Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.


fom
Posts:
1,968
Registered:
12/4/12


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


On 2/17/2013 10:08 AM, Shmuel (Seymour J.) Metz wrote: > In <SvmdnSXO9rHvYLMnZ2dnUVZ_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 nonvoid subset of T with 0<k<v. Then D is called a
(v,k,l)difference set if it satisfies:
The list of nonzero differences, ((dd')=0), for d,d' in D contains each element of T precisely l times.
=========
T=Z_7, D={1,2,4}=2D=4D
11=012=614=4 21=122=024=5 41=342=244=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,
6543210  1234560 2345601 4560123
> 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
6321  1456 2560 4012
540  230 341 563
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 16element additive group from the vector space over the Galois field of order 2 and a 6element multiplicative group. The namespaces were fixed such that relative to the 5element 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 setasobject from that same set in the sense of >> setascollection. > > 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 orderisomorphically 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 settheoretic 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 orderisomorphically 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 "picketfence" 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 nonmetrizable 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.



