The discussion of internal sets and functions in the previous two chapters
raises some fundamental conceptual issues:
• In proving internal versions of induction, the least number principle,
order-completeness, etc., we reverted once more to ultrafilter calculations.
Could we instead obtain these results by a logical transfer
principle, involving an extended version of the formal language of
Chapter 4? A limited extension of this kind is provided by the statement
(t) of Section 11.7, but perhaps this can be taken further by
using a more powerfully expressive language that would allow the
quantifiers V, .:3 to range over collections of sets or functions rather
than just collections of numbers (cf. Section 4. 7.)
• Now that we see how to identify certain subsets and mnctfonfi in
*JR as being internal, can we do the same for other more complex
entities? Are there internal topologies on *JR? Or internal measures?
If A * is hyperfinite of internal cardinality N1 does it follow
that the power set of A is hyperfinite of cardinality 2N? Or is it the
collection of internal subsets of A that should be hyperfinite? This
would seem to require the notion of an internal function of the type
{1, ... , 2N} -:-+ P(A).
theoretic universe that can be erected on a set like by forming sets of
158 13. Universes and Frameworks
sets, sets of sets of sets, etc., and then consider how this universe may
be "enlarged" to admit nonstandard entities, by analogy with the enlargement
of JR. to *JR.. Ultimately this will provide a framework that allows the
methodology of nonstandard analysis to be applied to any kind of mathematical
structure (function spaces, measure spaces, infinite-dimensional
Hilbert spaces, ... ). It will also cause us to review what we have been
doing so far from a more abstract set-theoretic standpoint.
13.1 What Do We Need in the Mathematical
World?
In developing a mathematical theory, or analysing a particular structure,
access may be needed to a wide range of entities: sets, members of sets,
sequences, relations, functions, etc. We will posit the existence of a "universe"
1IJ that contains all such entities that might be required. This will
have an associated formal language .Cu whose sentences express properties
of the members of 1IJ. Then 1IJ will be enlarged to another universe *1U that
contains certain new (nonstandard) entities whose behaviour can be used
to establish results about 1U by the use of transfer and other principles.
Here now is some more detailed discussion about the entities and closure
properties that 1U should have.
• Individuals. Although a real number might be viewed as a set of
Cauchy sequences, or a pair of sets of rationals (Section i.3), when
studying real analysis we generally regard real numbers as individuals,
i.e., as "points" or entities that have no internal structure. The
same applies to the basic elements of any other structure that might
concern us, be they elements of an algebraic number field, complex
numbers, vectors in some Hilbert space, and so on.
The universe 1U will contain a set X of entities that are viewed as
individuals in this way. An element of X will be taken to have no
members within 1U. It will be assumed that JR. X.
• Functions. If two sets A and B belong to 1U, then we may wish to
have all functions f : A B available in 1U, along with the range of
j, the !-image f(C) B of any C oA, and the inverse image of any
subset of B under f. Moreover, the set BA of all functions from A to
B should itself be in 1U.
Also, we should be able to compose functions in 1U.
• Relations. An m-ary relation is a set of m-tuples (a1, ... , am), and
is usually presented as a subset of some Cartesian product A1 x
···
x Am, the latter being the set of all such m-tuples that have
13.2 Pairs Are Enough 159
a1 E A1, ... , am E Am· Thus 1U should be closed under the formation
of tuples, and of Cartesian products and their subsets.
For binary relations (m = 2) the domain and range should be avail
able, and the operations of composing and inverting relations should
be possible within our universe.
• Set Operations. All the usual set operations of intersection A n B,
union AU B, difference A-B, and power set P(A), when performed
on sets in 1U, should produce entities that belong to 1U. In fact, some
important constructions will require the union UY and intersection
nY of any (possibly infinite) collection y E u to be available. Also,
if a set A belongs to 1U, then all subsets of A should too.
