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fom
Posts:
1,969
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 languagerelative identity discussed (and rejected) by Deutsch in
http://plato.stanford.edu/entries/identityrelative/#5
Revision rules are like a convergence with respect to languagerelative 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.
I had been "stuck" at uniformities until we had 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/largecardinalsandpartitionlattices
Consistent theories without function or constant symbols and injective interpretation maps have separated proximities (on the terms of their language).
Firstorder logic in the strict sense portrayed by Deutsch, in criticism of Geach, corresponds with proximities. Convergent languagerelative 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 firstorder 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 firstorder 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 onehalf 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.



