From Newsgroup: sci.logic

On 05/18/2015 06:48 PM, George Greene wrote:

On Sunday, May 17, 2015 at 11:53:00 AM UTC-4, Charlie-Boo wrote:

There are only a finite number of pairs
that can be made from the elements of a finite set,

That has nothing to do with anything unless you are going to make it a length-lexicographic ordering, where every finite set is smaller than every infinite one and every n-element set is smaller than every n+1-element set.
There is just no reason to do that.

not enough to equal every even number.

That DOESN'T MATTER.

{1, 3, 5, 7, 9,...} comes before {1, 3, 6, 10, 15,...} because the former DOES contain 5 and so is lexicographically first.

It's lexicographically first BY YOUR DEFINITION, but I'm just disputing the left-right reading. You would make the biggest set (all of N) the smallest. That is counter-intuitive. If you think of every number ACTUALLY INCLUDED
as "a letter of the (infinite) word", THEN your definition really does LOOK lexicographic in analogy with the usual alphabetic definition of a lexicographic
order. BUT THERE IS ANOTHER WAY to look at it, a way that is more general in that it makes both finite and infinite sets look like THE SAME kind of thing,
namely, an INFINITELY-long (or wide) BIT-string.
You'd give it more of a place-value system where there is an 0 in the nth place if n is not in the set, and a 1 in the nth place if n is in the set. Then you look at the lexicographic ordering ON THOSE bit-strings.
AND THEN {1,3,6,...} would come SOONER and be LESSER because it starts with 101001 while {1,3,5,...} starts with
101010.
My point is that lexicographically ordering the bit-strings gives you
the preferred property that the empty set is least and the whole of N
is greatest. You haven't made it clear what yours does with finite sets but it is actually a flaw of the framework that you STILL NEED to.
One thing that is clear is that sets that leave out the vast majority of the numbers at the beginning are, despite leaving them out and having far fewer
members, still BIGGER (for you) -- Because their "first" element is "bigger".

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From Newsgroup: sci.logic

On 09/28/2026 11:27 AM, Ross Finlayson wrote:

On 05/18/2015 06:48 PM, George Greene wrote:

On Sunday, May 17, 2015 at 11:53:00 AM UTC-4, Charlie-Boo wrote:

There are only a finite number of pairs
that can be made from the elements of a finite set,

That has nothing to do with anything unless you are going to make it a
length-lexicographic ordering, where every finite set is smaller than
every infinite one and every n-element set is smaller than every
n+1-element set.
There is just no reason to do that.

not enough to equal every even number.

That DOESN'T MATTER.

{1, 3, 5, 7, 9,...} comes before {1, 3, 6, 10, 15,...} because the
former DOES contain 5 and so is lexicographically first.

It's lexicographically first BY YOUR DEFINITION, but I'm just
disputing the
left-right reading. You would make the biggest set (all of N) the
smallest.
That is counter-intuitive. If you think of every number ACTUALLY
INCLUDED
as "a letter of the (infinite) word", THEN your definition really does
LOOK
lexicographic in analogy with the usual alphabetic definition of a
lexicographic
order. BUT THERE IS ANOTHER WAY to look at it, a way that is more
general
in that it makes both finite and infinite sets look like THE SAME kind
of thing,
namely, an INFINITELY-long (or wide) BIT-string.
You'd give it more of a place-value system where there is an 0 in the nth
place if n is not in the set, and a 1 in the nth place if n is in the
set.
Then you look at the lexicographic ordering ON THOSE bit-strings.
AND THEN {1,3,6,...} would come SOONER and be LESSER because it starts
with
101001 while {1,3,5,...} starts with
101010.
My point is that lexicographically ordering the bit-strings gives you
the preferred property that the empty set is least and the whole of N
is greatest. You haven't made it clear what yours does with finite
sets but
it is actually a flaw of the framework that you STILL NEED to.
One thing that is clear is that sets that leave out the vast majority
of the numbers at the beginning are, despite leaving them out and
having far fewer
members, still BIGGER (for you) -- Because their "first" element is
"bigger".

"Borel versus Combinatorics" gives an account of contradiction in
set-theory, or, "independence" if you're generous.


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