Re: Distinct linear orderings on Z

From: aeo6 (aeo6_at_cornell.edu)
Date: 03/22/05 

Date: Tue, 22 Mar 2005 10:17:29 -0500

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....
>
>
> 

-- 
Smiles,
Tony

Relevant Pages

  • Re: Distinct linear orderings on Z
    ... I mean that cardinality means the ... It does indeed depend on bijections. ... Cardinality in universal terms means exactly the same as number of ... modern math approach to set analysis in terms of cardinality is done ...
    (sci.math)
  • Re: Distinct linear orderings on Z
    ... >> Cardinality in universal terms means exactly the same as number of ... >> modern math approach to set analysis in terms of cardinality is done ... >> cardinality in ordinary mathematics is universal in nature. ...
    (sci.math)
  • Re: Distinct linear orderings on Z
    ... > Cardinality in universal terms means exactly the same as number of ... > modern math approach to set analysis in terms of cardinality is done ... How do you compute the "tangent" ...
    (sci.math)
  • Re: Distinct linear orderings on Z
    ... I've defined cardinality in universal terms as identical ... >> as more akin to tangentiality than cardinality in universal terms. ... properties such as set, ordinal, mapping, and bijection. ... with but which mathematikers pretend are defined in terms universal ...
    (sci.math)
  • Re: Distinct linear orderings on Z
    ... On Tue, 22 Mar 2005 15:59:53 GMT, Lester Zick wrote: ... Cardinality has nothing to do with derivatives. ... Modern math is not ... Tangents have nothing to do with cardinalities, ...
    (sci.math)