Ross Finlayson on Foundations of the Continuous

[Ross A. Finlayson]

Well hello the group,

A few months without much input, basically the general reverie is as to: features of the numbers, of the numerical continuum, with regards to modern mathematics and set theory, and alternative notions, in foundations.

The rationals and irrationals are each dense in the reals. For a given positive real, there are either uncountably irrationals between it and zero or not, and if so, for each of those a rational, as they are dense in the reals and each other's complement, in trans-finite Dirichlet. (The rationals and irrationals are equivalent/equipollent.)

The particular function so defined as to map the natural integers to the reals of the unit interval in a constant monotonic and strictly increasing fashion, has that then: the antidiagonal argument, and nested intervals, as proofs of uncountability of the range, don't apply to it. As well, it was shown that the complement of each expansion in the range, is contained in the range. As well, it was shown where the range was the unit interval, of reals, that no element of the unit interval wasn't an element of the range. (The naturals and an interval's reals are equivalent/equipollent.)

Then as to foundations in plainly set-theoretic notions, there was built via axiomless deduction: a theory with theorems, in as to where, then, the axiomatized/defined/imbued object of the regular well-founded completed infinity, was shown to be inconsistent. As well, from universals (and expressible in accord with the collapse of the trans-finite hierarchy) it was described how the completed infinity would not be well-founded or regular. ZF: would contain itself, is a simple restatement of Russell's paradox as unresolved, to be resolved. (Consistent theories of the continuum have a natural continuum.)

Expanding the notion of a natural continuum, a geometry of points and spaces was described, and as to how it founds the Euclidean, with a spiral space-filling curve of points. (A natural continuum describes the fundaments of geometry.)

Well then, I'll continue to develop this generally, with now that as many of the fundamental particulars of these directions have been written, to then bringing the more interestingly synthetic of surprises (for mathematicians, beautiful surprises) from these quite totally analytic results and first cause as final principle.

Warm regards,

Ross Finlayson

[Virgil]

In article <48758bba-c50e-4066...@googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> Well hello the group,
>
> A few months without much input, basically the general reverie is as to:
> features of the numbers, of the numerical continuum, with regards to modern
> mathematics and set theory, and alternative notions, in foundations.
>
> The rationals and irrationals are each dense in the reals. For a given
> positive real, there are either uncountably irrationals between it and zero
> or not, and if so, for each of those a rational, as they are dense in the
> reals and each other's complement, in trans-finite Dirichlet. (The rationals
> and irrationals are equivalent/equipollent.)

In whose real number system is that supposed to hold true.

In the STANDARD reals, in intervals of positive length there are
countably many rationals and uncountably many reals, and removing those
rationals from those reals would necessarily leave uncountably many
irrational reals.

Their mere density does not translate into equipollency.
--

[Ross A. Finlayson]

Those reals would be the ones where each irrational either has uncountably many irrationals less than it, and for each a rational distinct from each other irrationals's: or not.

Else, this collection of reals wouldn't have for each irrational: another irrational, distinct from it.

These are the reals where between any two irrationals: there are rationals.

I understand that there are only countably many elements of the reals in their normal ordering, in a well-ordering, in ZF(C).

The "standard" reals are as Eudoxus/Cauchy/Dedekind - here with simply acknowledging trichotomy and density of the rationals in the reals before occluding those features with perceived would-be foundations, which speak not to those features except that they aren't.

This acknowledges the density of the rationals in the reals, the irrationals in the reals, and the rationals in the irrational's reals and the irrationals in the rational's reals.

Then, are there only countably many positive reals less than a given irrational? An interval is defined by its two endpoints, are there not uncountably many with zero as an endpoint?

This is of course rife with careful allusions to the directness of statement of features of the foundations: of the fundament, that is this continuum of real numbers.

Regards,

Ross Finlayson

[Virgil]

In article <02ced3ed-81c0-419d...@googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> I understand that there are only countably many elements of the reals in
> their normal ordering

Then you misunderstand.
--

[Peter Percival]

Ross A. Finlayson wrote:

>
> I understand that there are only countably many elements of the reals in their normal ordering, in a well-ordering, in ZF(C).

How can order affect cardinality?


--
Sorrow in all lands, and grievous omens.
Great anger in the dragon of the hills,
And silent now the earth's green oracles
That will not speak again of innocence.
David Sutton -- Geomancies

[Virgil]

In article <l05lqi$8t7$1...@news.albasani.net>,

Peter Percival <peterxp...@hotmail.com> wrote:

> Ross A. Finlayson wrote:
>
> >
> > I understand that there are only countably many elements of the reals in
> > their normal ordering, in a well-ordering, in ZF(C).
>
> How can order affect cardinality?

Disorder apparently can!

>
>
> --
> Sorrow in all lands, and grievous omens.
> Great anger in the dragon of the hills,
> And silent now the earth's green oracles
> That will not speak again of innocence.
> David Sutton -- Geomancies

--

[Ross A. Finlayson]

The idea there is that if there were uncountably many reals in their normal
ordering in a well-ordering, it would be simple to show that there are
rationals among each of those.

That ordering can indeed affect cardinality is where the ordering affects the
contents of what would be a set. The ordering-sensitivity of sets and their
cardinals is relevant here. Indeed, it is only a particular ordering of the
unit interval of reals as mapped from the naturals that has the property of
being onto.

Here we use sets to define structures, as of their elements. That a given
ordering represents the state of a mathematical process over the structure is
relevant then to, for example, the general process of induction over all the
elements of the structure. Sets are defined by their elements, but also, sets
as used to represent structures are defined by those structures, and as to
processes about those structures. Here, with the representation of a structure
like the continuum of real numbers, it was shown above that a particular
ordering relevant to induction over the elements of the interval, establishing
a direct and trivial (tautologous) inter-order relation (as good words) between
the naturals and unit, reasonably maintains structural features of each.

Sets are defined by their elements. Structures represented as sets, are
defined by those structures.

The idea then is yes that the reals, as the continuum of real numbers, are
ordering-sensitive, as represented as a set, or, as having various
representations as sets.

Order can affect cardinality, because order can affect representation.

Regards,

Ross Finlayson

[Virgil]

In article <25d5c10c-c4db-439a...@googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> On Tuesday, September 3, 2013 2:54:25 PM UTC-7, Peter Percival wrote:
> > Ross A. Finlayson wrote:
> >
> >
> >
> > >
> >
> > > I understand that there are only countably many elements of the reals in
> > > their normal ordering, in a well-ordering, in ZF(C).
> >
> >
> >
> > How can order affect cardinality?
> >
> >
> >
> >
> >
> > --
> >
> > Sorrow in all lands, and grievous omens.
> >
> > Great anger in the dragon of the hills,
> >
> > And silent now the earth's green oracles
> >
> > That will not speak again of innocence.
> >
> > David Sutton -- Geomancies
>
> The idea there is that if there were uncountably many reals in their normal
> ordering in a well-ordering, it would be simple to show that there are
> rationals among each of those.

That in normal ordering of the reals there is a rational between any two
reals does not imply that the same will hold for any sort of reordering
of the reals.

>
> That ordering can indeed affect cardinality is where the ordering affects the
> contents of what would be a set.

When you say that

"ordering affects the contents of what would be a set"

are you saying that in reordering an ordered set that some members
become non-members or that some non-members become members, or both?

> The ordering-sensitivity of sets and their
> cardinals is relevant here. Indeed, it is only a particular ordering of the
> unit interval of reals as mapped from the naturals that has the property of
> being onto.

There is no mapping from the naturals to the e unit interval of the
reals that is onto that interval.

>
> Here we use sets to define structures, as of their elements.

Everywhere else it is its elements that make up the structure of a set.
--

[graham...@gmail.com]

On Monday, September 2, 2013 4:37:32 PM UTC-7, Ross A. Finlayson wrote:
> Well hello the group,
>
>
>
> A few months without much input, basically the general reverie is as to: features of the numbers, of the numerical continuum, with regards to modern mathematics and set theory, and alternative notions, in foundations.
>
>
>
> The rationals and irrationals are each dense in the reals. For a given positive real, there are either uncountably irrationals between it and zero or not, and if so, for each of those a rational, as they are dense in the reals and each other's complement, in trans-finite Dirichlet. (The rationals and irrationals are equivalent/equipollent.)
>
>
>
> The particular function so defined as to map the natural integers to the reals of the unit interval in a constant monotonic and strictly increasing fashion, has that then: the antidiagonal argument, and nested intervals, as proofs of uncountability of the range, don't apply to it. As well, it was shown that the complement of each expansion in the range, is contained in the range. As well, it was shown where the range was the unit interval, of reals, that no element of the unit interval wasn't an element of the range. (The naturals and an interval's reals are equivalent/equipollent.)
>
>
>
> Then as to foundations in plainly set-theoretic notions, there was built via axiomless deduction: a theory with theorems, in as to where, then, the axiomatized/defined/imbued object of the regular well-founded completed infinity, was shown to be inconsistent.

1 Axiom then. The meta-math fuzzy notion that a theorem is true.

ALL(X):THEORY X



No need to prove consistency, AXIOM 1 enforces it!


ALL(X):THEORY NOT(EXIST(Y):THEORY)
X<-/->Y ....from [1]


NOW try defining Russell's Set!

As well, from universals (and expressible in accord with the collapse of the trans-finite hierarchy) it was described how the completed infinity would not be well-founded or regular. ZF: would contain itself, is a simple restatement of Russell's paradox as unresolved, to be resolved. (Consistent theories of the continuum have a natural continuum.)
>
>
>
> Expanding the notion of a natural continuum, a geometry of points and spaces was described, and as to how it founds the Euclidean, with a spiral space-filling curve of points. (A natural continuum describes the fundaments of geometry.)
>
>
>
> Well then, I'll continue to develop this generally, with now that as many of the fundamental particulars of these directions have been written, to then bringing the more interestingly synthetic of surprises (for mathematicians, beautiful surprises) from these quite totally analytic results and first cause as final principle.
>
>
>
> Warm regards,
>
>
>
> Ross Finlayson

Beware Virgil's Eternal Paradox!


1 e MYSET <-> 1 ~e YOUR 1st SET
2 e MYSET <-> 2 ~e YOUR 2nd SET
3 e MYSET <-> 3 ~e YOUR 3rd SET

AND SO ON...
UNTIL |PS(N)}>OO




Herc

[Ross A. Finlayson]

Sets are defined by their elements, then what are relevant to structures are
what those elements are. For the reals and integral analysis in the reals,
i.e. that of their parts or that of their wholes, for real analysis, what is
relevant of their parts is the integration over their differential areas
(patches over regions).

Modern mathematics is quite verbose and mute on this matter then of what is the
countable additivity of differential regions, in the summation over the
infinitely many regions in each unit (for Riemann or Lebesgue, or for that
matter Newton and Leibniz). The elements of the course of integration between
the bounds of integration is as of induction over the differential regions as
they go from difference areas to differential areas. (Note the exact analog to
the construction of "EF" the equivalency function.) The set of differential
patches is the relevant structure, and they build as ordered.

Then, for standard real analysis as built on measure theory, the values may
well be Cauchy but the course-of-values is inductive, over exhaustion of
refinement of the differences delta to differential d.

And, they are the same elements of the structure of concern: the unit line
segment, of the continuum of real numbers.

What's in fact quite standard in standard real analysis is countable
additivity, and that the limit is the sum, and only no different in the
infinite refinement as above.

Regards,

Ross Finlayson

[Virgil]

In article <83da2bd4-071b-48c4...@googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> Modern mathematics is quite verbose and mute on this matter

Neatest trick of the week to be both!
--

[WM]

On Tuesday, 3 September 2013 23:54:25 UTC+2, Peter Percival wrote:
> Ross A. Finlayson wrote:
>
>
>
> >
>
> > I understand that there are only countably many elements of the reals in their normal ordering, in a well-ordering, in ZF(C).
>
>
>
> How can order affect cardinality?

The rational numbers are countable when well-ordered. In the Binary Tree however, which is constructed from all rational numbers only, we get an uncountable number of paths. How can order influence the presence or absence of the irrationals (and with that cardinality)?

Regards, WM

[Ben Bacarisse]

You meant to say "how can taking the limit influence the presence or
absence of the irrationals". Then you'd see the answer for yourself.

--
Ben.

[WM]

The set of all rationals does not contain irrationals. If reordered (or simply left as it is, the set of all rationals has no canonical ordering), so if reordered in form of a tree, the irrationals enter the scene unavoidably and without further ado? That sounds like magic or matheology. Who will believe that? I certainly not. You? Really?

