Rev.Acad.Canar.Cie nc., IX (Núm . 1) , 141- 171 ( 1 997 )
Abstract .
A NEW APPROACH TO NONSTANDARD ANALYSIS
Manuel Suárez Fernández
Profesor de Matemáticas de la U.C.L.M.
The aim of this paper is to presentan new and simple approach
to nonstandard analysis.
Preface.
The nonstandard analysis is the modern version of Newton and
Leibniz's old mathematical analysis in which unlimitedly big or
unlimited and unlimitedly small or infinitesimal numbers are
used. The most significant rediscoverer of such numbers is
Abrahám Robi nson [3] and , later on, there are relevant people
such as Edward Nelson [2] and Georges Reeb.
In my opinion, apart from its theoretical value that, doubtless,
nonstandard ideas have, such ideas have a paramount practical
interest.
But for it to be shown and so, first, the above mentioned ideas,
come to be known, then accepted and, finally, normally used not
only by mathematicians but also by other professionals that as
it is the case with physicists, chemists, engineers, economists,
etc., use mathematics and, even, nonstandard analysis come to be
u sed i n a systematic way in Uni ver si ty mathematical analysis
programs, it is essential to simplify more and more its approach.
To presentan new, simple and didactic approach to nonstandard
analysis , so that the nonstandard ideas are easy and attractive
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to reader, it is the main objetive of this paper, which contains
three parts.
The first part aim is to fundament, in the theory of sets, the
non standard theorems of the second part, which, on its part ,
fundaments the non standard principles of the third in which
there appear non standard definitions of sorne basic concepts of
infinitesimal calculus.
Posing and reasoning of everyone of the above mentioned parts,
are original, as well as the non standard principles stated in
the third of thern (though this principles could be proved also
in [2]) and, in this paper (as in [2] ), the non standard universe
of sets is the same as the standard universe of sets.
And if what the reader wishes is just to know with the slightest
possible effort a non standard theory to, far example, use it in
infinitesimal calculus, can pay attention only to the third part
(without reading either the first or the second) notwithstanding
he must know ZF ( Zerrnelo-Fraenkel set theory). And, to understand
the fist part, he must know ZFC (Zermelo-Fraenkel set theory with
axiom of choice1 ).
1The axioms of ZFC are (according to [l]) the extensionality
axiom, the power set axiom, the union set axiom, the class of
axioms of subti tution, the axiorn of infini ty, the statement
«there exists sorne set» and the axiom of choice. Specific sings
that we use (in a simple way, as rnuch as possible) are, far TN
model, and, in arder to syrnplify the denotations, also are far
TN model, the usual signs,
" --, V== 3 V E f/= = .. e et. u n ( ) \ { } 1 :S < + - • /
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First part: A model of ZFC valid for nonstandard analysis. 2
In this Firs part we presentan original proof that if Tw is a
consistent model of ZFC then there exists a consistent model T"
of ZFC valid for the nonstandard analysis because the classic set
of natural numbers ( of T" model), besides the classic ( or
standard) "limited". natural numbers, has "unlimited" natural
numbers3 , so that «zero is a limited natural number», «if nis
a limited natural number then n+l is a limited natural number»
and «there exist unlimited natural numbers» (by virtue of which
and of the classic recurrence principle, the limited natural
numbers do not constitute a set of T" model).
The T N theory.
We suppose that Tw is a model of ZFC for which W is the set of
natural number, Fw is a Fréchet's ultrafilter' of model Tw and,
from the befare mentioned Tw model, we define a "theory of
sets", that we call T", in the following way:
We suppose that if a,,(t1 , •• ,tJ) is an expression whichever in
which there are j expressions t 1 , •• , tJ (bearing in mind that
jEw•, w·={x l (xEW)A(x;,eQw)} and Ow is the zero of Tw model) and
~ 1 , •• , ~J are j expressions, then a,,(~1 , •• ,~J) is the
expression with ~1 , •• , ~J where in a(t1 , •• ,tJ), respectively,
2Even this first part, which is the least simple of the
three parts, it is easy to understand because its basic idea is
nepeated from beginning to the end.
