Re: Cantor Confusion

In article <457c1249@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

David Marcus wrote:
Tony Orlow wrote:

Ugh, yes, I guess I have. The von Neumann ordinals appear to be the
vehicle connecting set membership and order this way. Okay. I don't like
it, but it works in its way.

It is just supposed to work. No one is saying zero really is the empty
set (whatever "really is" means).


Then no one is saying the von Neumann successor ordinals "really are"
the naturals? Good.

But everyone says they make an adequate model for the naturals, and for
the ordinals.

It seems like it would be better to have
another primitive, such as Peano's successor, than to use this strange
definition of the naturals, but I'll have to think about that.

The idea is to be parsimonious. Since you can define the relation you
want, there is no reason to be redundant by including another primitive.
Doing so just makes things more complicated without any benefit.


Not when the alternative is a construction that establishes the
equivalent of a new primitive through a dubious connection between
succession and containment.

The only thing dubious about it is whether TO understood it.


What is more "parsimonious" in inventing
some weird model of the naturals and declaring that it exists, rather
than having succ() be a primitive relation? I don't see the advantage.

As we already have sets, membership and inclusion, all the stuff for the
vN system in every set theory, parsimony says don't add something else
when what is already there is sufficient.


If the sequence consists of segments of the form (0,x) or (x,0), there
is no segment which is diagonal in direction. If every segment in the
sequence is of the form (x,x), there is no segment which is not. This
information is not evident when the pairs describing the curve are
locations, because locations don't have direction.

So, my question remains. If this is a valid formulation of the two
objects, and an explanation for Chas' counterexample to infinite-case
induction, where does this fit with set theory? I don't think the von
Neumann ordinals as a model of a sequence can suffice, since they only
allow finite values until a leap is made to the limit ordinals, and
continuity is violated. This is the problem I have with the vNO's, and a
large part of my problem with transfinitology. The notion that a
sequence must be "countable" simply is not correct in the bigger picture.

So, the point set approach has that the same object has two different
measures, because it cannot distinguish between two objects which are
locationally the same, and directionally different.

Then come up with an approach that does what you want. By "approach", I
mean definitions (of objects, convergence, etc.) that let you prove the
theorems you want. That's what everyone does. For example, Einstein came
up with Brownian motion, but it wasn't clear how to mathematically model
it. Weiner figured out how.


Well, I'm suggesting a definition of the curve as a sequence of pairs
which denote xy offsets, rather than a set of pairs of xy coordinates.

The standard definition of a curve is as the continuous image of a real
interval. Such curves have 'directions' only at points at which the
function describing them is differentiable, so in the limit, TO's step
function would have no direction at any point.
.

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