# Re: [Jchat] J, APL, or calendars?

```Raul wrote:

> Hmm... here was my approach. I think I got different results:
.
.
.
> My way of summarizing this would be:
>
> "Lucy is a vampire",
> "Minna is human",
> "They are either both sane or both inane".```
```
responding to a post where Roger wrote:

> Jose, I agree with you exactly.  Here is my calculation:

(I also agree with him) responding to a post  where I wrote:

> [...] they have to be either both sane or both
> insane.  [...]
.
.
.
>    find o #: o i. 2^4
> 0 0 1 0
> 0 1 1 1
>
>    NB. That is, Lucy (either sane or insane) is a vampire and
>    NB. Minna (either sane or insane) is a human

So, what makes you think you got different results from Roger and me?  (I
cannot see any difference.)

On Tue, Jun 12, 2018 at 7:09 AM, Raul Miller <rauldmil...@gmail.com> wrote:

> Hmm... here was my approach. I think I got different results:
>
> lucy=: {."1
> minna=: {:"1
> species=: 2&|
> diagnosis=: 1&<
> human=: 0=species
> vampire=: 1=species
> sane=: 0=diagnosis
> insane=: 1=diagnosis
> invalid=: ~:/"1@(2 2&#:)
> both=:*/"1
>
> showSpecies=: ;@{&(;:'human vampire')@species
> showDiagnosis=: ;@{&(;:'sane insane')@diagnosis
> show=: showSpecies, ' and ', showDiagnosis
> possible=: 3 :0"1
>  echo 'Lucy is ',(show lucy y),' and Minna is ',show minna y
> )
>
> NB. all possibilities
> S=: 4 4 #:i.16
>
> NB. He knew that one was a vampire and one was a human
> S=: S=:S#~~:/"1 species S
>
> NB. Lucy: "We are both insane"
> S=:S#~(invalid lucy S)~:both insane S
>
> NB. Minna: "We are not both insane"
> possible S=:S#~(invalid minna S)~:-.both insane S
>
> Running this script gives me two plausible seeming results:
>
> Lucy is vampire and sane and Minna is human and sane
> Lucy is vampire and insane and Minna is human and insane
>
> In the ordinal fraction representation, the second possibility was
> expressed as 2221 and rejected at the final step of the process:
>
> "Now we know that Minna is an insane human, and so they are not both
> insane,"
>
> But Mina's statement was that they are not both insane and insane
> humans always make false statements according to the rules here.
>
> I am not sure what to make of that part of the ordinal fraction approach.
>
> My way of summarizing this would be:
>
> "Lucy is a vampire",
> "Minna is human",
> "They are either both sane or both inane".
>
> Thanks,
>
> --
> Raul
>
> On Tue, Jun 12, 2018 at 4:30 AM roger stokes <rogerstokes...@gmail.com>
> wrote:
> >
> > Jose, I agree with you exactly.  Here is my calculation:
> >
> > NB. There are four independent propositions: Lucy human, Lucy sane, Minna
> > human, Minna sane.
> >
> > NB. Therefore there are 16 possibilities, each representable as a 4-bit
> > string.
> >
> > P =: #: i. 16 NB. set of all possibilities: 0000, 0001, 0010 etc
> >
> > NB. A proposition is modelled as a function to select a single bit.
> >
> > Lh =: 0&{ NB. Lucy is human
> >
> > Ls =: 1&{ NB. Lucy is sane
> >
> > Mh =: 2&{ NB. Minna is human
> >
> > Ms =: 3&{ NB. Minna is sane
> >
> > truthteller =: = NB. x= human, y = sane : both or neither
> >
> > Ax =: Lh ~: Mh NB. Axiom: if Lucy human, then Minna not, and vice versa
> >
> > Lucy =: (Lh truthteller Ls) = (Ls +: Ms) NB. "we are both insane"
> >
> > Minna =: (Mh truthteller Ms) = (Ls +. Ms) NB. "not both insane"
> >
> > (Minna " 1 # ]) (Lucy "1 # ]) (Ax "1 # ]) P
> >
> > 0 0 1 0
> >
> > 0 1 1 1
> >
> >
> > On Tue, Jun 12, 2018 at 12:12 AM, Jose Mario Quintana <
> > jose.mario.quint...@gmail.com> wrote:
> >
> > > Friday night I decided to try to fall asleep by thinking about the
> puzzle
> > > and I thought I had solved it. This evening I wrote a wicked script to
> > > verify my thought process and I got the same result but it does not
> seem to
> > > match your conclusion.