3Consequently, the classic sets of integer, rational, real
and complex numbers (of T" model), also have unlimited elements.
4See Appendix of First part.
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there are l;.i, .• , l;,J.
We suppose that the sets of T" theory are the sequences of
sets of Tw model. 5
If f is a set of T" theory then, for each element n of w·,
we call "component n of f" to the set f(n) of Tw model. 6
We call "variables" of T" theory, to the same signs that we
call "variables" of Tw model.
We suppose that the formulas of T" theory are the expressions
that result if we replace in the formulas of Tw model, the
constants7 (of Tw model) appearing in them, for constants of
T" theory (So, for instance, if A is constant of Tw model, f
is a constant of T" theory and x is a variable then 3x(xEA) is
a formula of Tw model with the constant A and 3x(xEf) is a
formula of T" theory with the cons~ant f where in 3x(xEA)
there is the constant A).
If a is a formula of T" theory and nis an element of w• then,
if a is with no constants then we say that "a,, is the
component n of a" if and only if a,, is the expression
identical to a.
if a is with only p constants f 1 , •• , fP (bearing in mind
that pEw•), then we say that "a,, is the component n of a"
5So, f is a set of T" theory if and only if f is a sequence
A1 , A2 , A3 , ••• of Tw model. That is, f is a set of T" theory if and
only if f is a map frorn w· into a set E of Tw model (So, for each
element n of w· there exists an element An of E such that
f( n) =An).
6That is, is f is the sequence A1 , A2 , A3 , ••• of Tw model and
n is an element of w· then we call "component n of f" to An.
7\l!e call ''.constants" to the signs which denote sets,
"constants of Tw model" to the signs which denote sets of Tw
model and "constant of T" theory" to the signs which denote sets
of T" theory.
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if and only if a,, is the expression such that in a,, there
are the constants f 1(n), .• , fP(n) of Tw model where in a,
- respectively, there are the constants f 1 , •• , fP of T"
theory (that is, if a(f1 , •• ,fP) is a formula of T" theory
wi th only p constants f 1 , •• , fP and nEw• then the expression
a(f1(n), •• ,fP(n)) is the component n of a(f1 , •• ,fP)).
We suppose (as in Tw model) that the statements of T" theory
are the formulas of T" theory with no free variables (So, if
a is a formula of T" theory then a is a statement if and only
if for each element n of w·, the component n of a, is a
statement of Tw model).
We suppose that if a is a statement of T5 theory then a is
true if and only if the set of the elements n of w• such that
the component n of a is a true statement of Tw model, is an
element of (ultrafilter) Fw.
If a, ~ are statements of T5 theory then (in a similar way to
Tw model) we say that "a is equivalent to~", if and only if
either a is true and ~ is true ora is false and ~ is false.
And if a(x), ~(x) are formulas of T" theory with the same
variables and (at least) there is a free variable x then we
say that "a(x) is equivalent to ~(x)" if and only if, for all
constant f of T" theory, a(f) is equivalent to ~(f).
(So, for example, if f, g are constants of T" theory then the
statements f~g, ff/.g, f~g are equivalent to, respectively, the
statements ~(f=g), ~(fEg), ~(fCg)).
Theorem 1.1. If a is a statement of T" theory then either a is
true ora is false.
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Proof. If a is a statement of T" theory and for each element n of
w•, a,, is the component n of a, then for all element n of w·, a 0
is a statement of Tw model. So, since Fw is an ultrafilter, one
and only one of the two following formulations is true:
The set of elements n of w• such that a,, is a true statement
of Tw model, is an element of Fw.
The set of elements n of w• such that a,, is a false statement
of Tw model, is an element of Fw.
Therefore, one and only one of the two following formulations is
verified:
a is a true statement of T" theory.
a is a false statement of T" theory.
Theorem 1.2. If a is a statement of T" theory then a is true if
and only if ~a is false.
Proof. Taking into account now that Fw is a filter (because Fw is
an ultrafilter), similar proof to that of Theorem 1.1.