• Transitivity. If a set A is in 1U, we will want all members. of A to
be present in U as well, i.e., A \U. This condition is usually called
transitivity of U, because it takes the form
a E A E lU implies a E U.
This has an important bearing on the interpretation of a bounded
quantifier (Vx E A). We naturally read this as "for all x in A", but
when used to express a property of an entity of U, there is a potential
issue as to whether this means "for all x in A that belong to U"n, or
whether the variable x is ranging over all members of A absolutely.
When lU is transitive, this is not an issue: the members of A that belong
to lU are simply all the members of A that there are. Transitivity
thus ensures that quantified variables always range over members of
lU when given their natural interpretation.
Subset and Relation Closure
Transitivity of lU together ith closure under the power set operation will
guarantee that lU has the property mentioned above of closure under subsets
of its members. For then if A B E 1U, we get A E P(B) E U, and hence
A E 1U by transitivity.
Then closure of lU under Cartesian products will lead to closure under
relations between given sets in general. Thus if A, B are sets in 1U and
R .Ax B, then if Ax BE U, it follows that R E 1U by the argument just
given for subset closure.
13.2 Pairs Are Enough
The more we assume about the entities that exist and constructions that
can be performed within 1U, the more powerful will be this universe as a
160 13. Universes and Frameworks
tool for applications. On the other hand, for demonstrating properties of
1I.J itself or showing that it exists (and *'[]ndoes too), it is desirable to have
very few primitive concepts, so that we can minimize the number of cases
and the amount and complexity of work required in carrying out proofs.
Studies of the foundations of mathematics have shown that these opposing
tendencies can be effectively balanced by basing our conceptual
framework on set theory. To see this we will first show that apart from
purely set-theoretic operations, the other notions just described in Section
13.1 can be reduced to the construction of sets of ordered pairs:
• Functions. A function f : A ---+ B can be identified with the set of
pairs
{(a, b) : b = j(a)},
which is a subset of the Cartesian product set Ax B. Set-theoretically,
we define a function from A to B to be a set f of pairs satisfying
(i) if (a, b) E f then a E A and b E B;
(ii) if (a, b), (a, c) E j, then b = c (functionality);
(iii) for each a E A there exists bE B with (a, b) E f (the domain of
f is A).
• m-Tuples. Given a construction for ordered pairs (2-tuples), the case
m > 2 can be handled by defining
(ab ... , am) = {(1,a1), ... , (m, am)}.
Thus an m-tuple becomes a set of ordered pairs (and actually is a
function with domain {1, ... , m}).
Note: an alternative approach would be to inductively put
(al,··· ,am+!)= ((al,· .. ,am),nam+l),
so that an m-tuple beomes a pair of pairs ofn·n·· of pairs. This works
just as well, but would be more complex set-theoretically than the
definition given.
• Relations. An m-ary relation is a set of m-tuples (a1, ... , am), and
hence becomes a set of sets of ordered pairs. The Cartesian product
A1 x · · · x Am is a particular case of this, being the set of all such
m-tuples that have a1 E A1, .o.
. , am E Am.
13.3 Actually, Sets Are Enough
But what is an ordered pair? Well, one of the most effective ways to explain
a mathematical concept is to give an account of when two instances of the
13.4 Strong Transitivity 161
concept are equal, and for ordered pairs the condition is that
(a, b) = (c, d) iff a= c and b =d.
In fact, this condition is all that is ever needed in handling pairs, and it
can be fulfilled by putting
(a,ob) = { {a},o{a,b} }.
In this way pairs are represented as certain sets, and therefore so too are mtuples,
relations, and functions. When it comes to the study of a particular
structure whose elements belong to some given set X, all the entities we
need can be obtained by applying set theory to X. This demonstrates the
power and elegance of set theory, and explains the sense in which it provides
a foundation for mathematics.
Exercise 13.3.1
(i) Verify that {{a}, {a, b}o} = { {c}, {c, d}o} iff a= c and b =d.