> > >
> > > This is what I found...
> > >
> > >
> > >
> > >
> > >
> > > *** POTENTIAL SPOILER FOLLOWS ***
> > >
> > >
> > >
> > >
> > >
> > > Given the conditions of the puzzle and since the sisters' statements
> > > contradict each other then they have to be either both sane or both
> > > insane.  That leaves four possibilities:
> > >
> > > 0.  Lucy is a sane human and Mina is a sane vampire.
> > >     This one is inconsistent with Lucy's statement "We are both
> insane."
> > >
> > > 1.  Lucy is an insane human and Mina is an insane vampire.
> > >     This one is inconsistent with Mina's statement "of course not!"
> > >
> > > 2.  Lucy is a sane vampire and Mina is a sane human.
> > >     This one is consistent with both Lucy's statement and Minna's
> > > statement.
> > >
> > > 3.  Lucy is an insane vampire and Mina is an insane human.
> > >     This one is also consistent with both Lucy's statement and Minna's
> > > statement.
> > >
> > > However, either sane or insane, Lucy is a vampire (and Minna is a
> human).
> > >
> > > This is a verification in the form of a session corresponding to a
> wicked
> > > brute-force quick-and-dirty script (yet, the derived verb is not
> wicked)
> > > using the J Wicked Toolkit [0] (beware of line-wrapping)...
> > >
> > >    JVERSION
> > > Engine: j806/j64nonavx/windows
> > > Release: commercial/2017-11-06T10:01:33
> > > Library: 8.06.09
> > > Qt IDE: 1.6.2/5.6.3
> > > Platform: Win 64
> > > Installer: J806 install
> > > InstallPath: j:/program files/j
> > > Contact: www.jsoftware.com
> > >
> > >    NB. (Extra parentheses for clarity)
> > >
> > >    NB. Running the Wicked Tacit Toolkit first...
> > >    NB. (0!:0)<'/.../J Wicked Toolkit.ijs
> > >
> > >    'LH LS MH MS'=. 4 Fetch
> > >     NB. True and false are encoded as bits in the usual way (as 1 and 0
> > > respectively)
> > >     NB. The bits are in the following order
> > >     NB. Lucy       Minna
> > >     NB. human sane human sane
> > >
> > >    'or and not'=. [:+.*.-.]sb
> > >
> > >    NB. He knew that one was a vampire and one was a human...
> > >    ovoh=. (LH and not o MH) or (not o LH and MH)
> > >
> > >    NB. Lucy: We are both insane...
> > >
> > >    lbit=. (((LH and       LS) or (not o LH and not o LS)) and not o LS
> and
> > > not o MS)
> > >      NB. Lucy is not lying
> > >    lbil=. (((LH and not o LS) or (not o LH and       LS)) and       LS
> or
> > >       MS)
> > >      NB. Lucy is     lying
> > >    lbi=. lbit or lbil
> > >
> > >    NB. Minna: Of course not!
> > >    mont=. (((MH and       MS) or (not o MH and not o MS)) and       LS
> or
> > >       MS)
> > >      NB. Minna is not lying
> > >    moni=. (((MH and not o MS) or (not o MH and       MS)) and not o LS
> and
> > > not o MS)
> > >      NB. Minna is     lying
> > >    mon=. mont or moni
> > >
> > >    find=. ((ovoh and lbi and mon)"1 # ])f. NB. Finding the consistent
> > > possibilities
> > >
> > >    find o #: o i. 2^4
> > > 0 0 1 0
> > > 0 1 1 1
> > >
> > >    NB. That is, Lucy (either sane or insane) is a vampire and
> > >    NB. Minna (either sane or insane) is a human
> > >
> > >    NB. The wrapped linear representation of find is...