Theorem 1.3. If a, ~ are statements of T" theory then,
aA~ is true if and only if «a is true and ~ is true».
av~ is true if and only if «a is true or ~ is true».
Proof. In similar way to proof of Theorem 1, it is preved that,
aA~ is true if and only if a is true and ~ is true, taking
into account that Fw is a filter.
av~ is true if and only if a is true or ~ is true, taking into
account that Fw is an ultrafilter.
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Lemma 1.1. If, in Tw model, H is an element of Fw and
A1 , A2 , A3 , ••• is a sequence of sets such that for each element
n of H, "1'"4>w ( 4>w being the empty set of Tw model) then there
exists a set f of T" theory such that for each element n of H,
the component n off is an element of A..·
Proof. If A=Unew+A.. (that is, if A is the union of the An of
sequence) then, by virtue of the axiom of choice (that satisfies
the Tw model), there exists a map 'P from P(A)\{4>w} (set of
subsets non empty of A) into A such that for each element n of
H, 'l'(A.,) is an element of A.,. And let's b an element of A.
So, if f is the map from w· into A such that for each element n
of H, f(n)='l'(An) and for each element of w•\H, f(n)=b then f is
a set of T" theory such that for each element n of H, f(n)
(component n off) is an element of A.,.
Theorem 1.4. If, in theory T", E is a set and a(x) is a formula
with free variable x and with no other free variables then,
3x((xEE)Aa(x)) is true if and only if there exists sorne set
f of T" theory such that (fEE)Aa(f) is true.
Vx((xEE) = a(x)) is true if and only if, if f is a element
(whichever) of E then a(f) is true.
Proof. If for each element n of w·, En is the component n of E
and an(x) is the component n of the formula a(x) then,
3x((xEEn)Aan(x)) is the component n of 3x((xEE)Aa(x)) and
3x((xEE)Aa(x)) is true if and only if the set of elements
n of w• such that 3x((xEEn)A~(x)) is a true statement of· Tw
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model, is an element of Fw. So,
if there exists a set f of TN theory such that (fEE)Aa(f)
is true then the set of elements n of w• such that
(f(n)EEn)Aa~(f(n)) is a true statement of Tw model, is an
element of Fw. Then, the set of elements n of w· such that
3x( ( xEEn)A a n( x)) is a true statement of Tw model , is an
element of Fw and, consequently, 3x((xEE )A a (x)) is true.
if, reciprocally, 3x((xEE)Aa(x)) is true and H is the set
of elements of w· such that 3x((xEEn)A a n( x )) is true, then
H is an element of Fw. So, if for each element n of H,
An={x l (xEEn)Aan(x)} then, by virtue of Lemma 1.1, there
exists ( in Tw model) a map f from w• into U n,w.An ( union set
of En), such that for each element n of H, f(n)EAn is true
and, consequently, for each element n of H,
(f(n)EEn)A~(f(n)) is true. Therefore, the set of element
n .of w· such that (f(n)EEn)Aa(f(n)) is true, is an element
of Fw and, consequently, there exists a set f (of TN
theory) such that (fEE)Aa(f) is true.
For each element n of w·, ~3x((xEEn)A~an( x)) i s the component
n of ~3x((xEE)A~a(x)), ~3x((xEEn) A~an( x)) is equivalent to
Vx( (xEEn) = ~(x)) and this statement is the component n of
the statement Vx((xEE) = a(x)) (of TN theory). So,
Vx( (xEE ) = a(x)) is equivalent to ~3x( (xEE)A~a(x )). Then, the
statement Vx( ( xEE) = a( x)) is true if and only if for all set
f (of TN theory), a(f) is a true statement.
Theorem 1.5. If f , g are sets of TN theory then f=g if and only
if, if a(x) is a formula (whichever) of TN theory with a free
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variable x then the formula a(f) is equivalent to the formula
a(g). (So, if f is a set of T" theory then f=f and if f, g are
sets of T" theory then,
if f =g then f=g.
if h is a set of T" theory, f=h and h=g then f=g.
if f=g then f has the same elements and is an element of .the
same sets as g) .
