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fom

Posts: 1,031
Registered: 12/4/12

Re: distinguishability - in context, according to definitions
Posted: Feb 19, 2013 7:06 PM

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On 2/17/2013 10:08 AM, Shmuel (Seymour J.) Metz wrote:
> In <SvmdnSXO9rHv-YLMnZ2dnUVZ_oydnZ2d@giganews.com>, on 02/16/2013
> at 04:50 AM, fom <fomJUNK@nyms.net> said:
>
>> No problem here. No difference set contains the letter with
>> which it is associated.
>
> What is a difference set

From "Design Theory" - Beth, Jungnickel, Lenz

Definition:

Let (T,+) be a group of order v, and D be a non-void
subset of T with 0<k<v. Then D is called a

(v,k,l)-difference set if it satisfies:

The list of non-zero differences, -((d-d')=0), for d,d'
in D contains each element of T precisely l times.

=========

T=Z_7, D={1,2,4}=2D=4D

1-1=0|1-2=6|1-4=4
2-1=1|2-2=0|2-4=5
4-1=3|4-2=2|4-4=0

As a set, the 0's are treated as having
multiplicity 1

It is easiest to see the relation to
symmetric designs when it is presented
as the finite projective plane,

6|5|4|3|2|1|0
-------------
1|2|3|4|5|6|0
2|3|4|5|6|0|1
4|5|6|0|1|2|3

> and how is it relevant to 'it is intuitively
> reasonable to think of a letter as "a collection of letters"'. What
> are the letters that are elements of the letter L. If your answer is
> L, then you have problems with regularity.
>

In the finite plane above, the line labels 0,4,5
each contain a point labels of the same numeral.
The diagram partitions on that basis

6|3|2|1
-------
1|4|5|6
2|5|6|0
4|0|1|2

5|4|0
------
2|3|0
3|4|1
5|6|3

This gives a finitary sense of what is involved
for a system where

-xex and xex

The difference set for the constructed design is
different (I should be distinguishing between
the difference set, the generated design, and
the blocks of the design.). It is built from
a direct product of a 16-element additive group
from the vector space over the Galois field of
order 2 and a 6-element multiplicative group.
The namespaces were fixed such that relative
to the 5-element subspaces of the vector space,
that element used to generate each block of the
design would not be incorporated into the block
it generated.

So, for all 96 blocks,

-((given letter)e(block))

I call them letters because it is only syntax.

Regularity came to mind when I described it
because that is what I had in mind as I
formulated the construction.

>> If one looks at the statement of regularity closely, the class of
>> sets that do not contain the empty set all act to separate at least
>> one set-as-object from that same set in the sense of
>> set-as-collection.
>
> Theories such as ZFC don't have such a distinction.
>

Perhaps.

Let the variables for the language be indexed (as with Tarski)

Choose v_1

Form all wffs possible with only v_1

Take an enumeration of those wffs.

Associate those wffs of the enumeration without
a leading unary negation order-isomorphically
with the positive integers

Associate those wffs of the enumeration having a
leading unary negation with the negative integers
so that integers with the same absolute value
correspond to a wff and its negation.

Form the listing

<1, 2, 3, ... , v_1, ..., -3, -2, -1>

Now, choose v_1 and v_2

Form all wffs possible with only v_1 and v_2

Take the set-theoretic difference excluding the
wffs generated in the first iteration

Take an enumeration of the remaining wffs.

Associate those wffs of the enumeration without
a leading unary negation order-isomorphically
with the positive integers

Associate those wffs of the enumeration with a negation
sign with the negative integers so that integers with
the same absolute value correspond to a wff and its
negation.

Form the listing

<1, 2, 3, ... , v_2, ..., -3, -2, -1>

Now, choose v_1, v_2 and v_3

and so on.

If a class is what is given by a formula,
and if I can put a topology on the set of formulas,
what distinctions are there?

I don't claim to have successively done this.
Everyone who has ever programmed knows how
thorny a "picket-fence" error can be.

But, I do not see how anything I know about
ZFC prevents me from interpreting its classes
this way if I am able to perform this construction
successfully.

In that case, regularity may be a reasonable
proxy for "use"/"mention" distinctions.

As I pointed out to William Elliot. The back
of "Counterexamples in Topology" by Steen and
Seebach discusses non-metrizable topologies.

I believe one is called a tangent disc
topology. An identification map on the
Lindenbaum algebra might be similar.
After all, that is about what I have
tried to do here. I don't know. I would
have to check the definitions.