> > >
> > >       66 (-@:[ ]\ 5!:5@<@:]) 'find'
> > > (((0&({::) *. -.@:(2&({::))) +. -.@:(0&({::)) *. 2&({::)) *. ((((0
> > > &({::) *. 1&({::)) +. -.@:(0&({::)) *. -.@:(1&({::))) *. -.@:(1&({
> > > ::)) *. -.@:(3&({::))) +. ((0&({::) *. -.@:(1&({::))) +. -.@:(0&({
> > > ::)) *. 1&({::)) *. 1&({::) +. 3&({::)) *. (((2&({::) *. 3&({::))
> > > +. -.@:(2&({::)) *. -.@:(3&({::))) *. 1&({::) +. 3&({::)) +. ((2&(
> > > {::) *. -.@:(3&({::))) +. -.@:(2&({::)) *. 3&({::)) *. -.@:(1&({::
> > > )) *. -.@:(3&({::)))"1 # ]
> > >
> > >
> > > PS.  There is no need to run a Toolkit for this in Jx;
> > >      just replace 'LH LS MH MS'=. 4 Fetch by,
> > >        'LH LS MH MS'=. 4 ((i.@:[ <@:(((_3?:0) (_1?:0))&)"0 _ ])(_2?:0)
> > > {::)
> > >      and 'or and not'=. [:+.*.-.]sb by,
> > >        'or and not'=. [:+.*.-.]:
> > >      and define o=. @:
> > >
> > >
> > > [0] J Wicked Toolkit
> > >     http://www.2bestsystems.com/foundation/j/Jx.zip
> > >     \Jx\J\J Wicked Toolkit.ijs
> > >
> > >
> > >
> > > On Sat, Jun 9, 2018 at 9:23 AM, 'Bo Jacoby' via Chat <
> c...@jsoftware.com>
> > > wrote:
> > >
> > > > The Transylvanian problem, solved using ordinal fractions.
> > > > sane humans and insane vampires make only true statements;
> > > >
> > > > insane humans and sane vampires make only false statements.
> > > > This is the coding,
> > > > 0001 Minna is human
> > > >
> > > > 0002 Minna is vampire
> > > > 0010 Minna is sane
> > > > 0020 Minna is insane
> > > > 0100 Lucy is human
> > > > 0200 Lucy is vampire
> > > > 1000 Lucy is sane
> > > > 2000 Lucy is insane
> > > > These are the possibilities: 10#.>:#:i.2 8
> > > >
> > > > 1111 1112 1121 1122 1211 1212 1221 1222
> > > > 2111 2112 2121 2122 2211 2212 2221 2222
> > > > He knew that one was a vampire and one was a human, Discard 0101 and
> 0202
> > > > leaving 8 possibilities: 1112 1122 1211 1221 2112 2122 2211 2221
> > > > Lucy: "We are both insane" = 2020.
> > > > "sane humans make only true statements"
> > > > If Lucy is sane the statement is false. Discard 1100, leaving 6
> > > > possibilities: 1211 1221 2112 2122 2211 2221
> > > > "insane humans make only false statements". Discard 2120, leaving 5
> > > > possibilities: 1211 1221 2112 2211 2221
> > > > "insane vampires make only true statements". Discard 2210, leaving 4
> > > > possibilities 1211 1221 2112 2221
> > > > "sane vampires make only false statements". Then "We are both
> insane" is
> > > > false in any case.
> > > > Minna: "We are not both insane" If Minna is a sane human then Lucy is
> > > > insane: Discard 1011 leaving 3 possibilities 1221 2112 2221
> > > > If Minna is a sane vampire then contradiction. Discard 0012 leaving 2
> > > > possibilities: 1221 2221
> > > > Now we know that Minna is an insane human, and so they are not both
> > > > insane,
> > > > and so Lucy is sane. 1221.
> > > > Summary:
> > > >
> > > > 1221
> > > >
> > > > 0001 Minna is human
> > > > 0020 Minna is insane
> > > > 0200 Lucy is vampire
> > > > 1000 Lucy is sane
> > > >
> > > >
> > > >     Den 17:05 fredag den 8. juni 2018 skrev Don Guinn <
> > > dongu...@gmail.com
> > > > >:
> > > >
> > > >
> > > >  When in elementary school there was a chart showing the numbers.
> But the
> > > > zero was to the right of the nine. That confused me then. No wonder
> kids
> > > > have difficulty grasping the concept of zero.
> > > >
> > > > On Fri, Jun 8, 2018, 8:34 AM Björn Helgason <gos...@gmail.com>
> wrote:
> > > >
> > > > > beenary numbers
> > > > >
> > > > > https://m.phys.org/news/2018-06-scientists-bees-concept.html
> > > > >
> > > > > On Fri, 8 Jun 2018 07:40 'Bo Jacoby' via Chat, <c...@jsoftware.com
> >
> > > > wrote:
> > > > >
> > > > > > Cardinal numbers (0, 1, 2, . . .) are  including 0 (zero).
> > > > > > Ordinal numbers (1, 2, 3, . . .) are  starting with 1 (first).
> There
> > > is
> > > > > no
> > > > > > "zeroth".
> > > > > > There is arithmetic of cardinal numbers (including the J verbs +
> * ^
> > > !