Regards, WM

[Ben Bacarisse]

All of 3, 3.1, 3.14, 3.141, 3.1415... are rational. Put just them,
nothing else, into a binary tree using the 0/1 encoding suggested and
what happens? Either you stop at some point and leave the tree as
finite (containing only rational numbers) or you "complete" just this
one path and, bang!, an irrational is represented in it! Where did it
come from??? It's such a puzzle!!!

Of course you accept that a rational sequence can have an irrational
limit, but you don't see the complete binary tree as a limit. You think
of it as always unfinished. That's fine -- there are many unfinished
things in your world.

--
Ben.

[Virgil]

In article <f60771ce-f05f-4771...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Tuesday, 3 September 2013 23:54:25 UTC+2, Peter Percival wrote:
> > Ross A. Finlayson wrote:
> >
> >
> >
> > >
> >
> > > I understand that there are only countably many elements of the reals in
> > > their normal ordering, in a well-ordering, in ZF(C).
> >
> >
> >
> > How can order affect cardinality?
>
> The rational numbers are countable when well-ordered.

The ordering that they are in does not affect their countability.

The rationals are countable, whatever order one finds them in because
thec can be well-ordered with only one non-successor.

> In the Binary Tree
> however, which is constructed from all rational numbers only, we get an
> uncountable number of paths.

But a path is not a mere rational number in such a tree but an infinite
sequence of proper binary ratios:
{ b_n/2^n, n in |N} with b-N a non-negative integer
and b_0 = 0
and either b_(n+1) = 2*b_n or
b_(n+1) = 2*b_n + 1,
for each n on |N.

> How can order influence the presence or absence
> of the irrationals (and with that cardinality)?

While order type of set need not affect the type of numbers in it,
certain order types limit how many numbers can be in set of that type.

A well ordered set with only one non-successor is at most countable.
A well ordered set with only one non-successor and a last member is
finite.

Densely ordered sets, like |Q or |R, must be at least countably
infinite, like |Q, and may be uncountably infinite, like |R.

At least everywhere outside of WM's wild weird world of WMytheology.
--

[Virgil]

In article <7697a5af-7041-469f...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Thursday, 5 September 2013 15:45:52 UTC+2, Ben Bacarisse wrote:
> > WM <muec...@rz.fh-augsburg.de> writes:
> >
> > > On Tuesday, 3 September 2013 23:54:25 UTC+2, Peter Percival wrote:
> > >> Ross A. Finlayson wrote:
> > I understand that there are only countably many elements of the
> > >> > reals in their normal ordering, in a well-ordering, in ZF(C).
> >
> > >> How can order affect cardinality?
> >
> > > The rational numbers are countable when well-ordered. In the Binary
> > > Tree however, which is constructed from all rational numbers only, we
> > > get an uncountable number of paths. How can order influence the
> > > presence or absence of the irrationals (and with that cardinality)?

The tree structure doe not affect the cardinality of the set of binary
rationals as it is the nodes which represent those binary rationals but
it is sets of nodes in the form of paths which are uncountably frequent.

> The set of all rationals does not contain irrationals. If reordered (or
> simply left as it is, the set of all rationals has no canonical ordering), so
> if reordered in form of a tree, the irrationals enter the scene unavoidably
> and without further ado? That sounds like magic or matheology.

In a binary tree, irrationals appear only as those convergent (weakly
monotone increasing) infinite sequences of binary rationals called paths
which never become periodic.

The paths which become eventually constant (all 0 branchings from some
point onwards or all 1 branchings from some point onwards) all represent
the proper binary rationals, expressible as m/2^n with m <= 2^n.

The paths that never become constant, but do become periodic, produce
the non-binary rationals like 1/3 and 2/5. "Eventually periodic" means
that from some node onwards the same finite pattern of left versus right
branchings repeats forever.

> Who will
> believe that? I certainly not.

What no one outside WM's wild weird world of WMytheology will believe is
what goes on inside it.
--

[Ross A. Finlayson]

So, measure theory defines the analytical character of the reals, foundations
(as it is) is verbosely mute on the matter for leaving the particular
definition of the analytical character of the reals to measure theory. Then,
where analysis is over the reals, its elements as they are, as they would be as
a set, maintain _countable_ additivity, to preserve all known useful results of
standard real analysis.

Modern mathematics is explicitly mute on the analytical character of the reals:
else it would be tacitly deficient.

So: is measure theory part of foundations, or, are there fundamentally
structural features of the reals, that they have as a representation, of
elements (or course-of-values), an embodiment of their analytical character?

Regards,

Ross Finlayson

[WM]

Why make it so difficult, using pi?

Use the following sequence:

0.1
0.11
0.111
...

According to your view I get the limit 0.111... - unless I stop somewhere.
That is not a mathematical view. And it is not substantiated by any axiom. In analysis the sequence does *not* contain its limit. Same is true when I write every node of the Binary Tree: 0.-1-1-1-...

Why is the limit not contained in the sequence and in the tree? Simple answer: Because it does not exist other than by a finite definition "0.111..." or "0.-1-1-1-..."

But I understand that this is even more difficult to comprehend for a trained set theorist than my definition of continuity. (It is easy to understand for laymen.)

>
>
>
> Of course you accept that a rational sequence can have an irrational
>
> limit

in fact, that is true, but the limit is, in most cases, not a term of the sequence like |N is not in the sequence 1, 2, 3, ...

> but you don't see the complete binary tree as a limit.

The tree does not contain ist limits. This is the same with the above list. It does not contain its limit.

But it contains everything that can be considered a path.

> You think
>
> of it as always unfinished. That's fine -- there are many unfinished
>
> things in your world.

Yes, all infinite things are infinite or, in English, unfinished.

If you want to have them finished, you must give order - by a finite word.
Alas, you know ...

Regards, WM

[WM]

On Friday, 6 September 2013 04:40:46 UTC+2, Virgil wrote:
> In article <7697a5af-7041-469f...@googlegroups.com>,

>
> The tree structure doe not affect the cardinality of the set of binary
>
> rationals as it is the nodes which represent those binary rationals but
>
> it is sets of nodes in the form of paths which are uncountably frequent.

You can try to claim that for a set of all subsets of an infinite set like P(|N). There it is as wrong, but not so easy to prove. But in the Binary Tree there are not all subsets of the countable set of finite initial segments of paths. On the contrary, write a list of all finite initial segments of paths. This list is countable. In order to get the Binary Tree you have merely to remove some bits from the terms of this list. But by removing some bits, you can never get a larger cardinality.

Simple example:
Take the list

0.1000...
0.11000...

and write it as a tree

/000...
0.1
\1000...

Has the cardinality increased?

Can any sober mind accept that somewhere "in the limit" the cardinality will explode?

No. (Take this as the definition of "sober mind".)

Regards, WM

[Ben Bacarisse]

WM <muec...@rz.fh-augsburg.de> writes:

<snip>

> Use the following sequence:
>
> 0.1
> 0.11
> 0.111
> ...

My example pi is better because it addressed your point of being
surprised that the limit tree contains irrationals.

> According to your view I get the limit 0.111... - unless I stop
> somewhere. That is not a mathematical view. And it is not
> substantiated by any axiom. In analysis the sequence does *not*
> contain its limit. Same is true when I write every node of the Binary
> Tree: 0.-1-1-1-...

Yes, exactly the same. None of the trees you build as you add paths is
the limit tree.

But I'm quite happy to accept the in your kind of mathematics there is
no completed limit tree. Since you don't accept the infinite sets can
every be completed, you don't accept that there is a complete binary tree
so you should stop being surprised by its properties. If you want to
talk about it, you should use the normal meaning for the term of define
your own weird WMCompleteBinaryTree.

I'd be grateful if you'd pay me the respect of not putting words into my
mouth. Either address the point I make or

<snip>
--
Ben.

[Virgil]

In article <4c6d13f1-25e1-4d51...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Friday, 6 September 2013 04:40:46 UTC+2, Virgil wrote:
> > In article <7697a5af-7041-469f...@googlegroups.com>,
>
> >
> > The tree structure doe not affect the cardinality of the set of binary
> >
> > rationals as it is the nodes which represent those binary rationals but
> >
> > it is sets of nodes in the form of paths which are uncountably frequent.
>
> You can try to claim that for a set of all subsets of an infinite set like
> P(|N). There it is as wrong, but not so easy to prove. But in the Binary Tree
> there are not all subsets of the countable set of finite initial segments of
> paths. On the contrary, write a list of all finite initial segments of paths.

But no PATH is ever a FINITE initial segment, since no path can ever
have last node. so counting non-paths is irrelevant to counting paths.

> In order to get the Binary Tree you have merely to
> remove some bits from the terms of this list.

WRONG!

A COMPLETE binary tree, at least everywhere outside of WM's wild weird
world of WMytheology, uses all countably many the nodes that WM's
incomplete trees uses, but also uses each node in uncountably many
different paths.

> Simple example:
> Take the list
>
> 0.1000...
> 0.11000...
>
> and write it as a tree
>
> /000...
> 0.1
> \1000...
>
> Has the cardinality increased?

A finite list can be written as a finite tree, but a complete infinite
list cannot be rewritten as a binary tree as they cannot be made not
order isomorphic.

In a binary tree, irrationals appear only as those convergent (weakly
monotone increasing) infinite sequences of binary rationals called paths
which never become periodic.

The paths which become eventually constant (all 0 branchings from some
point onwards or all 1 branchings from some point onwards) all represent

proper binary rationals, expressible as m/2^n with m <2^n.

The paths that never become constant, but do become periodic, produce

non-binary rationals like 1/3 and 2/5. "Eventually periodic" means that
from some node onwards the same finite pattern of left versus right
branchings repeats forever.

That WM does not understand tis is a measure ofh ow far out of reality
he has become.
--

[Virgil]

In article <fc06e396-63dd-4bbe...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Friday, 6 September 2013 02:09:46 UTC+2, Ben Bacarisse wrote:
> > WM <muec...@rz.fh-augsburg.de> writes:
> >
> > > On Thursday, 5 September 2013 15:45:52 UTC+2, Ben Bacarisse wrote:
> >
> > >> WM <muec...@rz.fh-augsburg.de> writes:
> >
> > >> > On Tuesday, 3 September 2013 23:54:25 UTC+2, Peter Percival wrote:
> >
> > >> >> Ross A. Finlayson wrote:
> >
> > >> >> > I understand that there are only countably many elements of the
> > >> >> > reals in their normal ordering, in a well-ordering, in ZF(C).

There are uncountably many reals in any ordering everywhere, at least
outside of outre places like WM's wild weird world of WMytheology.

> >
> > >> >> How can order affect cardinality?

It can only effect the ease by which one determines cardinality, at
least for infinite sets.

For finite sets all orderings are essentially the same as far as
determining cardinality is conserned.

> > >> > The rational numbers are countable when well-ordered.

Changing ordering of a set does not change its cardinality.


> > >> > In the binary tree however, which is constructed from all

> > >> > rational numbers only, we get an uncountable number of paths.

A complete binary trees has a node for each binary rational, but has no
nodes for non-binary rationals lie 1/3 or 2/3.

> > >> > How can order influence the presence or absence of the
> > >> > irrationals (and with that cardinality)?

It doesn't

> > >>
> >
> > >> You meant to say "how can taking the limit influence the presence or
> >
> > >> absence of the irrationals". Then you'd see the answer for yourself.
> >
> > >
> >
> > > The set of all rationals does not contain irrationals. If reordered
> >
> > > (or simply left as it is, the set of all rationals has no canonical
> >
> > > ordering), so if reordered in form of a tree, the irrationals enter
> >
> > > the scene unavoidably and without further ado? That sounds like magic
> >
> > > or matheology. Who will believe that? I certainly not. You? Really?

The difference is between nodes and paths. One can identify the nodes of
a complete infinite binary tree with proper binary fractions, but we
cannot do that with the paths, because, among other reasons, there are
far more paths than nodes.

> >
> >
> >
> > All of 3, 3.1, 3.14, 3.141, 3.1415... are rational. Put just them,
> >
> > nothing else, into a binary tree using the 0/1 encoding suggested and
> >
> > what happens? Either you stop at some point and leave the tree as
> >
> > finite (containing only rational numbers) or you "complete" just this
> >
> > one path and, bang!, an irrational is represented in it! Where did it
> >
> > come from???

It is the entire set of those lower decimal approximations, or, if you
like, the LUB of them.

>
> Why make it so difficult, using pi?
>
> Use the following sequence:
>
> 0.1
> 0.11
> 0.111
> ...
>
> According to your view I get the limit 0.111... - unless I stop somewhere.

According to our view we get the convergent sequence
{ 0,1. 0,11, 0.111, ...}
which in a Cauchy model of the reals represents its limit, 1/(b-1) where
b is the base of the number system in which { 0,1. 0,11, 0.111, ...} is
being written.

> That is not a mathematical view.

It may not be a WMytheological view, but it is totally a mathemtical

view.



