Re: How man different paths can exist in the Complete Infinite Binary Tree?

[Ross A. Finlayson]

On Sunday, August 5, 2018 at 1:04:28 PM UTC-7, Ross A. Finlayson wrote:
> On Sunday, August 5, 2018 at 12:47:15 PM UTC-7, Shobe, Martin wrote:
> > On 8/5/2018 6:53 AM, transf...@gmail.com wrote:
> > > Am Sonntag, 5. August 2018 00:15:33 UTC+2 schrieb Shobe, Martin:
> > >> On 8/4/2018 5:44 AM, transf...@gmail.com wrote:
> > >
> > >>> As an example take this one: Every natural number is finite, i.e., less than aleph_0. Is that a brain-dead statement?
> > >>
> > >> No.
> > >>
> > >>> Same is true for the Binary Tree: There is no level where aleph_0 paths cross different nodes. Let alone where they cross uncountably many nodes.
> > >>
> > >> Correct. But the issue isn't how many nodes are crossed, it's how many
> > >> paths there are.
> > >
> > > How many paths there are differing by nodes at at least one level, to be precise.
> >
> > No. Just how many paths there are. It matters not where they differ nor
> > that they differ on a particular level. We just want to know how many paths.
> >
> > Martin Shobe
>
> Cantor would also be interested if there were
> any differences between "on the order of 2^n" and
> "uncounted" by cardinal 2^|N|, it's the Continuuum
> Hypothesis of Cantor which is established independent
> from ZF(C) by Cohen and a model of forcing.
>
> One wonders then whither and how there are examples
> of the ordinals of the cardinals as would so exist
> here between these two.
>
> Others and including myself have "line continuity"
> and "signal continuity" to so complement "field
> continuity" and all sorts of usual standard results.
>
> That mathematical intuition and curiousity of some
> isn't necessarily satisfied by "I don't have to
> think any more".
>
> Explore the structures of the paths of the infinite
> tree for themselves, this besides abstractions (and
> simplifications).

"Cantor would also be interested if there were
any differences between "on the order of 2^n" and
"uncounted" by cardinal 2^|N|, it's the Continuuum
Hypothesis of Cantor which is established independent
from ZF(C) by Cohen and a model of forcing.

One wonders then whither and how there are examples
of the ordinals of the cardinals as would so exist
here between these two.

Others and including myself have "line continuity"
and "signal continuity" to so complement "field
continuity" and all sorts of usual standard results. "

Cardinality is regular in ordinals.

[burs...@gmail.com]

Hi herpes boy, still confusing
ordinals and cardinals?

[Ross A. Finlayson]

"Cardinality is regular in ordinals",
this is as simply that "cardinality is
regular in ordinals, too, and with trichotomy",
cardinality is regular in ordinals if it's
not in cardinals (or functions) it's not in
ordinals, though clearly for far out is a
matter of fact.

The regularity of cardinals and ordinals is
quite uniform, here linear.

But, Burse, for Burse, why would you write that
when you know I already had time to set up?

Burse, no, "not" confusing cardinals and ordinals,
in matters of scale (here scale of cardinals
conditioning the large scale of ordinals).

Here some scale as of the larger scale, or frame,
of ordinals, and the small scale, of field divisions,
these are each maintaining nicely and neatly the
order statistics. This is where the "totally-ordered"
is sufficient to establish "order statistics" in order
of values that are sufficiently independent statistics.
Maintaining the scale and invariant then is for fields
in scale.

The scale of the measure of the line, a usual scale,
it's a projective space.

Then the "point at infinity" is after "not to scale",
eg with always "not to scale" between a point at infinity
and a usual integer point. This is with a diagram of a
projective space and its scale simple terms in partitions.

[burs...@gmail.com]

Herpes boy, cardinals are not that regular like
ordinals. For two ordinals we know if:

alpha < beta

Whethere there is something between alpha and
beta or not. There is nothing between alpha
and beta, if:

alpha = gamma+n

beta = gamma+n+1

https://en.wikipedia.org/wiki/Ordinal_arithmetic#Cantor_normal_form

On the otherhand, ZFC leaves undecided, whether
there is something between:

alpha_0 < 2^alpha_0

[Ross A. Finlayson]

Ha, it does.

The having the limit ordinals for the initial ordinals of
the cardinals less than c or 2^P(N) or 2^P(w) is, with
either, independent CH, or, Not CH. I.e., that's "regular".


Here then "amoebas" or "Burse's large sets equivalent to c",
those are regular or amoebas, and still regular.


Then, combine Continuum Hypothesis with "there exist large cardinals".

Or not.

All quite regular and ordinary, in ordinals, thank you.