> > > > )
> > > > > ,
> > > > > > but there is no arithmetic of ordinal numbers.
> > > > > > The codes of the UDC are important numerical objects, but they
> are
> > > > > neither
> > > > > > integers, nor decimal fractions, nor rational numbers, nor real
> > > > numbers,
> > > > > > nor complex numbers, nor quaternions, nor vectors, nor matrices,
> nor
> > > > > > functions, nor operators. They have been neglected by
> mathematicians.
> > > > > > A new kind of numbers must be considered. I dubbed them  'ordinal
> > > > > > fractions' .
> > > > > > A cardinal number, such as 'one', counts a set.
> > > > > > An ordinal number, such as 'the first', identifies an element in
> a
> > > set.
> > > > > > A cardinal fraction, such as 'one half', measures a part of a
> > > totality.
> > > > > > An ordinal fraction, such as 'the first half', identifies a part
> of a
> > > > > > totality.
> > > > > > Consider for simplicity the binary, rather than the decimal,
> > > notation.
> > > > > > one = 1 = 0001. The digit positions, right to left, indicate
> ones,
> > > > twos,
> > > > > > fours, and eights, and the digit values are one-digit binary
> cardinal
> > > > > > numbers, 0 and 1.
> > > > > > the first = 1 = 0001. This is the cardinal number corresponding
> to
> > > the
> > > > > > ordinal number in question.
> > > > > > one half = 0.1 = 0.1000. The digit positions after the binary
> point
> > > > > > indicate halfs, fourths, eights, and sixteenths, and the digit
> values
> > > > are
> > > > > > one-digit binary cardinal numbers, 0 and 1.
> > > > > > the first half = ?????
> > > > > > My solution to this problem is
> > > > > > the first half = 1 = 1000
> > > > > > the second half = 2 = 2000
> > > > > >
> > > > > > the first fourth = 11 = 1100
> > > > > >
> > > > > > the second fourth = 12 = 1200
> > > > > >
> > > > > > the third fourth = 21 = 2100
> > > > > >
> > > > > > the fourth fourth = 22 = 2200
> > > > > >
> > > > > > the odd fourths = 01 = 0100
> > > > > > the even fourths = 02 = 0200
> > > > > > the sixteenth sixteenth = 2222
> > > > > > Note that the digit positions indicate halfs, fourths, eights,
> and
> > > > > > sixteenths, and the digit values are either 1 meaning first, and
> 2
> > > > > meaning
> > > > > > second, or 0 meaning both. 1000 means: first half, both fourths,
> both
> > > > > > eights, both sixteenths. 'both' goes without saying, just as 0
> goes
> > > > > without
> > > > > > saying. 1000 = 1 = first half. That is one reason for choosing 0
> for
> > > > > > 'both'.
> > > > > > I did not know the words hyponymy and hypernymy. Thanks! That is
> just
> > > > > what
> > > > > > I need.
> > > > > > In logic I let 1 and 2 represent True and False. 0 means unknown
> or
> > > > > > unimportant.
> > > > > >
> > > > > > The Transylvanian problem:
> > > > > > 0001 Minna is human
> > > > > >
> > > > > > 0002 Minna is vampire
> > > > > > 0010 Minna is sane
> > > > > > 0020 Minna is insane
> > > > > > 0100 Lucy is human
> > > > > > 0200 Lucy is vampire
> > > > > > 1000 Lucy is sane
> > > > > > 2000 Lucy is insane
> > > > > > Check the 2^4 ordinal fractions from 1111 to 2222 against the
> data.
> > > (I
> > > > > > have not done it)
> > > > > > Thanks! Bo. .
> > > > > >
> > > > > >
> > > > > >
> > > > > >
> > > > > >
> > > > > >
> > > > > >
> > > > > >
> > > > > >
> > > > > >
> > > > > >
> > > > > >
> > > > > >
> > > > > >
> > > > > >
> > > > > >
> > > > > >    Den 5:58 fredag den 8. juni 2018 skrev Donna Y <
> > > dy...@sympatico.ca
> > > > >:
> > > > > >
> > > > > >
> > > > > >  Can we agree on definitions
> > > > > >
> > > > > > Ordinal numbers and Cardinal numbers are Natural numbers which
> do no
> > > > > > include the number 0. The Natural numbers are well ordered.
> > > > > > Whole numbers are the Natural numbers and 0. 0 is the least
> element
> > > of
> > > > > the
> > > > > > Whole numbers.
