Re: Cantor Confusion


David Marcus wrote:
Tony Orlow wrote:
<snip>

The infinite staircase comes to mind, where point set topology considers
the limit of the staircase from (0,0) to (1,1), as the number of steps
increases without bound, to be the same object as the diagonal line from
(0,0) to (1,1), since the locations of the corresponding points become
arbitrarily close. This produces a contradiction in measure, the object
being of length 2 for all staircases, but of length sqrt(2) for the
diagonal line. While the locations of the points in each set approach
each other with no lower limit, the directions of the corresponding
sub-segments of the two objects are always at a 45 degree angle to each
other, producing the error of sqrt(2)/2, the cosine of that angle. So,
what we have are a diagonal line of length sqrt(2) and a fractal "line"
or curve of length 2. In other words, characterizing the objects as sets
of points misses the distinction between the objects in terms of
measure, whereas characterizing them as sequences of segments preserves
the distinction in terms of direction and overall length.

Present your mathematics by itself. Then we can see if you are using
something other than what is in ZFC.

Now, sequences may be said to derive from ordered sets, but sets are
said to be determined solely by membership, with order unimportant. So,
the notion of a sequence derives really from an inductive definition
such as Peano's, and not from the one primitive in set theory,
membership, alone. The notion of order is not captured by "is an element
of". Do you disagree?

Of course I don't agree. You seem to be saying that infinite sequences
can't be handled in ZFC. Since ZFC has no trouble modeling the natural
numbers and defining functions, it clearly has no trouble acting as a
foundation for all of calculus and analysis.

But ZFC does have considerable difficulty dealing with infinite
sequences, when the Orlovian axioms are added in. I _think_ that Tony's
"infinite induction" thing means for a start that if a sequence of
elements has a particular property ("staircase length is 2") then the
limit of the sequence must have the same property. There doesn't seem
to be a definition of the Orlovian limit, except that in any particular
case Tony will construct an ad hoc story to make something having the
property being talked about. Thus the Tlimit of the staircase sequence
is a [search the archive for TO's words] "sort of infinitesimal
staircase-thingy of length 2".

I don't think ZFC will handle Orlovian "positive infinite quantities"
too well, either. Tony gets to infinite values by simply advancing
along the real line for, um, an infinite distance, through the tunnel
of love (where it's too dark to see properly). Despite the fact that
any finite quantity (integer) can be represented as a "finite length"
two-ended string of digits, and the fact that each new integer is
formed by adding one, the naive expectation that this would enable a
proof by induction that these "infinite quantities" were also simply
finite quantities, it doesn't work like this, because, um, there's a
principle of somethingorother that excludes this. I wonder if in fact
just as electromagnetic radiation is mediated by photons, induction is
mediated by inductons, and the tunnel of love just happens to block the
passage of inductons. Perhaps.

Brian Chandler
http://imaginatorium.org

.

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