> And it is not substantiated by any axiom. In
> analysis the sequence does *not* contain its limit.

In the Cauchy model the sequence IS the limit, or at least one
representation of it.

> Same is true when I write

>

> Why is the limit not contained in the sequence and in the tree? Simple
> answer: Because it does not exist other than by a finite definition

The Cauchy definition is a finite definition.
>
> But I understand

NO! You don not!

> >
> >
> >

>
> The tree does not contain ist limits.

The Complete Infinite Binary Tree does contain its paths as subsets of
the set of its set of nodes, and for every real in [0,1] there is at
last one such path, and for some of them two paths.

>
> > You think
> >
> > of it as always unfinished. That's fine -- there are many unfinished
> >
> > things in your world.
>
> Yes, all infinite things are infinite or, in English, unfinished.

In English, "infinite" and "unfinished" are not at all the same thing.


WM's mathematics is in many ways unfinished, rough, and
self-contradictory without being in any way infinite.


The paths in a Complete Infinite Binary Tree which become eventually

constant (all 0 branchings from some point onwards or all 1 branchings

from some point onwards) all represent binary rationals, each
expressible as m/2^n with m <= 2^n, m+1 and n in |N ( m can be 0).

The paths whose branching never become constant in one direction, but do
become periodic, produce the non-binary rationals like 1/3 and 2/5.
"Eventually periodic" means that from some node onwards a fixed finite
sequence of two or more left AND right branchings repeats forever.

In a binary tree, irrationals appear only as those convergent (weakly
monotone increasing) infinite sequences of binary rationals called paths

which never become either constant or periodic in the sense defined
above.

To the extent to which WM's notion of a COMPLETE Infinite Binary Tree
differs from the above, WM's notion is wrong!
--

[Ross A. Finlayson]

It's not unreasonable that there's infinite in value in the infinite in
quantity.

Consider for example Russell's paradox from set theory, that the set of all
regular sets contains itself. When there are only finite ordinals in the
theory, often built as they as each from the next: the collection contains the
collection. As it is not finite in the quantity it contains, it's not finite
in value as an element of the domain of discourse: natural integers as
ordinals.

As well, in the higher level, it is not alien to number theorists that there's
a point at infinity, here for example a prime or a (the) composite at infinity.

There are examples of this from Spinoza's (natural) integers as continuum to
Paris and Kirby's countable, non-standard integers (since Archimedes).

Then as well a reasonable person could consider a variety of structural
possibilities of relevant integers. We might start with zero and proceed via
induction: for induction, simply. The number theorists may have their prime or
composite at infinity (or other suitably large number) generally. The
continuum of integers may be seen as containing a single infinite value, or
that half of them are, or that most of them are. For some, they may have that
counting to infinity goes all the way back around to zero, the next zero, or
for that matter, that the ultimate or penultimate infinite value strongly
embodies what is "-1".

For usual methods of standard real analysis, the convergent sequence may not
contain its limit. While that may be so, in the usual definition, it _is_ that
limit (a representation of that value). The limit _is_ the sum. It doesn't
exist without it so being so, and, it exists.

As well, as above: of all the infinite sequences, the limit would be in the
convergent sequence.

In passing about the tree, there are countably many nodes and paths are only
pair-wise distinct by nodes. There are only countably many ways to distinguish
paths. As well, the sweep enumerates the paths at each level.

Then, the finitist view is a simplification of the infinite of the continuum to
keep notions of induction simple. And, there's more to it than that. The
finitist view is largely irrelevant to mathematics about the infinity of the
continuum, except that the continuum supports the finite as they each define
the other.

Then, the goal of analysis of the continuum beyond the standard is to retrieve
relevant results of interest: to the real. Then, modern (as it were)
mathematics has had a century to produce relevant results to the concrete and
there aren't any (as well-known). Banach and Tarski's re-composition of is a
nowhere measurable de-composition, Vitali's infinitesimal that would sum to
two, these touchstones in modern mathematics of features of the real numbers
are at once not only solely available via the stated foundations, but their
particular parameterizations as would make them concrete: aren't. This is
from density and symmetry from these structures, where, fairly enough,
well-founded transfinite cardinals aren't necessarily the structure(s) of the
continuum, where as Russell notes, reasonably enough: they're not
well-founded.

So, the goal of novelty in continuum analysis vis-a-vis the foundations, is not
to retreat to the finite, as that's not novel, nor retrench to the
trans-finite, as that's not relevant, but reveal that from first principles:
emergent features of the continuum provide mathematical grounds: for novel
application.

Regards,

Ross Finlayson

[Ross A. Finlayson]

About Vitali and Banch-Tarski, the point is that Vitali makes the unit line
segment everywhere discontinuous, re-assembles those points in a line: of
length two instead of one (or between one and three as Vitali). This where in
a sense the act of indviduating the points as discrete on the continuous line
makes them two-sided instead of one-sided, that again as one-sided they make
twice the point. (As it were.) (Parenthetically, parenthetically.)

Individual points are two-sided, as part of a line or ray: one-sided. In all
the dimensions the point is in, poly-dimensionally, its weight as individuated
corresponds increasingly to the dimensions that it is individuated from, from
the continuum and from the dimensions of the continuum.

Then with Banach-Tarski and the notion of re-composition of the ball from four
or five pieces, individuation of the points takes correspondingly more aspects
of delineation, as it were, than the two pieces of B-T line segment doubling.
Relevant to Vitali is that: that is the same individuation of the points. with
Vitali taking the points apart each and giving them weight and re-assembling
them, where B-T takes the points apart all and re-assembling those.

The point here about the least difference between discrete and continuous is
that it's reasonable to work up what happens in continuum mechanics in actually
going between the discrete and continuous, where this would build theorems
features of the real numbers, and where their structures are discrete and where
they are continuous. Basically about the measurement effect, in physics, these
what would be very real features, are currently beyond "modern" mathematics, in
the foundations.

Regards,

Ross Finlayson

[WM]

On Friday, 6 September 2013 12:29:46 UTC+2, Ben Bacarisse wrote:
> WM <muec...@rz.fh-augsburg.de> writes:
>
>
>
> <snip>
>
> > Use the following sequence:
>
> >
>
> > 0.1
>
> > 0.11
>
> > 0.111
>
> > ...
>
>
>
> My example pi is better because it addressed your point of being
>
> surprised that the limit tree contains irrationals.

It is the same. An infinite sequence of digits.

>
>
>
> > According to your view I get the limit 0.111... - unless I stop
>
> > somewhere. That is not a mathematical view. And it is not
>
> > substantiated by any axiom. In analysis the sequence does *not*
>
> > contain its limit. Same is true when I write every node of the Binary
>
> > Tree: 0.-1-1-1-...
>
>
>
> Yes, exactly the same. None of the trees you build as you add paths is
>
> the limit tree.

Small wonder. The limits of the paths cannot be represented by an infinity of digits. Who could write them in any discourse?

>
>
>
> But I'm quite happy to accept the in your kind of mathematics there is
>
> no completed limit tree.

You should try to write only one path without abbreviating it. Then you would learn something about your mathematics (and perhaps you would learn why many scientific and engineering departments have their students taught mathematics by scientists and engineers).

> Since you don't accept the infinite sets can
>
> every be completed, you don't accept that there is a complete binary tree
>
> so you should stop being surprised by its properties. If you want to
>
> talk about it, you should use the normal meaning for the term of define
>
> your own weird WMCompleteBinaryTree.

I accept as much as is possible. Again: Try to write an infinite sequence - do not only couterfactually believe that it could be done in mathematics discourse.

And if you have grasped the fact that it is impossible, other than as a finite word like pi or 1/9, then try to find out how many finite words can be used in mathematics discourse.

>
>
>
> I'd be grateful if you'd pay me the respect of not putting words into my
>
> mouth.

I am not aware to have done so. But hell and devil! Take the list of all *infinite* digit sequences of the unit interval. Note: infinite sequences are assumed to exist!

0.1000...
0.11000...

and write some or many or all of these infinite sequences as a tree

/000...
0.1
\1000...

Has the cardinality increased?

Make the experiment yourself, if you have an auditory of unperverted newbies. Tell them this example or what enumerating the positive rationals requires. And ask how many are ready to believe in Cantor's ideas.

Regards, WM

[WM]

On Friday, 6 September 2013 22:45:53 UTC+2, Virgil wrote:
> In article <4c6d13f1-25e1-4d51...@googlegroups.com>,

>
> > Simple example:
>
> > Take the list
>
> >
>
> > 0.1000...
>
> > 0.11000...
>
> >
>
> > and write it as a tree
>
> >
>
> > /000...
>
> > 0.1
>
> > \1000...
>
> >
>
> > Has the cardinality increased?
>
>
>
> A finite list can be written as a finite tree, but a complete infinite
>
> list cannot be rewritten as a binary tree as they cannot be made not
>
> order isomorphic.

Take the complete list of all terminating rationals of the unit interval.
Unite some or many of them into tree structure. Does the cardinality increase?

Regards, WM

[WM]

On Friday, 6 September 2013 23:28:31 UTC+2, Virgil wrote:


>
> The difference is between nodes and paths. One can identify the nodes of
>
> a complete infinite binary tree with proper binary fractions, but we
>
> cannot do that with the paths, because, among other reasons, there are
>
> far more paths than nodes.

Then map every node on a path that it belongs to and remove that node and that path from the Binary Tree. If there are more paths, then something should remain, because if you remove one path from the tree, at least some nodes of each other path remain there.

What will remain, after you have removed every node?

Regards, WM

[WM]

On Sunday, 8 September 2013 21:25:00 UTC+2, WM wrote:

Correction:

> Take the list of all *infinite* digit sequences

of RATIONAL NUMBERS

[Ross A. Finlayson]

How is this telling us something about the mathematical foundations
vis-a-vis the mathematical continuum?

There are plainly other threads (and plainly better forae) to discuss
the particulars you find of interest, I'd hope you'd find one more
suited for discussion of logic in sci.logic and then as to the
particulars in relevant threads.

That's where I can see that Muckenheim here has various points he sees
as relevant, but only to going backward in the development, I don't see
how that promotes the general development as to the novel in foundations
and here of foundations of the mathematical continuum of real numbers.

Then, I can agree that the infinite balanced binary tree has only
countably many nodes and the paths are through the nodes, but, the
question of relevance, is: given that we know modern mathematics'
theorems about the reals (or even the naturals) are at best incomplete,
that there are facts about these numbers yet to be known: how do you
justify a contemporary mathematics with relevant novel features of these
numbers, and furthermore, what is a direction for a general course to
elaborate on these facts as features of the continuum, for mathematics
and practice?

Because, that is the general theme of what I've written to sci.math and
sci.logic, and, I'm interested in that, and it wouldn't just be a
courtesy to discover and share real features of the continuum, but a
great service.

Regards,

Ross Finlayson

[WM]

On Monday, 9 September 2013 03:46:21 UTC+2, Ross A. Finlayson wrote:
> On Sunday, September 8, 2013 12:35:33 PM UTC-7, WM wrote:


> Then, I can agree that the infinite balanced binary tree has only
>
> countably many nodes

nobody doubts that

> and the paths are through the nodes,

and a path without nodes cannot exist. But if we remove every node and with each node one path, then nothing remains. That means, when removing node by node, we must have removed with some nodes more than one path. And since countable times countable is countable, we must have removed with some nodes uncountably many paths. Is it possible to believe that? Is it possible to obtain that from axioms of ZF?

> but, the
>
> question of relevance, is: given that we know modern mathematics'
>
> theorems about the reals (or even the naturals) are at best incomplete,

The axioms about the naturals are not incomplete.
1 in |N
if n in |N then n+1 in |N
nothing else in |N

>
> that there are facts about these numbers yet to be known: how do you
>
> justify a contemporary mathematics

There is no justification. It is actually completed rubbish.
The continuum cannot be resolved into points. But why should it? For any purpose it is enough to approximate a given point as precisely as you wish. Nothing else is possible in "modern mathematics", compare the efforts to approximate pi. They will never get the "true value", i.e., an infinite sequence of digits. So why should it be useful to believe that there is a "true value" other than the finite definition pi or one of its formulas.

Regards, WM

[Virgil]

In article <efaf39e5-e459-4119...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> Small wonder. The limits of the paths cannot be represented by an infinity of
> digits. Who could write them in any discourse?

Not everything that can exist in one's imagination can be completely and
explicitly written out. That does not prevent its use in matheatics.
--

[Virgil]

In article <5363208e-9818-4f9c...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> > Make the experiment yourself, if you have an auditory of unperverted
> > newbies.

I have often done so with an auditorium of such newbies, none of whom
ever expressed any problems with the need of an actual infinity of
numbers or points on a line or members of infinite sets in general.