Proof. If f, g are sets of T" theory and f=g then the set of
elements n of w· such that f(n)=g(n), is an element of Fw. So, if
a(x) is a formula of T" theory with a free variable x and for
each element n Of w·, a 0 (x) is the component n of a, then for all
element n of w·, x is free in a 0 (x) and the set of elements n of
w• such that a 0 (f(n)) is equivalent to a 0 (g(n)), is an element of
Fw 8 • Therefore, a(f) is equivalent to a(g).
Reciprocally, if f, g are sets of T" theory such that «if a(x) is
a formula (whichever) of T" theory with a free variable x then
a(f) is equivalent to a(g)», then, since x=g is a formula of T"
theory with a free variable x, the statement f=g is equivalent
to the true statement g=g and, consequently, f=g is true. 9
Theorem 1.6. If f, g are sets of T" theory then fCg if and only
if all element off is an element of g.
8Bearing in mind that if A, B are sets of Tw model, A=B and
y(x) is a formula of Tw model with a free variable x then y(A) is
equivalent to y(B).
90f the similar way it is preved that if, in general,
Ai, .. , AP, B1 , •• , BP are 2p set of T" theory, then ««A1=B1 , •• ,
AP=BP» if and only if, «if a( x1 , •• , xP) is a formula of T" theory
with p free variables x 1 , •• , xP then the formula a(A1 , •• ,AP)
is equivalent to the formula a(B1 , •• ,BP)»».
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Proof. Then, If fCg then the set of elements n of w· such that
f(n)Cg(n), is an element of Fw. And if hEf then the set of
elements n of w• such that h(n)Ef(n), is an element of Fw. So, if
fCg and hEf then the set of elements n of w• such that h(n)Eg(n),
is an element of Fw and, consequently, hEg. So, if fCg then all
element h off is an element of g.
If f(μ;J then the set of elements n of w• such that f ( n)(μ;J(n) is an
element of Fw. Then, if H is the set of elements of w• such that
there exists a set hn (of Tw model) such that hnEf( n) and hnftg(n),
then H is an element of Fw. Then, by virtue of Lemma 1 .1 , there
exists a set h of TN theory such that fer all element n of H,
h(n)Ef(n) and h(n)ftg(n). Consequently, hEf and fftg. So, if all
element off is an element of g then fCg.
Theorem 1.7. If f, g are sets of TN theory and f has the same
elements as g then f=g. w
Proof. If f~g and H is the set of elements n of w• s uch that
f(n)~g(n) then H is the set of elements n of w• such that there
exists a set hn of Tw model such that either «hnEf (n) and h nftg(n) »
or «hn(;l:f(n) and h nEg(n)», and H is an element o f Fw. Then, by
virtue of Lemma 1.1, there exists a set h of TN theory such that
fer each element n of H, either «h(n)Ef(n) and h(n )ftg(n) » or
«h(n)';l:f(n) and h(n)Eg(n)», and, consequently, e i ther «hEf and
hftg» or «h(;l:f or fEg».
Therefore, «if f ~g then f has not the same elements as g» and,
1ºThis theorem of Tw model is the extensionalit y axi om of TN
theory.
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consequently, if f has the sarne elernents as g then f=g.
Theorern 1.8. The constant sequence ~w, ~w, ~w, ••• of Tw rnodel (~w
being the ernpty set of Tw rnodel) is the ernpty set of T" theory,
which we denote~. 11
Easy proof.
Theorern 1.9. If f is a set of T" theory then there exists a set
P( f) ( of T" theory), that we cal! "power set of f", such that if
gis a set (of T" theory) then gEP(f) if and only if gCf. 12
Easy proof, taking into account that if f is a set of T" theory
then P(f) is a set of T" theory such that for each elernent n of
w+, the cornponent n of P(f) is P(f(n)), power set of f(n) (f(n)
being the set of Tw rnodel which is the cornponent n off).