(ii) Show that for m 2: 2,
Product Closure
Closure of 1U under Cartesian products can now be derived set-theoretically
from transitivity and closure under unions and power sets. If A, B E 1U and
(a, b) E Ax B, then both {a} and {a, b} are subsets of AUB, i.e., members
of P(A U B). Hence
(a,ob) = { {a},{a,b}} E PP(AUB).
This shows that Ax B PP(AUB), and so Ax BE PPP(AuB). Closure
under U and P and transitivity of 1U then give A x B E 1U.
13.4 Strong Transitivity
Before giving the axioms for a universe, there is a further important property
to be explained, which we do with the following example.
If a binary relation R belongs to 1U, then its domain dom R should be
available in U as well. Now, if a E dom R, then there is some entity b
with (a, b) E R. According to our new definition of pairs, we then have the
"membership chain"
a E {a} E (a, b) ERE 1U.
162 13. Universes and Frameworks
Transitivity of lU will ensure that it is closed downwards under such membership
chains, giving a E U. But this leads only to the conclusion that
domR U, whereas we want domnR E U Is domnR perhaps too "big" to
be an element of U?
Now, if R itself were transitive, we would get a E R, showing domnR
R E U, from which our desired conclusion would result by subset closure.
But of course R need not be transitive. On the other hand, it is reasonable
to suppose that R can be extended to a transitive set B that belongs to lU
(i.e., R B E U). Then we can reason that dom R B E U, leading to
domR E U, as desired, by subset closure.
The justification for this is that any set A has a transitive closure Tr(A),
whose members are precisely the members of members ofn··· of members
of A. Tr(A) is the smallest transitive set that includes A: any transitive
set including A will include Tr(A). We are going to require that lUbe "big
enough" to have room for the transitive closure of any set A E lU. For this
to hold it is enough that some transitive set including A belong to lU. Thus
our requirement is
• Strong Transitivity: for any set A in lU there exists a transitive
set BE lU with A B U.
Note that the stipulation that B lUis superfluous if lUis transitive, since
it then follows from B E U. But the definition of strong transitivity itself
implies that lU is transitive (since we get A lU when A E lU because
A B U), so this single statement captures all that is needed.
In a strongly transitive lU we can assume that any set we are dealing
with is located within a large transitive set. This will be the "key to the
universe", as will become apparent.
13.5 Universes
In the light of the foregoing discussion, we now define a universe to be any
strongly transitive set lU such that
• if a, bE U, then {a, b} E U;
• if A and Bnare sets in U, then AU BE U;
• if A is a set in U, then P(A) E lU.
Such a lU will be called a universe over X if X is a set that belongs to U
(X E lU), and the members of X are regarded as individuals that are not
sets and have no members:
(Vx EX) [x # 0 A (Vy E lU) (y x)].
13.5 Universes 163
It will always be assumed further that a universe contains at least one
set, and also contains the positive integers 1, 2, .n.
. to ensure that m-tuple
formation can be carried out. In practice we will be using universes that
haven E ll.J, with each member ofn being an individual, so these conditions
will hold.
Here now is a list of the main closure properties of such universes, many
of which have been indicated already. Uppercase letters A, B, Ai, etc. are
reserved for members of liJ that are sets.
Set Theory
• If a E ll.J, then {a} E liJ.
• A1, ... , Am E liJ implies A1 U · · U Am E liJ.
·
• liJ contains all its finite subsets: if A li.J and A is finite, then A E liJ.
• A B E liJ implies A E liJ.
• 0 E ll.J.
• If { Ai : i E I} A E ll.J, then UiEJAi E liJ. (Note: this uses strong
transitivity.)
• liJ is closed under unions of sets of sets: if B = { Ai : i E I} E li.J and
each Ai is a set, then UB = uiEJAi E 1U.
