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Topic: Product, Filters and Quantales
Replies: 31   Last Post: Oct 21, 2013 7:52 AM

 Messages: [ Previous | Next ]
 fom Posts: 1,968 Registered: 12/4/12
Re: Prime Interger Topology
Posted: Oct 20, 2013 11:51 PM

On 10/20/2013 9:58 PM, William Elliot wrote:
> On Sun, 20 Oct 2013, fom wrote:
>> On 10/20/2013 9:17 PM, William Elliot wrote:
>>>
>>> The prime integer topology for N is the topology generated
>>> by the subbase Bs = { (n + pN) /\ N | n in Z, p prime }
>>>
>>> What if the prime integer topology were applied to Z?

>>
>> The axiom for difference relations is given by,
>>
>> AwAxAyAz( -(w,x,y,z) <-> ( ( +(w,x,z,y) /\ -(w,y,x,z) ) /\ ( ~( -(w,y,z,x) )
>> /\ ( ~( -(w,x,z,y) ) /\ ( ~( -(w,z,x,y) ) /\ ~( -(w,z,y,x) ) ) ) ) ) )
>>
>> The first conjunct in the expression,
>> +(w,x,z,y) /\ -(w,y,x,z)
>>
>> asserts that there must exist an additive relation
>> by which z is the sum of x and y. The second conjunct
>> asserts that in relation to such a sum, the addends of
>> the sum are commutative with respect to difference relations.

>
> Why relations? Aren't functiosn used to manage the undue complexity of
> relations?
>

Categoricity.

http://plato.stanford.edu/entries/definitions/#CirDef

What Gupta is describing concerning revision rules
corresponds with the notion of language-relative
identity discussed (and rejected) by Deutsch in

http://plato.stanford.edu/entries/identity-relative/#5

Revision rules are like a convergence with respect
to language-relative identity. Deutsch is correct
in that they are not the same. But, how to understand
the relationships is topological.

A series of revision rules "converge" in the same
sense as the relations containing the diagonal in
a entourage uniformity (Willard uses a different
term -- surroundings?) Uniformities do not require
metrics and are thus amenable to 'x = y' without
numerical valuations.

that little discussion about proximities. Then,
the other day I proved the statement at the top
of this post:

http://math.stackexchange.com/questions/526335/large-cardinals-and-partition-lattices

Consistent theories without function or constant
symbols and injective interpretation maps have
separated proximities (on the terms of their
language).

First-order logic in the strict sense portrayed
by Deutsch, in criticism of Geach, corresponds with
proximities. Convergent language-relative identity
may be thought of in terms of uniformities.

With all of this said, let's think about what
Deutsch is saying about identity in first-order
logic.

A "system of objects" is equally a "system of
relations"

Consider the syntax,

AxAy( x = y <-> ~( x < y \/ x > y ) )

It would seem as if the identity relation is
eliminable.

But, that is not compliant with first-order logic
with identity.

The four *necessary* relations are: the full
relation, the empty relation, the identity
relation and the diversity relation.

Now, consider this quote from Padoa *defining*
what it means for objects in logical relation,

"If x and y are individuals, then x=y or ~x=y.
For us, these are the only relations that we
can consider between individuals without
transgressing the boundaries that separate
general logic from particular deductive theories."

When you join these statements together,
a model should *not* be built from *sets*
as in structures from universal algebra.

They should be built from *relations*. To be
exact, four of them.

In order for identity to be "eliminable" in the sense
of the statement above, the diversity relation would
have to be augmented by *two* relations. For each
pair of elements, one relation would contain,

< x, y >

while the other would contain

< y, x >

Moreover, they would have to be *consistent* with the
notion of identity.

It is ridiculous to have to prove the consistency
of the logic intended to provide the measure by which
theories are deemed consistent or inconsistent.

Relations may be hard. But, it is important to
understand their role.

By the way, both of my theories -- the class theory
and, now, the arithmetical theory -- are grounded on
a strict transitive relation. That is, they are
grounded on one-half of the order relation above.

Both theories also use the same "membership" relation
and depend on a sentence expressing filter convergence.

>> This expresses,
>>
>> 8 - 3 = 5
>> 8 - 5 = 3
>>
>> in relation to
>>
>> 3 + 5 = 8
>>
>> The expression
>>
>> ( ~( -(w,y,z,x) ) /\ ( ~( -(w,x,z,y) ) /\ ( ~( -(w,z,x,y) ) /\ ~( -(w,z,y,x) )
>> ) ) )
>>
>> shows the relationship of the difference relations to
>> order on the domain. Since difference relations are understood
>> with respect to sums, the sum of two terms may not occur in
>> specific indices of the relation.
>>
>> That is, the fourth term in a difference relation is always greater
>> than the second and third terms.

>
> What a difficult way to define <=.
>

Indeed.

Date Subject Author
10/9/13 William Elliot
10/10/13 Victor Porton
10/11/13 William Elliot
10/11/13 Victor Porton
10/12/13 William Elliot
10/12/13 Victor Porton
10/12/13 William Elliot
10/14/13 Victor Porton
10/15/13 William Elliot
10/15/13 Victor Porton
10/16/13 William Elliot
10/16/13 Victor Porton
10/17/13 William Elliot
10/17/13 Victor Porton
10/17/13 William Elliot
10/18/13 Victor Porton
10/18/13 William Elliot
10/19/13 Victor Porton
10/19/13 William Elliot
10/19/13 William Elliot
10/20/13 fom
10/20/13 William Elliot
10/20/13 fom
10/20/13 William Elliot
10/20/13 William Elliot
10/20/13 fom
10/20/13 William Elliot
10/20/13 fom
10/21/13 fom
10/21/13 William Elliot
10/21/13 fom
10/20/13 William Elliot