Re: Cantor Confusion

Tony Orlow wrote:
Is it sufficient to show that there are conclusions derived from
application of set theory that may not be mathematically correct in all
senses?

No. You have to present mathematics that can't be formalized in ZFC.

If a conclusion based on premises of set theory does not match
the conclusion based on other mathematical methods, then is there not a
contradiction between the premises, and therefore premises which are not
subsumed under set theory?

No. You have to present mathematics that can't be formalized in ZFC.
Simply present your "other mathematical methods". Then we can see if
they really need techniques that can't be modelled in ZFC.

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.

--
David Marcus
.

Relevant Pages

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