Perhaps WM is just a bad teacher.
--

[Virgil]

In article <0c20527f-b20c-4f9f...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Friday, 6 September 2013 23:28:31 UTC+2, Virgil wrote:
>
>
> >
> > The difference is between nodes and paths. One can identify the nodes of
> >
> > a complete infinite binary tree with proper binary fractions, but we
> >
> > cannot do that with the paths, because, among other reasons, there are
> >
> > far more paths than nodes.
>
> Then map every node on a path that it belongs to and remove that node and
> that path from the Binary Tree. If there are more paths, then something
> should remain, because if you remove one path from the tree, at least some
> nodes of each other path remain there.

Removing any one node from a COMPLETE INFINITE BINARY TREE cuts off
uncountably many of its paths and infinitely many of its nodes from the
tree, but any finite number of removals that does not cut off all paths
will leave uncountably many remaining. At least outside of WM's wild
weird world of WMytheology.
>

> What will remain, after you have removed every node?

Unless removing both a parent node and its child node can somehow leave
the connection between them still existing, nothing at all.
--

[Virgil]

In article <2571aa7b-5de0-454f...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Monday, 9 September 2013 03:46:21 UTC+2, Ross A. Finlayson wrote:
> > On Sunday, September 8, 2013 12:35:33 PM UTC-7, WM wrote:
>
>
> > Then, I can agree that the infinite balanced binary tree has only
> >
> > countably many nodes
>
>
> nobody doubts that
>
> > and the paths are through the nodes,
>
>
> and a path without nodes cannot exist. But if we remove every node and with
> each node one path, then nothing remains.

Removing a non-root node from a COMPLETE INFINITE BINARY TREE will
remove at as many paths as it leaves intact as there are as any paths
through any node as emanate from the root node the entire tree.

> That means, when removing node by
> node, we must have removed with some nodes more than one path.

Since in the Complete Infinite Binary Tree there are the same number of
paths through any one node as any other, including the root node,
removing any node, and necessarly al ots progeny along with it.
eliminates uncountably many infinite paths, as many as in the entire
tree.

> And since
> countable times countable is countable, we must have removed with some nodes
> uncountably many paths.

So long as there are any paths left (finite chains are not paths in a
Infinite Binary Trees) removing any node in a path removes uncountably
many paths and countably many other nodes
--

[Virgil]

In article <eb81cdaa-c069-4295...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> Take the complete list of all terminating rationals of the unit interval.

What is a "terminating rational"?

Is it anything like the value of a terminating decimal?

If so, how is a list of all terminating decimals essentially different
from a list of all terminating binaries or a list of all termniating
octals or a list of all terminating hexadecimals?
--

[Zeit Geist]

On Monday, September 2, 2013 4:37:32 PM UTC-7, Ross A. Finlayson wrote:
> Well hello the group,
>

Hey Ross,

Playing catch up here.

>
> A few months without much input, basically the general reverie is as to: features of the numbers, of the numerical continuum, with regards to modern mathematics and set theory, and alternative notions, in foundations.
>
> The rationals and irrationals are each dense in the reals. For a given positive real, there are either uncountably irrationals between it and zero or not, and if so, for each of those a rational, as they are dense in the reals and each other's complement, in trans-finite Dirichlet. (The rationals and irrationals are equivalent/equipollent.)
>

Just because they are both dense subsets of the reals does Not imply 1:1 correspondence.

>
> The particular function so defined as to map the natural integers to the reals of the unit interval in a constant monotonic and strictly increasing fashion, has that then: the antidiagonal argument, and nested intervals, as proofs of uncountability of the range, don't apply to it. As well, it was shown that the complement of each expansion in the range, is contained in the range. As well, it was shown where the range was the unit interval, of reals, that no element of the unit interval wasn't an element of the range. (The naturals and an interval's reals are equivalent/equipollent.)
>

What particular function?

There is no function mapping the Naturals onto the Reals.

>
> Then as to foundations in plainly set-theoretic notions, there was built via axiomless deduction: a theory with theorems, in as to where, then, the axiomatized/defined/imbued object of the regular well-founded completed infinity, was shown to be inconsistent. As well, from universals (and expressible in accord with the collapse of the trans-finite hierarchy) it was described how the completed infinity would not be well-founded or regular. ZF: would contain itself, is a simple restatement of Russell's paradox as unresolved, to be resolved. (Consistent theories of the continuum have a natural continuum.)
>

That is resolved. V, the Cumulative Hierarchy, is Not a Set in ZF(C). This is a provable theorem.

V is the "Collection" of all Sets. ZF "talks" about Sets, which are parts of V. It cannot tells us everything about V, for then V could be described as a Set.

>
> Expanding the notion of a natural continuum, a geometry of points and spaces was described, and as to how it founds the Euclidean, with a spiral space-filling curve of points. (A natural continuum describes the fundaments of geometry.)
>

The continuum of geometry is one such that a segment of the continuum "looks"just like any other segment. Moreover, if the other segment is contained in the first segment they "look" the same, no mater how small we get. The continuum is perfectly uniform down to any detail.

Here I am talking pragmatically about what we want from the Real Continuum. We also want the continuum to have one other property. We want it to be Complete. That is, it must have no gaps.

This is the Object that Modern Mathematics calls the Set of Real Numbers.

>
> Well then, I'll continue to develop this generally, with now that as many of the fundamental particulars of these directions have been written, to then bringing the more interestingly synthetic of surprises (for mathematicians, beautiful surprises) from these quite totally analytic results and first cause as final principle.
>

Looking forward to it.

>
> Warm regards,
> Ross Finlayson

ZG

[Zeit Geist]

On Monday, September 2, 2013 11:48:20 PM UTC-7, Ross A. Finlayson wrote:

> Those reals would be the ones where each irrational either has uncountably many irrationals less than it, and for each a rational distinct from each other irrationals's: or not.
>
> Else, this collection of reals wouldn't have for each irrational: another irrational, distinct from it.
>
> These are the reals where between any two irrationals: there are rationals.

>
> I understand that there are only countably many elements of the reals in their normal ordering, in a well-ordering, in ZF(C).
>

First of all ZF cannot prove that the Reals can be Well-Ordered.
In ZFC the Reals, like any other set, can be Well-Orderd.

Also, just because a Set is Well-Ordered does Not mean that it is Countable.
There are Uncountable Well-Orderings. Such as the Ordinal corresponding to the
Cardinality of the Reals.

> The "standard" reals are as Eudoxus/Cauchy/Dedekind - here with simply acknowledging trichotomy and density of the rationals in the reals before occluding those features with perceived would-be foundations, which speak not to those features except that they aren't.
>

The Foundations, ZFC, provide a setting in which the Reals can be defined with the properties we desire them to have. The Foundation is not there to Define the Reals, but to make sure our definitions are on firm footing. Here we produce a Set which possess those features.

>
> This acknowledges the density of the rationals in the reals, the irrationals in the reals, and the rationals in the irrational's reals and the irrationals in the rational's reals.
>
> Then, are there only countably many positive reals less than a given irrational? An interval is defined by its two endpoints, are there not uncountably many with zero as an endpoint?
>
> This is of course rife with careful allusions to the directness of statement of features of the foundations: of the fundament, that is this continuum of real numbers.
>

Not sure what you're trying to say here.

>
> Regards,
>
> Ross Finlayson

ZG

[Zeit Geist]

On Tuesday, September 3, 2013 8:47:04 PM UTC-7, Ross A. Finlayson wrote:

> On Tuesday, September 3, 2013 2:54:25 PM UTC-7, Peter Percival wrote:
>
> > Ross A. Finlayson wrote:
>

> > > I understand that there are only countably many elements of the reals in their normal ordering, in a well-ordering, in ZF(C).
>

> > How can order affect cardinality?

> > --
>
>
> The idea there is that if there were uncountably many reals in their normal
> ordering in a well-ordering, it would be simple to show that there are
> rationals among each of those.
>
> That ordering can indeed affect cardinality is where the ordering affects the
> contents of what would be a set. The ordering-sensitivity of sets and their
> cardinals is relevant here. Indeed, it is only a particular ordering of the
> unit interval of reals as mapped from the naturals that has the property of
> being onto.
>

I believe you think that Well-Ordering implies Countable Cardinality.
That is Not true.

>
> Here we use sets to define structures, as of their elements. That a given
> ordering represents the state of a mathematical process over the structure is
> relevant then to, for example, the general process of induction over all the
> elements of the structure. Sets are defined by their elements, but also, sets
> as used to represent structures are defined by those structures, and as to
> processes about those structures. Here, with the representation of a structure
> like the continuum of real numbers, it was shown above that a particular
> ordering relevant to induction over the elements of the interval, establishing
> a direct and trivial (tautologous) inter-order relation (as good words) between
> the naturals and unit, reasonably maintains structural features of each.
>

Structures are made of Sets, yes. But any Structure containing all Real Numbers must be Uncountable.

I don't think you showed that the Reals can be Ordered isomorphically with respect to the Order of the Naturals.

>
> Sets are defined by their elements. Structures represented as sets, are
> defined by those structures.
>

The definitions of the Structure define the Structure.
Structure a Set does Not change its content.

>
> The idea then is yes that the reals, as the continuum of real numbers, are
> ordering-sensitive, as represented as a set, or, as having various
> representations as sets.
>

No, they can never be put in a countable order.

>
> Order can affect cardinality, because order can affect representation.
>

How does order effect representation?

>
> Regards,
> Ross Finlayson

ZG

[Virgil]

In article <41ace296-31bd-473f...@googlegroups.com>,

Zeit Geist <tucso...@me.com> wrote:

> > Order can affect cardinality, because order can affect representation.

Wrong again! Each member of a set remains one member regardless of
"representation" or any order relation on the set.
--

[fom]

On 9/9/2013 6:37 PM, Zeit Geist wrote:
>>
>> Order can affect cardinality, because order can affect representation.
>>
>
> How does order effect representation?
>

I have quoted Hilbert before concerning the
presupposition of a delimited domain in formalistic
axiomatics. Hilbert also acknowledges that formalistic
axiomatics ought to be guided by contentual axiomatics.

I will no longer try to explain issues as if
I am correct -- or, even sensible. However, the
content of set theory in relation to the definitions
of number via Cantor and Frege have to do with individuation,
descriptions, and names. The latter two arise more from
Frege's work. The former from Cantor.

In Wittgenstein, descriptions are rejected, but names are
taken to have primitive correspondence with objects.

If you look at Bolzano's attempts to define simple
substance, you will see that the goal of giving
mathematics a better foundation through logic is
precisely related to how mathematics is represented.

Now, Hilbert's formal axiomatics happens to be amenable
to the algebraic standpoint. If there are no restraints
on how objects of a domain are delimited, then one can
generalize the notion of language to meaningless, uninterpreted
syntax and assume that a domain of application is
meaningful.

You will find this stated expressly in Tarski's 1933. He
actually rejects the notion of a formal language in the
sense of pure syntax. He speaks of formalized languages
as corresponding with established scientific jargons.

You have said nothing in response to Ross that is not
standard mathematics. It is all correct. But, there
have been two competing paradigms.

Ignoring the winning paradigm for the moment, suppose
I have "objects" which are not purely grammatical forms.
With this restriction, I exclude "names" from being
"objects".

Suppose I "name" the first "object".

If I "name" the second "object", I must take
into acount that the "second object" ought not
have the "same name" as the "first object".

So, I "name" the second "object" with a
"different name" so that the "second object"
is clearly not the "first object".

I suppose I should not have really commented on
your question without knowing what you mean
by representation. What I have just described
is a simple scenario which I believe expresses
Bolzano's admonition against "doing violence to
language".

I will concede every criticism everyone has ever
thrown at me.

But, I do not see how ordinal structure is not
intimately related to naming and identity. It
is these latter two which is emphasized by the
losing paradigm.

What could the winning paradigm possibly mean
by the word "representation" that isn't made
empty by its presuppositions?

[Ross A. Finlayson]

About a well-ordering of the real, there aren't uncountably many
elements in their normal ordering, because, a well-ordering of a
superset is a well-ordering of the set, so there's no uncountable subset
of the reals in their normal ordering, in the well-ordering. Yet,
between any two elements in their normal ordering in the well-ordering,
there would be uncountably many between those two, later in the
well-ordering. For any given member, via transfinite induction, here
over the elements of the well-ordering, there are uncountably many
elements in the well-ordering, in the normal ordering to it. So, there
are well-orderings, as well-orderings of an uncountable set where that
exists, that have more than countably many in the normal ordering. But
then, the rationals are dense in the reals, which is why it's said
there's no well-ordering of the reals with uncountably many elements in
their normal ordering, as the rationals (and other sets dense in the
reals) are countable.