• li.J is closed under arbitrary intersections: if { Ai : i E J} ll.J, then
niEJAi E liJ, whether or not the set {Aio: i E /}oitself belongs to liJ.
Relations and Functions
• If a, bE ll.J, then (a, b) E ll.J.
• If A,B E li.J and R Ax B, then R E ll.J.
• If ab ... , am E li.J (m > 2), then (a1, ... , am) E ll.J.
• 1U is closed under finitary relations: if A1, ... , Am E liJ and R C
A1 X X Am, then R E lU.
· ·
·
• If R E 1U is a binary relation, then liJ contains the domain dom R, the
range ran R, the R-image R' (C) of any C dom R, and the inverse
R-1, where
domR {a: 3b((a,b) E R)},
rannR -{b: 3a((a,b) E R)},
164 13. Universes and Frameworks
R'(C) {bo: 3a E C ((a, b) E R)n},
R-1 -{ (b, a) : (a, b) E R}.
• If R, S E 1IJ are binary relations, then 1IJ contains their composition
R o S = {(a, c) : 3b( (a, b) E R and (b, c) E S)}.
• Ifnf : A ----7 B is a function with A, B E 1U, then f E 1IJ. Moreover, for
any C A and D B, 1IJ contains the image
j'(C) = {f(a) :oa E C}
and the inverse image
j-1(D) ={a E A: f(a) EoD}.
• If A, B E 1U, then the set BA of all functions from A to B belongs to
1IJ.
• If {Ai : i E J} E 1IJ and IE 1U, then (IliEI Ai) E 1IJ.
13.6 Superstructures
It is time to demonstrate that there are such things as universes. Let X
be a set with 1R X. The nth cumulative power set 1Un(X) of X is defined
inductively by
so that
lUo(X) -X,
lUn+l (X) 1Un (X) U P(lUn(X)),
The superstructure over X is the union of all these cumulative power sets:
The rank of an entity a is the least n such that a E lUnn(X). The rank 0
entities (members of X) will be regarded as individuals:
(Vx E 1Uo(X)) [x of. 0 A (Vy E 1U(X)) (y ¢:. x)].
All other members ofnlU(X) (those with positive rank) are sets, and so lU(X)
has just these two types of entity. We can show:
(1) 1Un+I(X) =XnU P(lUn(X)).
13.6 Superstructures 165
(2) lUn
(X) E 1Un+1(X). Hence lUn
(X) E lU(X), and in particular, X E
lU(X).
(3) lUn+l (X) is transitive. Indeed, a E BE lUn+l (X) implies a E lUn(X).
(4) If a,b E lUn
(X), then {a,b} E lUn+l
(X).
(5) If A, BE lUn(X)
, then AU BE lUn+l
(X).
(6) A E lUn(X) implies P(A) E Un+2(X).
From (3) it follows that U(X) is strongly transitive, since every element of
U(X) belongs to some Un+l (X). Properties (4)-(6) then ensure that lU(X)
is a universe, and by (2) it is a universe over X.
In fact, U(X) is the smallest universe containing X, in the sense that if
any universe U has X E lU, then U(X) lU. Another description of this
superstructure over X is that it is the smallest transitive set that contains
X and is closed under binary unions AU Bnand power sets P(A).
Exercise 13.6.1
Verify results (1)-(6) above, and the observations that follow them. Show
further that if X Y, then U(IR) U(X) U(Y). D
A universe is not closed under arbitrary subsets: if A lU, it need not
follow that A E 1U (e.g., consider A = lU). In the case of a superstructure,
A will belong to U(X) iff there is an upper bound n E N on the ranks of
the members of A, i.e., iff A Un(X) for some n. All the entities typically
involved in studying the analysis of X can be obtained in lU(X) using only
rather low ranks. If A, B E lUn (X), then any subset of A x B, and in
particular any function from A to B, is in 1Un+2(X). So constructing a
function between given sets increases the rank by at most 2. Using this, we
see that:
• A topology on X is a subset of P(X), hence a subset of llh (X), so
belongs to U2(X). Thus the set of all topologies on X is itself a member
of U3(X).
