Re: Relative Boundedness

On Tue, 6 Jan 2009, John Jones wrote:

William Elliot wrote:
On Mon, 5 Jan 2009, John Jones wrote:

Each element of a particular series can be bounded or not bounded for that same series. Being bounded is an extrinsic property or relationship of a series and is independent of the elements of a series.

Define a series.


WHAT IS A SERIES?

I see you have no idea what a series is.

Let a:N -> S be a function from the positive integers N, into a set S.
a is a sequence and by definition a_n = a(n) is the value of a at n, ie the nth element of the sequence.

Now let S be an additive group such as (R,+), the reals with addition.
Then the sequence s:N - R defined by
s_n = sum(j=1,n) a_j
is a series and lim(n->oo) s_n is the sum of the series,
a1 + a2 +..+ a_n +..

Now that you know the difference between a series and a sequence we can proceed.

a) A series is built on an assembly of atomic (singletons) elements. The concept of an assembly can be defined as relatedness. An assembly is constructed from a single view, whether that view is physically/empirically direct, e.g. visual, or made empirically implicit in a Platonic realm. See also 2) above.

You are talking about the elements of S, the codomain of the sequence a.

b) A series 'has' the property of adjacency. Adjacency, and the differentiation of atomic elements, occurs together as the construction of 'many' views. Adjacency is not a subset of an assembly because the property of adjacency is not a property of an assembly.

This adjacency gibberish is the property of the positive integers N, which is the domain of every sequence. If you want more that just the positive integers, then you need to learn what a net is. The property of adjacency may not hold in the domain of the net.

c) Neither a series, nor the elements of which it is constructed, have properties, and the context in which I say that a series 'has' properties is a transcendentally ideal one. That is, because a series is constructed by viewings its ontology is predicated on that viewing, hence there is nothing we have indicated that can have internal properties of its own. This also shows that the distinction I make between 'internal' and 'external' properties is also a transcendentally ideal one, in that internal and external do not refer to properties associated with a construction that is independent of the method of its construction.

Thou transcendentally ascends unto white noise gibberish.

d) Series have no properties. Rather, the properties of a series are the methods of construction of a series. Another way of putting it is to say that a series has no ontology except as that ontology is a description of its method of construction. By phrasing it this way we also eliminate the spectre of subjectivism with which the term 'viewing' threatened our account.

A sequence is a map from the integers.

Is this constructivism? I do not think so. I am sure that the constructivists (correct me if I am wrong) require an absolute, stand-alone, transcendentally real ontology for their tools, an assumption which must, I would surmise, contaminate their applications. After all, we couldn't expect to use a transcendentally real tool to create a transcendentally ideal object, like a series.

No it's neither constructive nor intelligent nor humorous.

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