It's said there aren't uncountably many reals in their normal ordering
in a well-ordering, because that would contradict the countability of
the rationals, due their density in the reals. Then, as above, given a
well-ordering of the reals, a transfinite induction schema identifying
for each uncountably many following, holds else that would contradict
the uncountability of the reals, due their density in the reals.

A similar method has to pick for a positive irrational any of the
uncountably many irrationals less than it, and a rational between them
as a pair. Does each positive irrational have uncountably many
irrationals less than it, or not? A separate method using properties of
density of each of the rationals and irrationals, and their being each
others' complement, to have for each a unique other.

Then the particular function mentioned is the "natural/unit equivalency
function".
a) antidiagonal argument doesn't apply
b) nested intervals doesn't apply
c) it's shown the range is dense in [0,1]
d) it's shown for each f(n) as expansion that ~f(n) is in the range
e) it's shown that no element of [0,1] is not in the range

Vis-vis V the set-theoretic universe and L the constructible universe,
keep in mind Fefermann's note on well-ordering the reals that V = L.
Also, ZF's universe as Russell set contains itself.

The real numbers as Hardy would note also intrinsically have measure, as
geometric.

Warm regards,

Ross Finlayson

[Ross A. Finlayson]

The point includes that the continuum of real numbers has all its
properties, and, given a rationale as to why Eudoxus/Dedekind/Cauchy is
not always sufficient to represent the continuum, and that modern
mathematics demurs the definition of measure of the real numbers which
is an intrinsic quality, that there's a justified placement in the
foundations (of logical structure to defined objects of mathematics) of
all the properties of the continuum.

As well it is to that similarly as to how sometimes ordinals are a
representation of integers, and it's possible then to extrapolate
features of ordinals to relevant features of integers, and that is again
not necessarily within the canon except as development, the careful
allusion is as to preservation or conservation of these properties,
while maintaining the preservation or conservation of the surrounds.

From ZF(C) and modern foundations for of course the tremendous value of
the completeness results, as an example, that is to notions of the
transfer principle (that some features that hold true for the finite
hold true for the infinite, or for the elements, the collection), and
domain principle (that for constructing from the e-minimal, there yet
exists the e-maximal), that the real foundations maintain these
features, as relevant to ZF (and its universe) as it is to them (and no
more).

Regards,

Ross Finlayson

[Ross A. Finlayson]

Did you miss the memo?

https://groups.google.com/d/msg/sci.logic/C1l6ihQI62k/psJUBpdHe5sJ

[Ross A. Finlayson]

Use the integral calculus, integrate over a bounded region. Is that not: in order?

That's also called continuum analysis.

Also its founders called it the analysis of infinitesimals.

This, all of the real numbers.

Regards,

Ross Finlayson

[Ross A. Finlayson]

Nice to hear from you, fom, ZG, good day et. al.

The integral calculus is used every day for most matters
of real analysis.

Set theory as transfinite cardinals has yet to find a single
application in physics.

I'm not quite sure what you see as winning.

Regards,

Ross Finlayson

[Zeit Geist]

On Monday, September 9, 2013 7:36:01 PM UTC-7, Ross A. Finlayson wrote:
>
>
> About a well-ordering of the real, there aren't uncountably many
> elements in their normal ordering, because, a well-ordering of a
> superset is a well-ordering of the set, so there's no uncountable subset
> of the reals in their normal ordering, in the well-ordering.

Any Ordering of the Reals is an Uncountable Order.

> Yet, between any two elements in their normal ordering in the well-ordering,
> there would be uncountably many between those two, later in the
> well-ordering. For any given member, via transfinite induction, here
> over the elements of the well-ordering, there are uncountably many
> elements in the well-ordering, in the normal ordering to it. So, there
> are well-orderings, as well-orderings of an uncountable set where that
> exists, that have more than countably many in the normal ordering. But
> then, the rationals are dense in the reals, which is why it's said
> there's no well-ordering of the reals with uncountably many elements in
> their normal ordering, as the rationals (and other sets dense in the
> reals) are countable.
>

A Well-Ordering of the Reals would not necessarily have either the Rationals or the Irrationals as an Everywhere Dense Set.

And, what do you mean by "reals in their normal ordering in a well-ordering" as you keep saying.

>
> It's said there aren't uncountably many reals in their normal ordering
> in a well-ordering, because that would contradict the countability of
> the rationals, due their density in the reals. Then, as above, given a
> well-ordering of the reals, a transfinite induction schema identifying
> for each uncountably many following, holds else that would contradict
> the uncountability of the reals, due their density in the reals.
>

See above.

> A similar method has to pick for a positive irrational any of the
> uncountably many irrationals less than it, and a rational between them
> as a pair. Does each positive irrational have uncountably many
> irrationals less than it, or not? A separate method using properties of
> density of each of the rationals and irrationals, and their being each
> others' complement, to have for each a unique other.
>

You know, a Well-Ordering of the Reals could have all the Integers first in the Ordering, then the rest of the Rationals and finally all the Irrationals in some extremely complex fashion.

>
> Then the particular function mentioned is the "natural/unit equivalency
> function".
>
> a) antidiagonal argument doesn't apply
>
> b) nested intervals doesn't apply
>
> c) it's shown the range is dense in [0,1]
>
> d) it's shown for each f(n) as expansion that ~f(n) is in the range
>
> e) it's shown that no element of [0,1] is not in the range
>

That function does Not exist!

>
> Vis-vis V the set-theoretic universe and L the constructible universe,
> keep in mind Fefermann's note on well-ordering the reals that V = L.
> Also, ZF's universe as Russell set contains itself.
>

ZFC + V=L does prove the existence of a Well-Ordering on the Reals, via proving Choice. However, it also prove that the Reals are Uncountable.

V does Not contain itself, as V contains only Sets and V is Not a Set.

>
> The real numbers as Hardy would note also intrinsically have measure, as
> geometric.
>

Not too clear on Hardy's concept of Measure.
Could you fill me in some.

[Zeit Geist]

No, I didn't!

You are simply WRONG regardless of the structure if that structure contains all reals then that structure is an uncountable structure.

ZG

[Virgil]

In article <0b802685-a701-44be...@googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> On Monday, September 9, 2013 6:13:06 PM UTC-7, Virgil wrote:
> > In article <41ace296-31bd-473f...@googlegroups.com>,
> >
> > Zeit Geist <tucso...@me.com> wrote:
> >
> >
> >
> > > > Order can affect cardinality, because order can affect representation.
> >
> >
> >
> > Wrong again! Each member of a set remains one member regardless of
> >
> > "representation" or any order relation on the set.
> >
> > --
>
> Did you miss the memo?

If your "memo" claimed that one could change the cardinality of a set
without either adding to is membership or subtracting from its
membership, then everyone should miss it.
--

[Ross A. Finlayson]

Hardy has that real numbers are as the points of geometry, from which we
can build methods of exhaustion for the integral calculus, so the real
numbers are as much Euclid's as Eudoxus'.

Then, the equivalency function quite well does exist, standardly modeled
by real functions a la the step of Heaviside or impulse of Dirac (each
widely used throughout real analysis).

Hardy's text "Pure Mathematics" is of course well-known, and Euclid's
influence on geometry goes without saying. Of course you know that:
what's relevant to convey is that over the course of its study, the
continuum has as well been the course-of-values from point a to point b,
while at the same time having the properties of self-similiarity at all
scales, and of completeness or gaplessness at all scales.

Then, what is of regular interest for those who study mathematics and
wonder at its most fundamental truisms: what is the difference between
discrete and continous, and what is the point: on the line, or
alternatively, in the line.

What is the point?

What all is there, for there to be a point? What is the continuum, but
of points, and vice versa?

The goal is to least axiomatics and supporting theorems to maintain the
useful results of real analysis, and as well mathematics of the
discrete, and: mathematics of the continuous vis-a-vis the discrete,
and what surprising results may follow that provenance an avenue for
progress.

Regards,

Ross Finlayson

[Ross A. Finlayson]

No, it says the reals have complete representations as different sets.

But, the different representations maintain trichotomy of their elements.

This is of course R^bar^dots: R^crown.

Heh and Virgil the known time you cobbled field axioms together for something novel I presented one from whole cloth, ad hoc. And I'm still generous!

Novel, useful directions for mathematics: get some.

Regards,

Ross Finlayson

[Ross A. Finlayson]

Do you agree that the integral calculus sees only countable additivity?

Because the differential isn't any more than point width.

And the reason I state those facts as so is having found them.

Regards,

Ross Finlayson

[Virgil]

In article <70833dd3-8d8c-459a...@googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> Do you agree that the integral calculus sees only countable additivity?
> Because the differential isn't any more than point width.

There are a lot of things wrong with that,
including at least two assumptions.
--

[Virgil]

In article <b6aa31e9-46ff-4975...@googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> > If your "memo" claimed that one could change the cardinality of a set
> >
> > without either adding to is membership or subtracting from its
> >
> > membership, then everyone should miss it.
> >
> > --
>
> No, it says the reals have complete representations as different sets.

That there are different models which satisfy all the criteria required
by the standard real number field requirements does not mean that the
'sets of reals' in those different models do not biject in such a way as
to produce an isomorphism of ordered fields..
--

[Virgil]

In article <ed352fc5-18c8-4dfc...@googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> The goal is to least axiomatics and supporting theorems to maintain the
> useful results of real analysis,

With that as its start, there is no way to end that sentence
successfully.
--

[WM]

On Monday, 9 September 2013 20:57:06 UTC+2, Virgil wrote:
> In article <0c20527f-b20c-4f9f...@googlegroups.com>,
>
> WM <muec...@rz.fh-augsburg.de> wrote:
>
>
>
> > On Friday, 6 September 2013 23:28:31 UTC+2, Virgil wrote:
>
> >
>
> >
>
> > >
>
> > > The difference is between nodes and paths. One can identify the nodes of
>
> > >
>
> > > a complete infinite binary tree with proper binary fractions, but we
>
> > >
>
> > > cannot do that with the paths, because, among other reasons, there are
>
> > >
>
> > > far more paths than nodes.
>
> >
>
> > Then map every node on a path that it belongs to and remove that node and
>
> > that path from the Binary Tree. If there are more paths, then something
>
> > should remain, because if you remove one path from the tree, at least some
>
> > nodes of each other path remain there.
>
>
>
> Removing any one node from a COMPLETE INFINITE BINARY TREE cuts off
>
> uncountably many of its paths

Let the tails of the paths that are not removed continue to exist. Remove only *one* infinite path with each node - one path of your choice that contains the node to be removed.

Since you can imagine many things that cannot be written or expressed otherwise you should be able to imagine that.

Regards, WM

[fom]

It is not important Ross.

When Skolem wrote his critique of set theory, he effectively
put the nails in the coffin of set theory. Skolem had been
trained in algebraic methods, and, model theory is little
more than the algebraic methodology needed to support Hilbert's
failed research programs. In mathematics, "truth" is a
completely meaningless phrase.

The "losing" paradigm is the logical foundation in the
tradition of Bolzano, Weierstrass, Dedekind, Cantor,
Frege, and even Russell.

Read Skolem. When you find the paper in which primitive
recursive arithmetic is introduced, you will find that
Skolem, like WM, would have us all believe that Kronecker
received the natural numbers at the same burning bush
where Moses received the ten commandments.

If you wish some insight as to what role transfinite numbers
may play in applied mathematics, I suppose you should read
Feferman. To a large extent he directs his research toward
the minimal predicative mathematics required in support of
scientifically relevant mathematics.

Feferman distinguishes between "conceptual reduction" and
"proof-theoretic reduction". He contrasts these as oppositely
directed with respect to what each emphasizes in relation to
the transfinite hierarchy.

I know that the kind of predicativism that is espoused by
Feferman does involve the transfinite. I have seen it
discussed in papers. But, as with set theory, its involvement
generally has to do with the presuppositions of analysis
rather than the actual calculations of applied mathematics.

[Virgil]

> On Monday, 9 September 2013 20:57:06 UTC+2, Virgil wrote:
> > In article <0c20527f-b20c-4f9f...@googlegroups.com>,
> > WM <muec...@rz.fh-augsburg.de> wrote:
> > > On Friday, 6 September 2013 23:28:31 UTC+2, Virgil wrote:
> > >
> > > > The difference is between nodes and paths. One can identify the nodes
> > > > of
> > > > a complete infinite binary tree with proper binary fractions, but we
> > > > cannot do that with the paths, because, among other reasons, there are
> > > > far more paths than nodes.
> > > Then map every node on a path that it belongs to and remove that node and
> > > that path from the Binary Tree.

Removal of any one node eliminates all uncountably many paths through
that node, since through every node pass as many paths as in the entire
tree.

> > > If there are more paths, then something
> > > should remain, because if you remove one path from the tree, at least
> > > some nodes of each other path remain there.

Removal of any one node cuts out (as a path) every path through that
node.