• An IR-valued measure on X is a function J.L: A
---+ IR with A a collection
of subsets of X, so A is of rank 2 and J.L of rank 4. Thus the set of all
measures on X is also an element of U(X)
, of rank 5.
• A metric on X is a function d : X x X ---+ IR of rank 5 (since X x X has
rank 3). The set of all metrics on X has rank 6.
• The Riemann integral on a closed interval [a, b] can be viewed as a
function
J: : R[a, b] ---+ IR,
166 13. Universes and Frameworks
where R[a, b] is the set of integrable functions f : [a, b] -t R Such an
f is of rank 3, since [a, b] and IR have rank 1, so R[a, b] has rank 4
and therefore the integral J: is an entity of rank 6.
13.7 The Language of a Universe
Given a denumerable list of variables, a language £u associated with the
universe liJ is generated as follows:
Cu-Terms
• Each variable is an £u-term.
• Each member of 1U is a constant £u-term.
• IfnT1, ... ,Tm are £u-terms (mn;::::: 2), then (TI, ... ,Tm) is an £u-term,
called a tuple.
• If T and u are £u-terms, then T•(u) is a function-value £u-term.
Notice that our rules allow iterated formations of tuples of terms, such as
((T, a), T, (Tb ... , Tm)).
A term with no variables is closed, and will name a particular entity of U
if it is defined (recall the discussion of undefined terms in Section 4.3.1).
The rules for determining when a closed tuple is defined, and what it
names, are as follows:
• If T1, ... , T m name elements a1n, ... , am, respectively, then ( T1n, ... , T m)
names the thenm-tuple (ai,o· .. ,am)·
• ( T1, ... , T m) is undefined if one of T1, ... , T m is undefined.
For a closed function-value term, the rules are:
• If T names a function f and u names an entity a that belongs to the
domain of J, then T(u) names the entity f(a).
• T(a") is undefined if one ofnT and CJ is undefined, or if they are both
defined but T does not name a function, or if T names a function but
CJ does not name a member of its domain.
The language L()l. of Chapter 4 allowed formation of terms f(TJ, ... , T m)
where f is an m-ary function. This is catered for here because of tuple
formation. Any finitary function on IR belongs to liJ because IR X, so f is
a constant of £u, and j(T1n, ... , Tm) can be taken to be a simplified notation
13.7 The Language of a Universe 167
for the £u-term f(( 71, ... , 7 m)). More generally, we can write 0'(71, ... , 7 m)
for 0'((71, ... , 7m)) when 0' is an arbitrary £u-term (this is in line with
common practice: an m-ary function on a set A is just a one-placed function
on Am).
It follows that all L!)t-terms are .Cu-terms.
Atomi
c Cv-Formulae
These have one of the forms
7 = 0',
7 E 0',
where 7 and 0' are .Cu-terms. For example, if P E 1U is a k-ary relation,
then· there are atomic formulae
which may also be written P(71
, ... , 7k) as in the notation of Section 4.3,
or, in the case k = 2, using infix notation, as in 71 < 72, 71 =/= 72, etc.
Since a function is a special kind of relation, symbols for functions may
occur in atomic formulae in two ways. For example, the two formulae
have the same intended meaning.
Formulae
• Each atomic .Cu-formula is an .Cu-formula.
• If t..p and 'lj; are .Cu-formulae, then so are t..p 1\ 'lj;, t..p V 'lj;, •t..p, t..p ---t 'lj;,
t..po'lj;.
• If c.p is an .Cu-formula, then so are (Vx E 7)t..p and (:3x E 7)c.p, where 7
is any .Cu-term and x is any variable symbol that does not occur in
7.
A sentence is, as usual, a formula in which every occurrence of a variable
is within the scope of a quantifier for that variable.
