Ultrafilter! - Insert coin

The definition of an ultrafilter

Firstly, we require the notion of an indexing set "I". That is, an allocation of an integer placeholder for the elements of the set to which we wish to show the ultrafilter's presence upon. (We do not wish to form the opinion that the ultrafilter changes the set, merely that it is present) we are interested in the elements  as referenced from the indexing set of typically natural numbers {1,2,3,4,...}

 The definition for an ultrafilter U is as follows.

1) For any subset S of the indexing set I, either S or the complement of S, (I ∖ S) belongs in the ultrafilter U, but never both.

2) The empty set φ is never a member of the ultrafilter U

3) If sets X and Y are in U, then so is their intersection X ∩ Y.

4) If X is a set in U, then so is every set that is a superset of X, (All sets that properly contain X as a subset)

These sets S, X, Y etc... are subsets of the indexing set I, and are not to be confused with the set upon which is the index.

I admit it is hard to see what difference it makes to the original set that is indexed, as the ultrafilter does not appear to be in "direct contact" with it. The ultrafilter could be imagined as the "connected" portion, or "traversable route" through the indexed set, in the sense that "dead ends" are not allowed. Of course, this is only possible if the indexing set I is related to the "shape" of its indexed set. The set has to be what is called a "poset", short for "partially ordered set". Such a poset consists of a relation, or ordering upon certain pairs of elements in the set, such that one is "greater than" the other. There is no requirement that every pair is related in this way - this is perhaps the clearest reason why ultrafilters are considered to apply "almost everywhere". In the choicee of properties above, each is chosen to make the ultrafilter a "bigger than not" set of subsets of I.

Finite or infinite ultrafilters appear in two known forms, as follows;

Principal Ultrafilters
Such an ultrafilter contains a "least element x" an "indivisible subset of I", by virtue of (3), x is in every subset in U. With a little consideration by (4), U consists of all subsets of I containing x. All finite ultrafilters are principal.

Non-Principal Ultrafilter
Infinite ultrafilters that have no such least element as above are non-principal, plain and simple.

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