Since every non-terminal node lies in more than one path,
removing any non-terminal node removes more than one path.

And Complete Infinite Binary Trees do not have any terminal nodes.

In any COMPLETE INFINITE BINARY TREE, at least outside of WM's wild
weird world of WMytheology, there are uncountably many paths through
each and every one of its nodes, and so removing any node, except the
root, leaves a tree with uncountably many paths through each of the
other nodes of the same generation.

If you remove the root node, by making them into less than compete paths
through the root, you remove ALL paths originally passing through the
root node, which removes the entire tree.

If you remove a child of the root node, you remove "half" of the paths
by disconnecting them from the root, but that still leaves all
uncountably many paths through the other child of the root node.

Cutting out a grandchild of the root cuts off "1/4" of the paths.
and cutting out a great-grandchild cuts out "1/16" of them.

But an positive rational fraction of an uncountable set is still
uncountable.

> > >
> > Removing any one node from a COMPLETE INFINITE BINARY TREE cuts off
> > uncountably many of its paths
>
> Let the tails of the paths that are not removed continue to exist. Remove
> only *one* infinite path with each node - one path of your choice that
> contains the node to be removed.

To "remove" Just *one* infinite path requires the removal of ALL
infinitely many of its nodes, as a path is nothing more that a suitable
infinite set of nodes, and that disrupts the pathhood of any path
sharing any of its nodes.

And that would require deleting as paths every path sharing any of those
nodes.

> Since you can imagine many things that cannot be written or expressed
> otherwise you should be able to imagine that.

Much of what WM imagines in that wild weird world of WMytheology of his
is unrealistic everywhere outside of his wild weird world of
WMytheology.

Like being able to remove just one path from a tree of more than one
path without messing up the treeness of what remains,


WM's maundering about infinite binary trees show how little he
understands of then, even about finite binary trees.


000
/
00
/ \
/ 001
0
\ 010
\ /
01
\
011

The above is a FINIE binary tree of 3 generations, 7 nodes and 4 paths.
The removal of a terminal node will eliminate a single path, but
the removal of any non-terminal node will eliminate MORE than one path.

And in any binary tree, removal of a a non-terminal node will,
at least for all such trees outside of WMytheology,
necessarily eliminate more than one path.

Perhaps when WM finally earns how binary trees REALLY work he can begin
to draw some true conclusions about them.

But don't bate your breath.
--

[Ross A. Finlayson]

I built an ordered field [-1,1].

https://groups.google.com/forum/?hl=en#!topicsearchin/sci.math/ordered$20AND$20field$20AND$20authorname$3AFinlayson$20AND$20after$3A2012$2F08$2F01

https://groups.google.com/d/msg/sci.math/4RBNLj-Q4Mo/hsgK3usvIAcJ

And then, well, no: that doesn't say that they don't.

Though, it well does involve successive copies of the natural integers
or an inductive set.

Some might have simply that increasing the number of variables increases
the order, here of the logic, plainly enough as to mappings from N in
higher order onto R.

And there aren't any elements in the higher order's N not in N.

Regards,

Ross Finlayson

[Virgil]

> In article <a1566763-bb44-4ef1...@googlegroups.com>,

> "Ross A. Finlayson" <ross.fi...@gmail.com> wrote:
>

> I built an ordered field [-1,1].

There are no such ordered fields.

As far as I know every ordered-field has a subfield isomorphic to the
field of rational numbers.

At least since according to any standard definition of an ordered field,
the order relation must be such that the product and the sum of two
positive members is a positive member.
--

[WM]

On Wednesday, 11 September 2013 01:35:07 UTC+2, Virgil wrote:
> > On Monday, 9 September 2013 20:57:06 UTC+2, Virgil wrote:
>
> > > In article <0c20527f-b20c-4f9f...@googlegroups.com>,
>
> > > WM <muec...@rz.fh-augsburg.de> wrote:
>
> > > > On Friday, 6 September 2013 23:28:31 UTC+2, Virgil wrote:
>
> > > >
>
> > > > > The difference is between nodes and paths. One can identify the nodes
>
> > > > > of
>
> > > > > a complete infinite binary tree with proper binary fractions, but we
>
> > > > > cannot do that with the paths, because, among other reasons, there are
>
> > > > > far more paths than nodes.
>
> > > > Then map every node on a path that it belongs to and remove that node and
>
> > > > that path from the Binary Tree.
>
>
>
>
>
> Removal of any one node eliminates all uncountably many paths through
>
> that node,

No.
Select a node x that has not yet been removed.
Remove all nodes of *one* path that contains x, as far as ist nodes have not yet been removed.

Too difficult to unerstand for you?

> since through every node pass as many paths as in the entire
>
> tree.

Let's see what remains if all nodes and countably many paths have been removed.
>
>
Regards, WM

[Virgil]

In article <bb8d41a7-1cd0-4988...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> > Removal of any one node eliminates all uncountably many paths through
> >
> > that node,
>
> No.
> Select a node x that has not yet been removed.
> Remove all nodes of *one* path that contains x, as far as ist nodes have not
> yet been removed.

How does one remove a node yet keep any one of the uncountably many
paths through that node?

Removing the root node would result in two disjoint trees

Removing any non-root node and any prior node would then have to be
directly connected to 1 child node and two grandchild nodes so the tree
would no longer be binary.

Or else all paths originally through that deleted node would no longer
be paths.

>
> Too difficult to unerstand for you?

It is difficult for me to unDerstand how a node in such a tree can be
deleted without affecting every set of nodes containing that node, thus

every path through that node.
>
>

> > since through every node pass as many paths as in the entire
> >
> > tree.
>
> Let's see what remains if all nodes and countably many paths have been
> removed.

Removing all nodes removes all sets of nodes which removes all paths,

at least outside of WM's wild weird world of WMytheology

--

[WM]

On Wednesday, 11 September 2013 22:00:01 UTC+2, Virgil wrote:
> In article <bb8d41a7-1cd0-4988...@googlegroups.com>,
>
> WM <muec...@rz.fh-augsburg.de> wrote:
>
>
>
> > > Removal of any one node eliminates all uncountably many paths through
>
> > >
>
> > > that node,
>
> >
>
> > No.
>
> > Select a node x that has not yet been removed.
>
> > Remove all nodes of *one* path that contains x, as far as ist nodes have not
>
> > yet been removed.
>
>
>
> How does one remove a node yet keep any one of the uncountably many
>
> paths through that node?

Very easy. Here is an enumeration of the nodes:

0
1, 2
3, 4, 5, 6
7, ...

First remove the root node 0 and one path, 0137... say. Then all nodes of this path are gone, but there remain nodes belonging to all other paths. From each of these other paths infinitely many nodes remain. Next remove node 2 with some arbitrarily chosen path and so on. After all nodes have been removed, aleph_0 tails of paths have been removed too. If there were more paths definable by nodes, then there must be further nodes. But all have gone. This proves that "the other" paths are not definable by nodes but only by finite definitions (like, in fact, every infinite path, since my "0137..." above is also a finite definition.)

This should be easily comprehensible. Further every countable alphabet and the cartesian product of all countable alphabets yields only contably many finite definitions. This disproves the existence of uncountable sets in mathematics.

Well, matheology is another field. But I prefer mathematics.

Regards, WM

[Virgil]

In article <cc212bac-ac29-4dc7...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Wednesday, 11 September 2013 22:00:01 UTC+2, Virgil wrote:
> > In article <bb8d41a7-1cd0-4988...@googlegroups.com>,
> >
> > WM <muec...@rz.fh-augsburg.de> wrote:
> >
> >
> >
> > > > Removal of any one node eliminates all uncountably many paths through
> >
> > > >
> >
> > > > that node,
> >
> > >
> >
> > > No.
> >
> > > Select a node x that has not yet been removed.
> >
> > > Remove all nodes of *one* path that contains x, as far as ist nodes have
> > > not
> >
> > > yet been removed.
> >
> >
> >
> > How does one remove a node yet keep any one of the uncountably many
> >
> > paths through that node?
>
> Very easy. Here is an enumeration of the nodes:
>
> 0
> 1, 2
> 3, 4, 5, 6
> 7, ...
>
> First remove the root node 0 and one path, 0137... say.

Since every path must pass through 0 , removing 0 cuts off every path.
While there remain two subtrees, with roots 2 and 3 respectively, none
of the paths in either of them are paths in the original tree.

OUtside of WM's wild weird world of WMytheology. removing a node from a
tree eliminates every path containing that node, at least as a path in
the original tree, and what remains will be two subtrees, though if any
node but the root node is removed, one of the subtrees will not be a
COMPLETE Infinite Binary Tree.

>
> This should be easily comprehensible.

But is only valid as reuslting in a SINGLE tree inside WM's wild weird
world of WMytheology.
Outside of WM's wild weird world of WMytheology, it results in two
trees only one of which need be a binary tree since removing any but the
root node creates a node with only one child node.

>
> Well, matheology is another field. But I prefer mathematics.

If WM is thinks that what goes on in his wild weird world of WMytheology
is really mathematics, he is deceived.

>
> >
> > Removing the root node would result in two disjoint trees
> >
> >
> >
> > Removing any non-root node and any prior node would then have to be
> >
> > directly connected to 1 child node and two grandchild nodes so the tree
> >
> > would no longer be binary.
> >
> >
> >
> > Or else all paths originally through that deleted node would no longer
> >
> > be paths.
> >

> > It is difficult for me to unDerstand how a node in such a tree can be
> > deleted without affecting every set of nodes containing that node, thus
> > every path through that node.
> >
> > > > since through every node pass as many paths as in the entire
> > > > tree.
> >
> > > Let's see what remains if all nodes and countably many paths have been
> > > removed.

Removing any one node from a Complete Infinite Binary Tree removes all
the uncountably many paths through that node.
--

[WM]

On Thursday, 12 September 2013 23:41:54 UTC+2, Virgil wrote:
> In article <cc212bac-ac29-4dc7...@googlegroups.com>,

> > > How does one remove a node yet keep any one of the uncountably many
>
> > >
>
> > > paths through that node?
>
> >
>
> > Very easy. Here is an enumeration of the nodes:
>
> >
>
> > 0
>
> > 1, 2
>
> > 3, 4, 5, 6
>
> > 7, ...
>
> >
>
> > First remove the root node 0 and one path, 0137... say.
>
>
>
> Since every path must pass through 0 , removing 0 cuts off every path.

But it leaves the other nodes in their positions. A path is defined as long as every node beyond the level n is present.

>

> removing a node from a
>
> tree eliminates every path containing that node,

No. A path is defined as long as every node beyond a fixed level n is present.

Too dangerous to further think about?

Regards, WM

[Virgil]

In article <d088ceaa-d00a-4e44...@googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On Thursday, 12 September 2013 23:41:54 UTC+2, Virgil wrote:
> > In article <cc212bac-ac29-4dc7...@googlegroups.com>,
>
>
> > > > How does one remove a node yet keep any one of the uncountably many
> >
> > > >
> >
> > > > paths through that node?
> >
> > >
> >
> > > Very easy. Here is an enumeration of the nodes:
> >
> > >
> >
> > > 0
> >
> > > 1, 2
> >
> > > 3, 4, 5, 6
> >
> > > 7, ...
> >
> > >
> >
> > > First remove the root node 0 and one path, 0137... say.
> >
> >
> >
> > Since every path must pass through 0 , removing 0 cuts off every path.
>
> But it leaves the other nodes in their positions. A path is defined as long
> as every node beyond the level n is present.

Every path in a Complete Infinite Binary Tree contains the root node of
that Complete Infinite Binary Tree.

Every path in a Complete Infinite Binary Tree contains the parent node
of every non- root node in that path in that Complete Infinite Binary
Tree.

What goes on in WM's NOT=Complete=Infinite=Binary=Trees is irrelevant
to the properties of Complete Infinite Binary Trees

So removing any one node from a path destroys the pathness of that path.

In any Complete Infinite Binary Tree, a path is defined to be any subset
of the tree's set of nodes which:
1. Contains the root node, and
2. contains exactly one child of each of its nodes

>
> > removing a node from a
> >
> > tree eliminates every path containing that node,
>
> No. A path is defined as long as every node beyond a fixed level n is
> present.

Removing any node from any path leaves some connected sets of nodes
either without the root node or without a child for one of its chains of
node.

Neither of those types of node sets is a path any more.

>
> Too dangerous to further think about?