If t..p is a formula in which only the variable x occurs freely, and 7 is a
closed term denoting the set A E 1U, then the sentence (Vx E 7)c.p asserts
that c.p(a) is true for every a E A, while (::lx E 7)c.p asserts that c.p(a) is true
for some such a. As explained in Section 13.1, transitivity of 1IJ ensures
that quantified variables always range over members of 1IJ when given their
natural interpretation.
168 13. Universes and Frameworks
As we will see later in applications of the language .Cu to mathematical
reasoning, the term T in a quantifier form (Vx E T) or (3x E T) is usually
a variable or a constant.
Having observed above that the .Cu-terms include all L!Jt-terms, and that
.Cu allows the atomic formation P(T1, ... ,Tk), we can now conclude that
the .Cu-formulae include all .C!R-formulae: any subset P of lR is in 1U, so the
formation rules of .Cu admit the bounded quantifier forms (Vx E P) and
(3x E P).
13.8 Nonstandard Frameworks
Let 1U 1U1 be a mapping between two universes, taking each a E 1U to an
element *a of 1U'. Then each .Cu-term T has an associated *-transform *T,
which is the £u,-term obtained by replacing each constant symbol a by *a.
A constant a occurring in an .Cu-formula cp will do so as part of a term
T that appears either in an atomic formula or within one of the quantifier
forms ('i/x E T) and (3x E T). Applying the replacement a *a to all such
constants transforms cp into an £u,-formula *cp. If cp is a sentence, then so
too is *cp.
A nonstandard framework for a set X comprises a universe 1U over X and
a map 1U lU' satisfying:
• *ao= a for all a EX .
• *0 = 0.
• 'Transfer: an .Cu-sentence cp is true if and only if *cp is true.
Such a map will be called a universe embedding or transfer map. It preserves
many set-theoretic operations:
• a = b iff *a = *b. Hence a *a is injective.
• a E B iff *a E *B.
• A B iff *Ao *B.
• If A X, then A *A *X. In particular, X *X.
• *(AnB)=*An*B.
• *(AUB)o=*AU*B.
• *(Ao-B) =*A-*B.
• *{a1, ... , am}o= {*ab ... , *am}· Thus *A= {*ao: a E A} if A is finite.
13.8 Nonstandard Frameworks 169
• All members of *X are individuals in 1IJ'.
• preserves transitivity: if A is a transitive set in 1IJ, then *A is
transitive.
• *P(A) P(*A).
• If R E 1IJ is an m-ary relation, then so is *R.
• *(A1 X··· X Am)o= *A1 X··· X *Am. Hence *(Am)= (*A)m fornmE N.
• If R E 1IJ is a binary relation, then
*(domnR) domn*R,
*(ran R) ran*R,
*(R-1) (*R)-1,
*(R'(C)) (*R)'(*C) for C domnR,
*(R-1n(C)) (*R)-1(*C) for C rannR.
• If RnandnS are binary relations, then *(R S) = *R o *S.
o
• If a function f : A ---7 B belongs to 1IJ, then * f is a function from *A to
*B, with *(!(a))o= *!(*a) for all a EnA. Also, f is injective/surjective
iffn*f is.
To show that A *A *X whenever A X, observe that if A X, then
*A *X, and if a E A, then a E X, and so a = *a E *A. Also, by transfer
(using *0 = 0) we have
(Vx E *X) [x =f. 0/\ -{3y Ex) (y Enx)]
true, so if b E *X, then b is not the empty set and has no members, and
therefore is an individual.
For preservation of transitivity by . let A E 1IJ be transitive, i.e.,
(Vx E A) (Vy E x) y E A.
This transforms to
(Vx E *A) (Vy E x)y E *A,
showing that any set belonging to *A is a subset of *A, i.e., *A is transitive
asdesired.