Theorern 1.10. If f is a set of T" theory then there exists a set
U ufx ( of T" theory) that we cal! "union set of elernents of f",
such that if g is a set of T" theory then gEU ufx if and only if
there exists sorne set h (of T" theory) such that hEf and gEh. 13
11By virtue of this theorern of Tw rnodel, the staternent «there
exists the ernpty set» is an axiorn of T" theory and is easy prove,
in a classical way, that this axiorn is equivalent to staternent
of T" theory «there exists sorne set».
12This theorern of Tw rnodel is the power set axiorn of T"
theory.
13This theorern of Tw rnodel is the union set axiorn of T"
theory.
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Easy proof, taking into account that if f is a set of TN theory
then U nfx is a set of TN theory such that for each element n of
w•, the component n of U adx is U nt<n>x, union set of elements of
f(n) (f(n) being the set of Tw model which is the component n of
f) and taking into account the Lemma 1.1 as in above proofs.
Theorem 1.11. If a( x, u) is a formula of TN theory wi th free
variables x, u and no other free variables, a ( x,u ) is a
functional formula of function u 14 and f is a set of TN theory
then there exists a set f' of TN theory such that i f g ' is a set
of TN theory then g' is an element off' if and only i f there
exists an element g off such that the statement a ( g , g' ) (of TN
theory) is true. 15
Proof. It a( x, u) is a formula of TN theory wi th free variables x ,
u, no other free variables and for each element n of w•
'
the
component n of a(x,u) is a formula ~(x,u) then, for each element
n of w·, a n(x , u) is a formula of Tw model with free vari ables x ,
u and no other free variables. If is notan element of Fw the set
of element n of w· such that an(x,u) is a functional formula of
function u then is an element of Fw the set of element of w• such
that ( t being a variable no en a( x, u)) the statement ( of Tw
model) 3x3u3t(an(x,u)Aa(x,t)A(u~t)) is true. Then the statement
14If a( x , u) is a formula of TN theory wi th free va r iables
x, u and no other free variables, then (in a similar way to Tw
model) we say that "a(x,u) is a formula functional of function
u" if and only if for all set g (of TN theory) it does not exist
more than one set g' (of TN theory) such that the statement
a(g,g') (of TN theory) is true.
15This theorem of Tw model is the class of axi oms of
sustitution of TN theory.
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(of TN theor•) 3x3u3t(a(x,u)Aa(x,t)A(u;,0t) )is true and,
consequently, a(x,u) is nota functional formula of fuction u.
Therefore, if a(x,u) is a fuctio~al formula of fuction u (of TN
theory) then the set of element n of w· such that an(x,u) is a
fuctional formula of fuction u (of Tw model), is an element of
Fw. So, if the formula a(x,u) is a fuctional formula of fuction
u and f is a set of TN theory, then is an element of Fw the set
H of elements n of w· such that there exists a set f 'n of Tw model
such that if g' n is a set of Tw model then g'n is an element of
f'n if and only if there exists an element gn of f(n) such that
a(gn,g'n). Then, by virtue of Lemma 1.1, there exists a set f' of
TN theory such that if g' is a set of TN theory then g' is an
element off' if and only if there exists an element g off such
that a(g,g') is a true statement (of TN theory) .
Lemma 1.2. If f is a set of TN theory then there exists a set
of TN theory, which we denote 11 {f} 11 , whose only element is f. 16
Easy proof taking into account that {f} is the set of TN theory
such that for each element n of w• , the component n of {f} is
{f(n)}.
Lemma 1.3. If f, g are sets of TN theory then there exists a set
( of T N theory) which we denote II fU g II and we call II set f un ion
16This theorem of Tw model is a theorem of TN theory ( i t is
proved in the mentioned TN theory, in a classical way,
considering the power set axiom, the class of axioms of
substitution and that the empty set~ is a set of TN theory).
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g", whose elernents are the elernents of f and the elernents of g. 17
Easy proof taking into account that fUg is the set of TN theory
such that for each elernent n of w·. the cornponent n of fUg is
f(n)Ug(n).
Definition 1.1. If A is a set of T~ rnodel or of TN theory then we
denoted s(A) and we say "successor of A" to AU{A}.