Only dangerous for WM tho think about, as no matter how hard he thinks
about usch trees he just does not understand them.
--

[WM]

On Saturday, 14 September 2013 01:54:59 UTC+2, Virgil wrote:
> In article <d088ceaa-d00a-4e44...@googlegroups.com>,
>
> WM <muec...@rz.fh-augsburg.de> wrote:
>
>
>
> > On Thursday, 12 September 2013 23:41:54 UTC+2, Virgil wrote:
>
> > > In article <cc212bac-ac29-4dc7...@googlegroups.com>,
>
> >
>
> >
>
> > > > > How does one remove a node yet keep any one of the uncountably many
>
> > >
>
> > > > >
>
> > >
>
> > > > > paths through that node?
>
> > >
>
> > > >
>
> > >
>
> > > > Very easy. Here is an enumeration of the nodes:
>
> > >
>
> > > >
>
> > >
>
> > > > 0
>
> > >
>
> > > > 1, 2
>
> > >
>
> > > > 3, 4, 5, 6
>
> > >
>
> > > > 7, ...
>
> > >
>
> > > >
>
> > >
>
> > > > First remove the root node 0 and one path, 0137... say.
>
> > >
>
> > >
>
> > >
>
> > > Since every path must pass through 0 , removing 0 cuts off every path.
>
> >
>
> > But it leaves the other nodes in their positions. A path is defined as long
>
> > as every node beyond the level n is present.
>
>
>
> Every path in a Complete Infinite Binary Tree contains the root node of
>
> that Complete Infinite Binary Tree.

And every path is uniquely defined by its tail, even when finitely many nodes are missing.

>
>
>
> Every path in a Complete Infinite Binary Tree contains the parent node
>
> of every non- root node in that path in that Complete Infinite Binary
>
> Tree.
>
>
>
> What goes on in WM's NOT=Complete=Infinite=Binary=Trees is irrelevant
>
> to the properties of Complete Infinite Binary Trees

But it is highly relevant for mathematicians and psychologists who wish to investigate the idea that matheologians believe in paths that are not defined by nodes.
>
Regards, WM

[Virgil]

The definition of paths is based on nodes:

A path in a Complete Infinite Binary Tree is a minimal set of nodes that
(1) Contains the root node of the tree, and
(2) contains exactly one child of each node in the tree.

In fact, each path in any Complete Infinite n-ary Tree, for n in |N, is
a maximal Complete Infinite Unary Tree and is order-isomorphic to the
set of naturals.

Thus when one removes ANY node from a CIBT, one destroys the treeness of
that tree and the pathness of any path in that tree containing that node.

At least everywhere outside WM's wild weird world of WMytheology.
--

[WM]

On Saturday, 14 September 2013 23:01:52 UTC+2, Virgil wrote:


>
> Thus when one removes ANY node from a CIBT, one destroys the treeness of
>
> that tree and the pathness of any path in that tree containing that node.

In clear words: If one removes countably many paths from the Binary Tree then one has "destroyed the 'pathness'" of uncountably many paths and the delusions of finitely many matheologians.

Regards, WM

[FredJeffries]

The only interesting thing in this correspondence (which goes
totally unnoticed by Herr Professor) is that when you "remove"
node 0 you have two (isomorphic) complete binary trees, one with
root 1 and the other with root 2.

If you then "remove" node 1 you then have three complete binary
trees, with roots 2, 3, and 4

It's hilarious that "removal" of his "arbitrarily chosen" path
results in infinitely many complete binary trees like a Hydra,
except that here cutting out the root results in two taking its
place...

[Ross A. Finlayson]

On Saturday, September 14, 2013 3:22:01 PM UTC-7, FredJeffries wrote:
>
>
> It's hilarious that "removal" of his "arbitrarily chosen" path
>
> results in infinitely many complete binary trees like a Hydra,
>
> except that here cutting out the root results in two taking its
>
> place...

It is the sweep or breadth-first traversal that removes or marks only
one path at a time, simply enough for any finite tree and in the limit.
(Besides that, rays through the paths are each through between a pair of
countably ordinal points, 2^w many.)

WM's world and Virgil's discursions and aspersions therein are _not
directly relevant_ to the matter at hand: what is the nature of the
continuum that it is of the discrete, and of the discrete, the
continuous?

Jeffries, this is back to the notions of uniform probability
distributions over the the unit of reals, and over the naturals: with
the notions that there are various ways to construct these types of
numbers: from the naturals.

Here for example with lim d->oo lim n-> d n/d as mapping the naturals to
R[0,1], notice twice as n and d, or the semi-infinite for n, then
constructing the set {n/d, n < d, d -> oo} that is the rationals Q[0,1].

Natural/Unit Equivalency Function <-> Natural Rational Unit Set-Builder

Then, as to the discrete and continuous, and rationals as discrete but
uniformly dense in the reals (and as are the algebraics, which is then
to an extension of the numbers of copies and rangings through N), here
it is clearly seen that what are alternative foundations (and
foundations for the alternative) may well be more direct and explicit in
their construction of the eventual self-same structures as sets for
higher algebra.

Regard,

Ross Finlayson

[Virgil]

In article <827d3d8a-a605-490a...@googlegroups.com>,

In even clearer words, if one removes even one node from a Complete
Infinite Binary Tree, one destroys the property of being a complete path
in that original tree from each of the of the uncountably many sets of
nodes which were paths in that original tree.

But since in WM's wild weird world of WMytheology, no such trees are
allowed to occur, it is not surprising that WM cannot figure out how
they actually work.
--

[FredJeffries]

On Saturday, September 14, 2013 4:19:21 PM UTC-7, Ross A. Finlayson wrote:
>
> It is the sweep or breadth-first traversal that removes or marks only
> one path at a time, simply enough for any finite tree and in the limit.
> (Besides that, rays through the paths are each through between a pair of
> countably ordinal points, 2^w many.)
>
> WM's world and Virgil's discursions and aspersions therein are _not
> directly relevant_ to the matter at hand: what is the nature of the
> continuum that it is of the discrete, and of the discrete, the
> continuous?
>
> Jeffries, this is back to the notions of uniform probability
> distributions over the the unit of reals, and over the naturals: with
> the notions that there are various ways to construct these types of
> numbers: from the naturals.
>
> Here for example with lim d->oo lim n-> d n/d as mapping the naturals to
> R[0,1], notice twice as n and d, or the semi-infinite for n, then
> constructing the set {n/d, n < d, d -> oo} that is the rationals Q[0,1].
>
> Natural/Unit Equivalency Function <-> Natural Rational Unit Set-Builder

> Then, as to the discrete and continuous, and rationals as discrete but
> uniformly dense in the reals (and as are the algebraics, which is then
> to an extension of the numbers of copies and rangings through N), here
> it is clearly seen that what are alternative foundations (and
> foundations for the alternative) may well be more direct and explicit in
> their construction of the eventual self-same structures as sets for
> higher algebra.

Tell me: Does your software company use your faux-post-modern gibberish
in its advertising campaigns?

Have you yet figured out how to use this alleged uniform probability
distribution to solve an actual problem? Like finding the area of
a triangle?

Have you ever actually used any of the nonsense techniques you post
in your job? Would you still have a job if your employers read this
kind of junk? Would you ever get a job if you included it as part
of your submitted portfolio?

You, like the Herr Professor, have got no skin in the game. You are
a fraud.

[Ross A. Finlayson]

Oh, a novel development of foundations from simple first principles
generally goes over well in face-to-face communications.

You ask again to derive or show the area of a triangle, a sketch of a
proof was provided from these, as well a note on the historical
development in as to why Euclid reserved the development of the area of
the triangle in his general treatise, that of course isn't totally
linear.

As far as getting paid goes I generally rely on engineering skills as
there isn't much of a market for novel developments of mathematical
foundations. At least one program I wrote is deployed on millions of
machines and used for high-volume applications throughout the industry
of documents. No, Jeffries, I do mathematics for free.

Then I'll respectfully disagree on being anything less than sincere.

So, tell me: what is the continuous and discrete? Or, if you'd rather,
point me to a respected mathematical authority on the continuous
vis-a-vis the discrete including applications in, casually, the nature
of things between continuous and discrete.

While you're at it, please bring forward any application of modern
mathematics to continuum analysis, built as it is in the countable from
measure theory.

Regards,

Ross Finlayson

[FredJeffries]

On Monday, September 9, 2013 7:58:26 PM UTC-7, Ross A. Finlayson wrote:

> Set theory as transfinite cardinals has yet to find a single
> application in physics.

There is more to science than (theoretical) physics. There are
more applications of mathematics than science.

There is more to useful mathematics than direct application.

If you don't understand set theory, including transfinite
ordinals, you'll write rotten SQL.

If you don't understand (at least subconsciously) the principles
of transfinite collections (the collection has properties beyond
those obtained by combining the properties of the individuals)
you ain't gonna' understand emergent phenomena or accomplish
anything involving teamwork.

[fom]

On 9/14/2013 9:51 PM, FredJeffries wrote:
> On Monday, September 9, 2013 7:58:26 PM UTC-7, Ross A. Finlayson wrote:
>
>> Set theory as transfinite cardinals has yet to find a single
>> application in physics.
>
> There is more to science than (theoretical) physics. There are
> more applications of mathematics than science.
>
> There is more to useful mathematics than direct application.
>
> If you don't understand set theory, including transfinite
> ordinals, you'll write rotten SQL.
>

Well, that is a curious statement.

How so?

[Virgil]

In article <4_adncm_CYBJq6jP...@giganews.com>,

I, too, would be most interested in Ross' theories on how familiarity
with, or ignorance of, for that matter, the theory of transfinite
ordinals, would have any influence whatsoever on one's ability to write
good SQL.
--

[WM]

On Sunday, 15 September 2013 00:22:01 UTC+2, FredJeffries wrote:


>
> The only interesting thing in this correspondence (which goes
>
> totally unnoticed by Herr Professor) is that when you "remove"
>
> node 0 you have two (isomorphic) complete binary trees, one with
>
> root 1 and the other with root 2.

If you really are as dense as Virgil, then replace "to remove" by "to colour".


Regards, WM

[Ross A. Finlayson]

Who writes SQL anymore? Mostly those queries are ad-hoc, typical
CRUD/FLWR, the data access is much moved to the object layer /
annotation driven. I'd recommend you read "Data Logic", a book from the
70s on data. (C/C++/Java/J2EE/Spring/XML/Eclipse/Perl/zsh/vim, into
things.) Don't get me wrong, Matlab loads data. Why you no use APL? Z
is for formalists.

http://strategoxt.org/

Those are tools. I'm familiar with the work of many software engineers.

If you don't understand that the trans-finite ordinal as defined as
well-founded isn't necessarily consistent (in quantifying over finite
ordinals, eg as Russell or Burali-Forti), that the structure arrived at
via quantification instead of definition (axiomatization, fiat) would be
non-well-founded, then, why is it so easily understood as to apply to
all the well-founded ordinals, that their collection, isn't a
well-founded collection?

Then, where the transfinite has yet to find application in continuum
analysis, and, standard real analysis is quite well put together with
the additivity of countably-many elements, then what do I proffer as
application of this alternative view of a structure of real numbers
besides that it first naturally fits and supports real analysis?
Measuring the triangle indeed, how does it measure the path integral?

https://en.wikipedia.org/wiki/Path_integral_formulation

"The integration variables in the path integral are subtly
non-commuting. The value of the product of two field operators at what
looks like the same point depends on how the two points are ordered in
space and time. This makes some naive identities fail."

Regards,

Ross Finlayson

[FredJeffries]

I am not saying that one actually comes across omega^2 + 5*omega
when writing a query. (After all, a database table is an
un-ordered collection)

It is a case of skills carrying over from one area to another.

The importance of understanding data as a whole, not as individual
records, can be found in almost any SQL reference book. For instance
http://sqlmag.com/t-sql/t-sql-foundations-thinking-sets

The similarity or metaphor with ordinals comes from the notion
that you have some completed (potentially(?) infinite) collection
and then you use that completed collection as an intermediate
step in the solution of your problem -- like omega in
omega^2 + 5*omega

One place this is apparent in SQL is in the use of Common Table
Expressions (CTE)
http://en.wikipedia.org/wiki/Hierarchical_and_recursive_queries_in_SQL#Common_table_expression

So you start out with a CTE which is a complete SELECT statement, but
(like in lazy evaluation) you don't have to actually execute
it until you need it. It's a potentially infinite set. And then
your main SELECT statement uses the CTE, like in omega*5 the omega
is our CTE and the 5 our main SELECT and we get the result by
substituting an omega in at every point in the 5.

When you have multiple CTE's that recursively call each other
and themselves you get something with a resemblance to a
(countable) transfinite ordinal.

No, you don't HAVE to know transfinite set theory to become
an SQL programmer. But if you understand transfinite ordinals
you will have a better chance of "getting it" with CTE's and
correlated subqueries. And your first thought when given a
difficult problem will NOT be to write a cursor loop and do
an evaluation for every row in the table and then throw most
of the results away, in the process writing your own execution
plan when your company has spent thousands of dollars buying
a fancy database management system whose main job is to figure
out execution plans.

