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Discussion of "kancu"
[parent] [root]
Comment #4: Re: Issues
Curtis W Franks (Mon Jan 2 18:24:45 2017)

krtisfranks wrote:
> I think that there really are two words here.
>
> The first is "cardinality", which we already have. It is simple "x1 is
the
> number of elements in/cardinality of set x2". This uses on kancu2 and
> kancu3 (antirespectively). This requires no counter (person doing the
> counting) and is independent of the method used to count.
>
> The second word is something along the lines of "enumerate", "tabulate",
> "tally", or "count up". This is a task performed by someone (a counter)
and
> depends on the method. I personally think that it would be fine to
> straight-up delete kancu3 for this purpose, but perhaps we should just
> retool it (I am not sure how, though). Instead of trying to figure out
the
> cardinality of a set (kancu2), it would just be producing a running tally

> of objects/divisions/units (kancu2'). In this way, it is not actually
tied
> to the grand total, only what has thus far been counted.

Maybe this is a good option:

The cardinality of a set is equal to the cardinality of another set, by
definition, iff there exists a bijection between the sets. A set A is
called "countable" iff there exists a (possibly improper) subset of the set
of natural numbers B such that there exists a bijection between A and B. If
a set A is finite (therefore also countable), then its cardinality is the
nonnegative integer n such that there is a bijection between A and
ci'ai'u(n); A is countable and infinite iff its cardinality is
aleph-null.

The first word which I described would, for countable A, produce the n so
described (or, iff appropriate, aleph-null), regardless of which bijection
is used in order to relate A to the other set.

Note that, if there were a counter, then the wrong n may be produced by a
human method of counting (attempting but failing to produce a correct
bijection). This does not change the cardinality of the set, it just means
that the claim is false. For example, one can count by threes and come to
the conclusion that there were six Beatles, but that would not make them
correct. (In the current definition of kancu, they cannot possibly be
incorrect because their method of counting allows for them to come to
whichever conclusion they like.)

I offer a third possibility in replacing kancu now. This possibility does
not have a counter terbri (so it is like the word for cardinality in this
way) but it also does not have a terbri for the number at which the
counting would arrive (so it is like the second word which I described in
my first comment in this thread, which emphasizes the ongoing process of
tallying). This word merely describes /which/ bijection or injection, in
particular (of all of the possibilities), is being used/has been selected
in order to attempt to enumerate (some subset of) the set, where the
function maps from the set being counted to a subset of the natural numbers
united with the singleton of 0. In fact, in order to allow for joke
counting, repetition, or unforeseen uses, the function need not even be
injective nor mapping to N; it just needs to exist/be well-defined and map
to the set of extended real numbers (not even positive finite ones). Note
that, in partovular, ci'ai'u is not necessarily involved and that
ordering the range of the function is not necessary.

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