.

Every real number has a decimal representation. Some reals have two
decimal representations: The nonzero reals that have a decimal
representation ending in all zeros have another representation ending in
all nines.

--
David Marcus
.

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