[Marshall]

On Sunday, September 15, 2013 3:46:39 AM UTC-7, Ross A. Finlayson wrote:
>
> Who writes SQL anymore?

Me. Thousands of other developers. Anyone who wants their database
application to perform acceptably.

> Mostly those queries are ad-hoc, typical
> CRUD/FLWR, the data access is much moved to the object layer /
> annotation driven.

Yeah, I know the stuff you mean. It's why a lot of database applications
don't perform very well.


Marshall

[FredJeffries]

On Sunday, September 15, 2013 3:46:39 AM UTC-7, Ross A. Finlayson wrote:
>
> Who writes SQL anymore?

In the real world, quite a few people.

But go back to your ivory tower and write more of your gibberish.

[FredJeffries]

Hey -- it's job security for me having to rewrite hotshot code so
that it actually works for our department's needs.

[Marshall]

It appears you and I have had some similar experiences in the job world.
Based on your posts, I'd expect we'd get along well professionally.


Marshall

[Marshall]

On Sunday, September 15, 2013 7:50:34 AM UTC-7, FredJeffries wrote:
>
> And your first thought when given a
> difficult problem will NOT be to write a cursor loop and do
> an evaluation for every row in the table and then throw most
> of the results away, in the process writing your own execution
> plan when your company has spent thousands of dollars buying
> a fancy database management system whose main job is to figure
> out execution plans.

Yeah, I've had to rewrite a lot of code that was done this way.
Usually it turns into a single SQL statement that performs
hundreds or thousands of times faster than what it replaced.

It can be tough trying to explain the importance of this to
programmers used to OOP. But this is sci.logic, and not one
of the many comp.database.theory vs. comp.object flamewars
of ten years ago.

Ah, it takes me back.


Marshall

[fom]

chuckle

I still need to read Fred's response. Since it has been
some time, I expect to learn something. I will read his
links with great interest. I did expect something
like what he wrote. SQL is characterized as a set-based
query language and is contrasted with other programming
languages on that basis.

My career in information technology ended with
the dot-com bust.

I still harbor a certain resentment that a database
administrator with 10 years of experience, "current
experience" with Oracle 8.0, and installation experience
with Oracle 8i could not get an interview one month after
the release of "internet ready" Oracle 8i (=8.1). (And,
lest I forget, 99 percentile certifications in Sun Solaris
and Oracle).

It had been a bad time for everyone in information
technology (except, perhaps, Java programmers and SAP
developers). In addition to the general economic decline
in the sector (caused by interest rate increases immediately
after successful Y2K upgrades) systems administrators and
database administrators watched businesses attempt to
expand that labor market into a global competition
because the internet permitted remote management and
those jobs had been among the highest salaried.

I hope none of you ever have to experience such things.

As I had not participated in newsgroups while in
my career, I cannot attest to the nature of those
flame wars. But, I recall the many senseless arguments
about what is "best" among options or the "right way"
to do things with little consideration for the needs
of a particular implementation or the limitations
imposed by software, hardware, budgets, and company
policies.

I do not really miss that aspect of being part of
information technology.

By the way... cursor loops?

My biggest problem had been trying to explain
the need for accommodating multiple disk accesses
to minimize bottlenecks from the physical limitations
associated with data transfer -- or, trying to explain
that unaccessed records and rarely accessed records
should be taken into account because their inclusion
with current records affects response times.

I still remember the dumb stares I received. It was
like looking at deer in car headlights. This had been
especially the case when discussing disk subsystems.
The fault tolerance of early RAID systems had nothing
to do with the needs of database optimizations.

To you guys, this is all old technology.

Maybe it will make you remember when life was simpler.

[Ross A. Finlayson]

On Sunday, September 15, 2013 7:50:34 AM UTC-7, FredJeffries wrote:

Then, wouldn't polynomials suffice, as the "tables" (images of
relations) are finite? Most see an RDBMS as table-oriented, vis-a-vis
relations, relational algebra, Codd, and the normal forms.

So you know, sqlldr? AL32UTF16 or WEISO8859P1 means something to you?
I'll admit using SQL casually and knowing my way around CTL files and a
passing knowledge of PL/SQL, as well then as to explain plans. As well,
I've written object/relational mapping (O/R M) setups with escapes to
SQL (for specialization of the table models, not "performance", per se).
Consistency, availability, and performance are regular concerns
throughout the distributed stack.

O/R-M (done well) is convenient for a) reducing errors, b) reducing the
learning curve with typical lowest-common-denominator as simple in
development for implementing data access in "the logic", c) reducing
points-of-change, d) re-using optimized connection pooling, cacheing,
queries, patterns, and error-handling and resource availability.

These are tools. And if you can find a copy of "Data Logic" it's very
good. Network I/O, CPU, RAM, Disk, Database I/O; develop, clock, and
calendar time: resources.

That said, the data in the RDBMS in its organization is generally of the
sample values and not as to summary for the range arithmetic. The
relevance here of SQL to continuum analysis is negligeable.

So, that's irrelevant to that transfinite cardinals don't give us, as
far as is known, foundations directly applicable to the path integral.
Modern mathematics (set theory with transfinite cardinals) is verbosely
mute on a connection between the sets as objects and the objects of the
infinitesimal analysis, as applied.

The path integral is physics _looking for mathematics_ to make it first
tractable, then rigorous toward consistency. And, an application of
modern mathematics for physics would certainly be of note, because:
there aren't any known, and, modern mathematics withs its reals as
uncountable disqualifies itself from putting the reals in their natural
order for their analysis. Then, an alternative foundation for the real
numbers that first supports standard real analysis then offers features
of these objects beyond the standard is of intrinsic interest to the
conscientious mathematician (or physicist).

Then, this is to be built from "the logic".

Regards,

Ross Finlayson

[Virgil]

In article <1ba2c77f-b06f-43a2...@googlegroups.com>,

Since no one else here is anywhere nearly as dense as WM,
everyone else else here thinks that merely coloring nodes has no effect
on any parent-child linkages, and, thus, has absolutely no effect on
any tree structure defined only by such links.

WM's wild weird world of WMytheology fails again!
--

[Ross A. Finlayson]

You speak for yourself.

No, only the sweep marks/colors the paths of the infinite tree, as for
all finite, and not as undone from the anti-diagonalization, in the
theory of only the tree and the infinite tree.

Then I'd note that you seem to have at once demanded that a set was
built of its elements, for R in any set suitable for representation of
its elements, but then that you demurred about that N^G has no elements
not in N but is equivalent to R.

In the digital, the discrete, it's said that modern mathematics shows
that Halts() can't be built (though it doesn't say that Halts_i() can't
be built and that Sum_i Halts_i() =/= Halts()). What does modern
mathematics give to real analysis? What is the representation of R in
real analysis, for results, besides that its structure is of the
re-composition of infinitesimal differential patches ordered-ly?

Then, the theory of well-founded infinite totalities in itself is all
well and fine (as it contains itself), you can see pink elephants all
day for as far as I care as pure mathematics: mathematics _owes_
physics why its numbers _are_.

Measure theory sits between modern mathematics and real analysis to
maintain that 1 = 1. If it took Russell and Whitehead hundreds of pages
to show 1 + 1 = 2, modern mathematics has shelves to declaim that
transfinite cardinals don't say 1 = 1.

So, add 'em up.

There's a lot ongoing in large cardinals, if but about large cardinals
and ordered by their discovery. There's a purpose for large cardinals
in application, but way above the foundations.

There's a purpose for mathematics: it includes the applied.

Regards,

Ross Finlayson

[FredJeffries]

On Sunday, September 15, 2013 11:00:56 AM UTC-7, Ross A. Finlayson wrote:
> On Sunday, September 15, 2013 7:50:34 AM UTC-7, FredJeffries wrote:

> > So you start out with a CTE which is a complete SELECT statement, but
> > (like in lazy evaluation) you don't have to actually execute
> > it until you need it. It's a potentially infinite set. And then
> > your main SELECT statement uses the CTE, like in omega*5 the omega
> > is our CTE and the 5 our main SELECT and we get the result by
> > substituting an omega in at every point in the 5.
>
> > When you have multiple CTE's that recursively call each other
> > and themselves you get something with a resemblance to a
> > (countable) transfinite ordinal.

>

> Then, wouldn't polynomials suffice, as the "tables" (images of
> relations) are finite?

You are free to create your own metaphors. But the only places
that multiplication is seen as replacing each item in a
collection with a (copy of) nother collection is in cardinal
and ordinal arithmetic. So you could possibly use polynomials
over the natural numbers and function composition in your
metaphor. Problem is that multiplication of natural numbers
and polynomials is commutative and database querying is in general
not commutative. So you use polynomials over some non-commutative
quasi-ring. But then the substitution analogy breaks down...

As I said, I work for a living and thus have not the time, even
if I had the inclination (which I don't) to try to figure out
your questions. You want a straight answer, put a little effort
into asking an understandable question.

Your ORM tools, which seem to create 847 lines of SQL for my
10 line query, are worthless if the programmer doesn't understand
the data to start with. GIGO.

I know nothing of continuum mechanics.

I repeat my challenge (which I also make to the good Herr Professor
regarding his complete binary tree): Show us how to solve a problem
with it.

[Ross A. Finlayson]

On Sunday, September 15, 2013 12:40:40 PM UTC-7, FredJeffries wrote:
>
...

>
> I repeat my challenge (which I also make to the good Herr Professor
>
> regarding his complete binary tree): Show us how to solve a problem
>
> with it.

The notion is to go from a spiral space-filling curve as natural
continuum, to the establishment of a geometry of points and space,
translated to Euclidean geometry: to use the proper tools as those with
the clearest semantic relevance, closure, directness, and
expressitivity: and shared understanding.

spiral-space filling curve -> geometry of points and spaces > geometry
of points and lines
spiral-space filling curve -> R as R_bar, R_dots > standard real
analysis

Standard problems are solved as they usually are: with the relevant
tools of the domain. For the formalist: the standard problem may
include all the way through foundations.

Then, as to solving a novel problem, with applications, that doesn't
already have a meaningful solution: this approach is of the
poly-dimensional perspective (that points define and are defined by an
infinity of dimensions at once) that individuation of the points from a
given perspective in that process yields new facts about these objects,
of the continuum of real numbers and R^N, and the universe of
theoretical objects.

Starting with simple fundamental properties of the natural integers (as
whole) and induction through them, as n/d, it is that discretizing the
points in the line, to one the line, has that Int_n EF(n) d(x from n) =
1, instead of 1/2, as the line under f(x)=x from 0 through 1.

Then, it's a ways to go from that as to why particular configuration of
experiment and the measurement effect in physics sees twice what's
expected, from the parallel transport of these points in the
polydimensional, where physics shrugs: "why, this is so."

Regards,

Ross Finlayson

[Virgil]

In article <a30e0721-a675-481f...@googlegroups.com>,

How do you work with database programs like Panorama?
--

[Virgil]

In article <afb823b3-2e51-417d...@googlegroups.com>,

The usual GIGO.
--

[Ross A. Finlayson]

Macsbug, windbag?

[Ross A. Finlayson]

"Garbage In, Garbage Out" - beginner's software credo
Typically, that means Virgil's doggeral generator puked up something
that would contradict something he already did and disagree with him
later when he dog-fooded himself (to regurgitate). He snips that in
disgust and pecks "GIGO".

The issue with transfinite cardinals for real analysis is
whatever in ->
transfinite cardinals ->
nothing out (because it would contradict t.c.s or r.a. or both).
measure theory ->
real analysis

And no, neither is it garbage nor usual: unlike picking the same nit
off Wolfgang, and nobody really cares about his hygiene (nor much
whether he tries to dumb-down budding engineers to his level as they
laugh in their sleeves), I put forward that:
a) there's a way to give geometry and real analysis a common
foundation
b) the usual standard results hold much more simply and in
shorter proofs
c) the would-be foundation is as well simpler than modern
mathematics
d) results in mathematical physics may be so tractable

That simply is, as it is. And the physicist looks at the path integral
and shrugs "why, is that so."

People have been looking for applications of foundations for more than a
hundred years. It's high time they found some.

Well then gentlemen (and ladies, as it were), good day,

Regards,

Ross Finlayson

[fom]

These links refer to hierarchical data models.

Are there specialized schemas that implement this?
Or is this merely a production of the query syntax
when column data has hierarchical relationships?

[Virgil]

In article <24a3a228-546d-4154...@googlegroups.com>,

You recognize your own style then?
--

[Virgil]

In article <2fff8fb4-c9c6-4dc2...@googlegroups.com>,

Neither of which work with Panorama databases, though the latter is very
much Ross' style.
--

[Ross A. Finlayson]

No, that's all well-nigh obsolete.

Oh, and they work.