A Kind of ‘All-at-Once Compactification’

Another technique is concurrence. This is a logical technique that guarantees that the extended structure contains all possible completions, compactifications and so forth. [14, p. 3]

Suppose that s is infinite. If I is also infinite and an injection g of I into s exists, it will suffice to require that no finite set belongs to the ultrafilter ж in order that gj = x a.e. for any x e s; thus g must differ from any function h from I to s which is a.e. constant, and nonstandard individuals exist! This is one way of making the nonstandard enlargement non-trivial (see [28, p. 52]).

Preliminary to the construction of a much richer nonstandard universe, [14, p. 34] defines concurrence. In our own, slightly readjusted terms:

Definition 10.3 Relative to a universe U, a dyadic relation r such that r e U and r U dom(r) c U is said to be concurrent if to every finite d c dom(r) there corresponds some b e U s.t. d x {b} c r. 4

Now let I be the set of functions ф such that dom^) is the set of all concurrent relations r e ^ and ф r is a finite subset of dom (r) for each such r. The ultrafilter ж will then be chosen so that I = U ж holds and the membership relation

also holds, for each f e i. Here comes a key theorem, due to Abraham Robinson:

Theorem 10.3 (Concurrence theorem) To every concurrent relation r e "s there corresponds some t e w such that [*a : a e dom(r)} x [t} c *r.

From this claim, [14, p. 36] draws the conclusion that nonstandard individuals exist: for, assuming N c s in order to slightly simplify the argument, one such is the ‘limit’ element t corresponding to the concurrent relation

in fact, t e IN s.

The third technique is internality. A set s of elements of the nonstandard universe is internal if s itself is an element of the nonstandard universe; otherwise, s is external. A surprisingly useful method of proof is one by reductio ad absurdum in which the contradiction is that some set one knows to be external would in fact be internal under the assumption being refuted. [14, p. 3]

Definition 10.4 We call

external set: every element of w W;

internal set: every element of W w. 4

After showing, with the aid of the transfer principle, that IN N is an external set, [14, pp. 39-41] provides criteria for demonstrating the internality of specific sets:

Theorem 10.4 (Internality theorem) Ifd c W is definable in W and a is an internal set, then a П d is an internal set.

Theorem 10.5 If a and b are internal sets, then so is a x b.

Theorem 10.6 (Internal function theorem) If f e ba, where a and b are internal sets, and for a suitable term t of Lw involving one free variable

then f is internal.

Along the way, [14, pp. 39-41] shows N to be an external set.

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