Re: Distinct linear orderings on Z

From: Jesse F. Hughes (jesse_at_phiwumbda.org)
Date: 03/20/05 

Date: Sun, 20 Mar 2005 20:38:26 +0100

"Allan C Cybulskie" <allan.c.cybulskie@yahoo.ca> writes:

> "Jesse F. Hughes" <jesse@phiwumbda.org> wrote in message
> news:87u0n6r7o2.fsf@phiwumbda.org...
>> stevendaryl3016@yahoo.com (Daryl McCullough) writes:
>>
>> > Tony says...
>> >
>> >>Yes, I understand how bijection and cardinality work. I have serious
> problems
>> >>with those results however.
>> >
>> > No, you have nonserious problems with those results. Your problem is
>> > the fact that people use the phrase "more elements" to mean "larger
>> > cardinality", when you want to use "more elements" to mean "larger
>> > order type". That is not a serious objection, it is a completely
>> > silly objection---nothing follows from a choice of words. If you
>> > want to reserve "more elements" to mean "larger order type", fine.
>> > Nobody cares what words you use.
>>
>> Nobody cares, with the caveat that in each post in which he writes
>> "more elements", he will have to explicitly state that he means
>> "larger order type". Otherwise, the same dull, semantic arguments
>> will be repeated forever.
>
> Um, and why don't the mathematicians have to explicitly state "cardinality"
> when they use "number of elements"? Your intellecutal arrogance is
> blinding.

In informal communication, the equivalence between "cardinality" and
"number of elements" may reasonably be taken for granted. Call it
"intellecutal arrogance" if you want, but there *are* good arguments
for cardinality as an appropriate generalization for counting and I've
seen no alternative definition that even comes close.

The fact is that mathematicians and the overwhelming majority of
philosophers of mathematics accept that "cardinality" is the right
generalization for "number of elements" (where the former is an
explicit, technical definition and the latter is really well-defined
only for finite sets). A careful presentation in a more formal
setting (like a journal or textbook) would probably abstain from using
terms involving "number of elements", but it is natural to conflate
the two ideas in less formal settings.

Put differently: why do you suppose that the mathematicians should
always assume that *you* mean something different (without knowing
*what* you mean) when you use the term "number of elements"? What
should they suppose you mean?

>
>>
>> > There is no content to your complaints.
>>
>> No mathematical content, that is. There may be philosophical content
>> in asking which relation ("has larger cardinality than" and "has
>> larger order type than") is the more plausible interpretation for "has
>> more elements than".
>>
>> (But here, cardinality seems to have the advantage, since it is a
>> fairly obvious generalization of counting elements.)
>
> But it leads to certain conflicting conclusions, and so is not as
> advantageous as it may seem.

There are no conflicting conclusions. There are only unintuitive
consequences.

So what *is* the definition of set size you prefer? I seem to keep
missing it. 

-- 
Jesse F. Hughes
"Would you please stop talking and start talking?"
   -- Vincent Price as the Saint

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