Re: Distinct linear orderings on Z

From: Jesse F. Hughes (jesse_at_phiwumbda.org)
Date: 03/24/05 

Date: Thu, 24 Mar 2005 19:59:28 +0100

Tony Orlow (aeo6) <aeo6@cornell.edu> writes:

> Jesse F. Hughes said:
>> Tony Orlow (aeo6) <aeo6@cornell.edu> writes:
>>
>> > Anyway, I need to refine what I am thinking with regard to simpler
>> > sets before I start trying to generalize.
>>
>> Great! Start refining here, if you don't mind. I posted this in
>> another thread (in reply to a different poster), but surely you can
>> answer these questions, too. Just replace "number of elements" with
>> "bigulosity".
>>
>>
>> If S is a set of natural numbers then let f(S) = { 2n | n in S }.
>> For instance:
>>
>> f({0,1}) = {0,2}
>> f({13, 200, 210}) = {26, 400, 420}
>>
>> Let's write #(S) for the number of elements in S, so that we avoid the
>> notation for cardinality.
>>
>> (1) Do you agree that for every finite set S, it is the case that
>>
>> #(f(S)) = #(S)?
> For any given set S of numbers such as this, buglulosity is defined as the
> product of the set domain and the set density. When you apply f to a finite
> set, you double the domain (range, if you prefer), and halve the density,
> therefore they are the same size.

This is really bizarre to me. I have to return to it.

Question: Is there or is there not an operator #:Set -> P for some
linear ordered class P such that #(S) < #(T) iff S is smaller than T?

Can we or can we not compare sizes of arbitrary sets or do you believe
that we can only compare sets of numbers to sets of numbers?

Let h(S) = { {n} | n in S }, so

h( { 0, 1 } ) = { {0}, {1} }.

What is your brilliant method of comparing the sizes of these two
sets?

Can you compare the sizes of the set of even numbers with the set of
all continuous functions f:R -> R?

Can you compare the sizes of the set of curves in R^2 with the set of
partial orders with finite domain?

Or is your "bigulosity" defined piecemeal and ad hoc so that this is
impossible? 

-- 
"A recruitment consultant I know thinks the most important quality in
a winner is to be lucky.  To avoid wasting his time with unlucky
applicants, he takes half the resumes piled on his desk and throws
them straight in the bin."  -- John Ramsden

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