Changes

An initial set of operations, required to establish the subsequent constructions, all have in common the property that they do exactly the opposite of what is normally done in abstracting sets from situations.  These operations reconstitute, though still in a generic, schematic, or stereotypical manner, some of the details of concrete context and interpretive nuance that are commonly suppressed in forming sets.  Stretching points back along the direction of their initial pointing out, these extensions return to the mix a well-chosen selection of features, putting back in those dimensions from which ordinary sets are forced to abstract and in their ordination to treat as distractions.

An initial set of operations, required to establish the subsequent constructions, all have in common the property that they do exactly the opposite of what is normally done in abstracting sets from situations.  These operations reconstitute, though still in a generic, schematic, or stereotypical manner, some of the details of concrete context and interpretive nuance that are commonly suppressed in forming sets.  Stretching points back along the direction of their initial pointing out, these extensions return to the mix a well-chosen selection of features, putting back in those dimensions from which ordinary sets are forced to abstract and in their ordination to treat as distractions.

In setting up these constructions, one typically makes use of two kinds of index sets, in colloquial terms, ''clipboards'' and ''scrapbooks''.

In setting up these constructions, one typically makes use of two kinds of index sets, in colloquial terms, "clipboards" and "scrapbooks":

 

<ol style="list-style-type:decimal">

1. The smaller and shorter term index sets, typically having the form I = {1, ... , n}, are used to keep tabs on the terms of finite sets and sequences, unions and intersections, sums and products.

<p>The smaller and shorter-term index sets, typically having the form <math>I = \{ 1, \ldots, n \},\!</math> are used to keep tabs on the terms of finite sets and sequences, unions and intersections, sums and products.</p>

In this context and elsewhere, the notation [n] = {1, ... , n} will be used to refer to a "standard segment" (finite initial subset) of the natural numbers N = {1, 2, 3, ... }.

<p>In this context and elsewhere, the notation <math>[n] = \{ 1, \ldots, n \}\!</math> will be used to refer to a ''standard segment'' (finite initial subset) of the natural numbers <math>\mathbb{N} = \{ 1, 2, 3, \ldots \}.\!</math></p></li>

2. The larger and longer term index sets, typically having the form J c N = {1, 2, 3, ... }, are used to enumerate families of objects that enjoy a more abiding reference throughout the course of a discussion.

<p>The larger and longer-term index sets, typically having the form <math>J \subseteq \mathbb{N} = \{ 1, 2, 3, \ldots \},\!</math> are used to enumerate families of objects that enjoy a more abiding reference throughout the course of a discussion.</p></li></ol>

Definition.  An "indicated set" j^S is an ordered pair j^S = <j, S>, where j C J is the indicator of the set and S is the set indicated.

Definition.  An "indicated set" j^S is an ordered pair j^S = <j, S>, where j C J is the indicator of the set and S is the set indicated.