170 13. Universes and Frameworks
The fact that *P(A) P(*A) follows by transfer of
(Vx E P(A)) (Vy Ex) (yoEoA).
This shows that if x E *P(A), then y Ex implies y E *A, and sonxo *A,
whence x E P(*A). The exact relationship between *P(A) and P(*A) will
be revealed in Section 13.12.
Exercise 13.8.1
Verify all the other properties of the transfer map listed above.
If lR X, all of the standard operations and relations on lR liken+, -, x, lxl,
sinx, <, 2':
, =/=-,netc. are entities in 1U, and so have corresponding entitiesn*+,
*sinnx, *=/=-, etc. for *JR in 1U'. We will continue the practice of dropping the
*-prefix from such familiar notions when the intention is evident. However,
while this is harmless for functions and relations between individuals, when
entities of nonzero rank are involved, a transformed function * f need not
agree with f where their domains overlap, so more caution is needed. In
general, if a E domnf, then *f(*a) =o*(!(a)), but even when *ao= a this will
reduce to *f(a) = f(a) only when *(!(a)) is equal to f(a). For an example
showing that this need not hold, let f: lR-+ P(JR) be defined by
f(r) = {x E lR: x > r }.
For a given r E JR, transfer of the sentence
(Vx E JR.) (x E f(r) x > r)
shows (since *r = r) that
*f(r) = *(f(r)) = {x E *JRo: x > r}.
In particular, f(O) =
JR.+
, whilen*f(O) = *JR+.
Exercise 13.8.2
If cp(x1, ... ,xm) is an .Cu-formula and A E 1U, show that
*{(all ... , am) E Am : cp(a1, ... , am)}o=
{(b1, ... , bm) E *Am : *cp(b1, ... , bm)}.
Explain how various of the above results about preservation of propertiffi
byn can be derived from this general fact.
13.9 Standard Entities
The members of 1U' of the form *a with a E 1U, will be called standard. The
other members of 1U' are nonstandard. Any element a of X is thus standard,
since in that case a = *a.
13.9 Standard Entities 171
The question of the existence of nonstandard entities will be taken up
in earnest in the next chapter. For now we will just assume that lU is a
universe over a set X that includes IR, and that
• there exists an element n E *N -N.
Then will be infinitesimal, and using transfer instead of ultrafilter calculations
we can derive in lU' the arithmetical properties of limited, unlimited,
appreciable, etc. numbers as in Chapter 5, and then develop the theory of
convergence, continuity, differentiation, integration for IR-valued sequences
and functions as in Chapters 69.
All standard members of *N belong tonN, for if *a E *N, then by transfer
a E N, and so *a = a E N. Thus any member of *N-N must be nonstandard.
More generally, this argument shows that if A X, then any member of
*A -A will be nonstandard.
We see then that standard sets *A may have nonstandard members. In
fact it turns out that *A has nonstandard members for every infinite set
A E 1U (cf. Section 13.14).
Examples of nonstandard sets are provided by initial segments of *N.
Consider the sentence
(Vn EN) (::3V E P(N)) (Vx EN) [x E V +--> x n],
which expresses "for all n E N, {1, ... , n} E P(N)"n. By transfer it follows
that for any N E *N, the subset {1, ... , N} of *N belongs to the standard
set *P(N). If N E *N-N, then {1, ... , N} cannot itself be a standard set.
For if *A is any standard subset of *N, then A N, so either A is finite
and hence *A = A, or else A is unbounded in N, and hence by transfer *A
is unbounded in *N. In either case *A :f= {1, ... , N}.
For another illustration, let Int E 1U2(X) be the set of open subintervals
of the real line:
Int =n{(a, b) IR: a, bE IR}.
It follows by a transfer argument that
*(a, b) = {X E *JR : a < X < b},
so the standard set *(a, b) will have nonstandard elements. By transfer of
the statements
(VA E Int) (::3a, bE IR) (Vx E JR) (x E A+--> a< x