Re: Distinct linear orderings on Z
From: Lester Zick (lesterDELzick_at_worldnet.att.net)
Date: 03/22/05
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Date: Tue, 22 Mar 2005 16:01:30 GMT
On Tue, 22 Mar 2005 10:17:29 -0500, Tony Orlow (aeo6)
<aeo6@cornell.edu> in comp.ai.philosophy wrote:
>Dave Seaman said:
>> On Mon, 21 Mar 2005 20:37:59 GMT, Lester Zick wrote:
>> > On Mon, 21 Mar 2005 17:50:25 +0000 (UTC), Dave Seaman
>>
>> > You might be interested in a collateral reply to Tony on this subject.
>> > Cardinality in universal terms means exactly the same as number of
>> > elements.
>>
>> A definition that works only for finite sets is not "universal".
>>
>> > When cardinality is applied to undefined numbers of elements
>> > the modern mathematical meaning of bijective mapping simply refers to
>> > the slope or tangent of the set instead of the set itself, which is a
>> > restricted parochial definition and not universal. In other words, the
>> > modern math approach to set analysis in terms of cardinality is done
>> > in non universal terms whereas the conventional interpretation of
>> > cardinality in ordinary mathematics is universal in nature.
>>
>> Cardinality applies to arbitrary sets. How do you compute the "tangent"
>> of the set of all continuous mappings f: [0,1] -> R^n, to take an example
>> that recently came up in sci.math? Or how about the "tangent" of the set
>> of all subsets of R? I have quoted you the definition of cardinality,
>> and it says nothing at all about "tangents".
>I think what Lester is referring to is the idea that the rate of increase of
>the members in a set defined by a function is inversely related to the size of
>the infinity. For instance the evens are defined by the function f(x)=2*x on
>the integers, which has twice the slope of f(x)=x, and therefore repesents half
>as large an infinity. When the ratio is oo/oo or 0/0 for x=oo, one should be
>able to apply L'Hospital's rule to resolve the slope and calculate a tangent.
>>
>> > Basically what I'm saying is that cardinality and number of elements
>> > are identical concepts in universal terms of ordinary mathematics and
>> > only become different when cardinality is interpreted in terms of
>> > slopes or tangents as it is in bijective matching in set theory in
>> > modern math. And of course it is this bifurcation between universal
>> > and parochial applications of the single term, cardinality, that leads
>> > directly to the nominally counterintuitive results in modern math.
>>
>> As best I can understand it, you are using "universal" to mean "applies
>> only to finite sets", and "parochial" to mean "applies to all sets,
>> whether finite or infinite". Huh?
>I think "parochial" is Lester's poetic way of saying "local", perhaps referring
>to local limits determining slopes! Or maybe not....
It's Lester's poetic way of saying moder math sucks non universally,
Tony.
Regards - Lester
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