> > > > > > Integers are Whole numbers and Negative signed Natural numbers.
> (-n +
> > > > n=
> > > > > > 0  Zero is not a positive or a negative integer—0 has no sign.)
> The
> > > set
> > > > > of
> > > > > > integers has no least element.
> > > > > > Real numbers are continuous and complete and can be rational or
> > > > > > irrational—rational numbers are integers and fractions,
> irrational
> > > > > numbers
> > > > > > cannot be expressed as a ratio of two integers.
> > > > > > Imaginary numbers are not real—they exist in another
> dimension—root
> > > > > > negative 1 or i, complex numbers have real and imaginary
> components.
> > > > > >
> > > > > > Tables are two dimensional arrays.
> > > > > >
> > > > > > The basic structure of UDC is hierarchy but it could be viewed in
> > > other
> > > > > > ways—as you said yourself.
> > > > > >
> > > > > > Are you using tables where Rows are records and Columns are
> > > attributes?
> > > > > Is
> > > > > > there a primary key?
> > > > > >
> > > > > > I am not sure where your Ordinal Fraction comes in but a computer
> > > > > > application of UDC would need full integration of information
> > > retrieval
> > > > > > (IR) features into a database management system (DBMS).
> > > > > >
> > > > > > Right truncation specifies hypernomy (is a—as in number theory
> is a
> > > > > subset
> > > > > > of mathematics)—I copied this table but from my view forms of
> higher
> > > > > degree
> > > > > > is not a subset of diophantine equations.
> > > > > >
> > > > > > Table 1:
> > > > > >
> > > > > > 5: mathematics and natural sciences
> > > > > > 51: mathematics
> > > > > > 511: number theory
> > > > > > 511.5: diophantine equations
> > > > > > 511.57: forms of higher degree
> > > > > >
> > > > > > What is the precise reason you chose to use 0 as a wild card?—why
> > > not *
> > > > > or
> > > > > > # or & or … ? What advantages are derived by using 0?  For
> example
> > > when
> > > > > you
> > > > > > use boolean logic the 0 and 1 can represent False and True and
> then
> > > the
> > > > > > result vector of 0 and 1 can be used for selection.
> > > > > >
> > > > > > hat does this even mean?
> > > > > >
> > > > > > > For example: 0=0 and 0>1 and 1<0 and 10<>01 and 1><2. (meaning
> that
> > > > the
> > > > > > whole is equal to the whole, and the whole comprises the first
> part,
> > > > and
> > > > > > the first part is part of the whole, and the first half is
> compatible
> > > > > with
> > > > > > the odd fourths, and the first part is disjoint with the second
> > > part).
> > > > > > > The notation for ordinal fractions makes ordinal fraction
> > > arithmetic
> > > > > > easy, just as the notation for cardinal numbers makes cardinal
> number
> > > > > > arithmetic easy.
> > > > > > > Keep asking!
> > > > > >
> > > > > >
> > > > > > Maybe you can show how Ordinal Fractions can be applied to the
> > > problem
> > > > > > below:
> > > > > >
> > > > > > PUZZLE BREAK
> > > > > >
> > > > > > Inspector Craig Visits Transylvania
> > > > > > Inspector Craig of Scotland Yard was called to Transylvania to
> solve
> > > > some
> > > > > > cases of vampirism. Arriving there, he found the country
> inhabited
> > > both
> > > > > by
> > > > > > vampires and humans. Vampires always lie and humans always tell
> the
> > > > > truth.
> > > > > > However, half the inhabitants, both human and vampire, are
> insane and
> > > > > > totally deluded in their beliefs: all true propositions they
> believe
> > > > > false,
> > > > > > and all false propositions they believe true.
> > > > > > The other half of the inhabitants are completely sane: all true
> > > > > statements
> > > > > > they know to be true, and all false statements they know to be
> false.
> > > > > Thus
> > > > > > sane humans and insane vampires make only true statements; insane
> > > > humans
> > > > > > and sane vampires make only false statements.
> > > > > > Inspector Craig met two sisters, Lucy and Minna. He knew that one
> > > was a
> > > > > > vampire and one was a human, but knew nothing about the sanity of
> > > > either.
> > > > > > Here is the investigation:
> > > > > > Craig (to Lucy): Tell me about yourselves.
> > > > > > Lucy: We are both insane.
> > > > > > Craig (to Minna): Is that true?
> > > > > > Minna: Of course not!
> > > > > > From this, Craig was able to prove which of the sisters was the
> > > > vampire.
> > > > > > Which one was it?
> > > > > > — From Logician Raymond Smullyan
> > > > > >
> > > > > >
> > > > > > Donna Y
> > > > > > dy...@sympatico.ca
> > > > > >
> > > > > >
> > > > > > > On Jun 7, 2018, at 5:55 PM, 'Bo Jacoby' via Chat <
> > > c...@jsoftware.com
> > > > >
> > > > > > wrote:
> > > > > > >
> > > > > > >
> > > > > > > "You do not show how to access particular rows or columns or
> > > elements
> > > > > > from a table".
> > > > > > > The table, and the table name, is addressed by 00.
> > > > > > > The left column, and the left column header, is addressed by
> 01.
> > > > > > > The right column, and the right column header, is addressed by
> 02.
> > > > > > > The upper row, and the upper row header, is addressed by 10.
> > > > > > > The upper left table entry, and its data content, is addressed
> by
> > > 11.
> > > > > > > The upper right table entry, and its data content, is
> > > > 12.
> > > > > > > The lower row, and the lower row header, is addressed by 20.
> > > > > > > The lower left table entry, and its data content, is addressed
> by
> > > 21.
> > > > > > > The lower right table entry, and its data content, is
> > > > 22.
> > > > > > >
> > > > > > > "without knowing the number of rows or columns."
> > > > > > > In the example there are two rows and two columns.
> > > > > > > If you need a third row, call it 30.
> > > > > > >
> > > > > > > "Why not the concept of arrays with an index?"
> > > > > > > Arrays may have different shapes. Any ordinal fraction has the
> > > shape
> > > > > > (_\$9).
> > > > > > > Arrays have elements. Ordinal fractions don't.
> > > > > > >
> > > > > > > Only array elements contains data. Any ordinal fraction may
> contain
> > > > > data.
> > > > > > >
> > > > > > > Arrays may have subarrays. Any ordinal fraction has subordinate
> > > > ordinal
> > > > > > fractions.
> > > > > > > Arrays and atoms have names. Ordinal fractions don't.
> > > > > > >
> > > > > > > "the set of Natural numbers begin with 1 yet we use base 10
> > > > > > representation that uses 0."
> > > > > > >
> > > > > > > Cardinal numbers and ordinal fractions have similarities and
> > > > > > differences.
> > > > > > > Cardinal number 0 (meaning "nothing") is not the same thing as
> > > > ordinal
> > > > > > fraction 0 (meaning "everything").
> > > > > > >
> > > > > > > Cardinal number 1 (meaning "one") is not the same thing as
> ordinal
> > > > > > fraction 1 (meaning "first part").
> > > > > > >
> > > > > > > Using "0" for wild card character does not get along with
> using "0"
> > > > for
> > > > > > counting to ten.
> > > > > > > A cardinal number is represented by a right-justified sequence
> of
> > > > > digits
> > > > > > with a finite number of nonzero digits. The cardinal number 1
> may be
> > > > > > written 00001.
> > > > > > >
> > > > > > > An ordinal fraction is represented by a left-justified
> sequence of
> > > > > > digits with a finite number of nonzero digits. The ordinal
> fraction 1
> > > > may
> > > > > > be written 10000.
> > > > > > >
> > > > > > > "However this was not an easy system in which to do arithmetic
> so I
> > > > > > cannot see how your base 9 system could be either."
> > > > > > > Cardinal numbers, A and B, are ordered such that either A=B or
> A<B
> > > or
> > > > > > A>B. For example: 0=0 and 0<1 and 1>0 (meaning "zero is equal to
> > > zero"
> > > > ,
> > > > > > and "zero is fewer than one", and "one is more than zero").
> > > > > > >
> > > > > > > Ordinal fractions, A and B, are ordered such that either A=B
> or A<B
> > > > or
> > > > > > A>B or A<>B or A><B. For example: 0=0 and 0>1 and 1<0 and 10<>01
> and
> > > > > 1><2.
> > > > > > (meaning that the whole is equal to the whole, and the whole
> > > comprises
> > > > > the
> > > > > > first part, and the first part is part of the whole, and the
> first
> > > half
> > > > > is
> > > > > > compatible with the odd fourths, and the first part is disjoint
> with
> > > > the
> > > > > > second part).
> > > > > > > The notation for ordinal fractions makes ordinal fraction
> > > arithmetic
> > > > > > easy, just as the notation for cardinal numbers makes cardinal
> number
> > > > > > arithmetic easy.
> > > > > > > Keep asking!
> > > > > > > Bo.
>
>
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