Theorern 1.12. There exists a set N of TN theory, whose elernents
we call "natural nurnber" (of TN theory) , such that,
<l>EN and we call "zero" (and "ernpty set" of TN theory) to
natural nurnber el>.
if pEN then s(p)EN.
(Recurrence principle of TN theory) if fCN, <l>Ef and «if pEf
then s(p)Ef», then f=N. u
Proof. We suppose that N is a set of TN theory such that for each
elernent n of w·, the cornponent n of Nis the set W of natural
nurnber of Tw rnodel (that is, we suppose that Nis the sequence
W, W, W, ... of Tw rnodel). Therefore,
Easy proof that <l>EN, taking into account that <l>w is an elernent
of W (since <l>w is the natural nurnber of Tw rnodel that we call
17This theorern of Tw rnodel is a theorern of TN theory ( i t is
preved in the rnentioned TN theory, in a classical way,
considering the power set axiorn, the union set axiorn, the class
of axiorns of sustitution and that the ernpty set el> is a set of TN
theory).
18This theoren of Tw rnodel is the axiorn of infini ty of TN
theory.
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"zero" and also we denote Ow).
If pisan element of N then the set of elements n of w· such
that p(n)EW, is an element of Fw. So, since if p(n)EW then
s(p(n) )EW, the set of elements n of w• such that s(p(n) )EW, is
an element of Fw. So, s(p) is an element of N.
If fC}¡, ~Ef, f~N and H is the set of elements n of w• such
that for each element n of H there exists the first element
Pn of f(n) such that s{pn) is not element of f(n) then H is an
element of Fw. So, if pis a set of T" theory such that for
each element n of H, p(n)=pn then pEf and s(p)~f.
Therefore, if f is a subset of N, ~Ef and «if pEf then
s(p)Ef», then f=N.
Theorem 1.13. If f is a non empty set of T" theory then there
exists a map \JI from P(f)\{~} (set of non empty subsets off) into
f such that if gEP(f)\{~} then \JI (as to map of T" theory) assigns
to g an element of g. 19
Proof. Since the Tw model satisfies the axiom of choice, for each
element n of w• such that f(n)~~w (~w being the empty set of Tw
model) there exists a map \JI" from P{f(n))\{~w} (set of non empty
subsets of f(n)) into f(n) such that, if g" is an element of
P(f(n))\{~w} then \JI" assigns to 9n an element of 9n· · And, since
f~~ (~ being the empty of T" theory), if H is the set of elements
n of w• such that f(n)~~w then H is a element of Fw. So, · by virtue
of Lemma 1.1, there exists a set \JI of T" theory such that for
19This theorem of Tw model is the axiom of choice of T"
theory.
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each element n of H, IJI( n) ( component n of IJI) is a map from
P(f(n))\{~w} into f(n) such that if gnEP(f(n))\{~w} then IJl(n)
assigns to 9n a element of 9n• So, since H is a element of Fw, IJI
is a map from P(f)\{~} into f such that, if gis an element of
P(f)\{~} then IJI, as to map of T" theory, assigns to g , an element
of g.
Therefore, if Tw is a model of ZFC then the empty set is a set of
T" theory and, the mentioned T" theory, satisfi es the
extensionality axiom, the power set axiom, the union set axiom,
the class of axioms of sustitution, the axiom of infinity and the
axiom of choice and, conseguently, if Tw is a model of ZFC then
T" theory is also a model of ZFC, that we call "T" model".
Definition 1.2. If pisan element of N (set of natural number
of T" model) then we say that,
"pis limited" if and only if there exists an element g o~ W
(set of natural number of Tw model) such that the set of
elements n of w· such that p(n)=g, is an element of Fw.
"pis unlimited" if and only if pis not limited.
Theorem 1.14. ~ (zero of N} is limited. If f is a limited natural
number then s(f) is limited. And there exists unlimited elements
of N.
Proof. ~ is limited since ~ is the seguence Ow, Ow, Ow, ··· of Tw
model ( Ow being the zero of W, Ow=~w and ~w the empty set of Tw
model) . If f is a limited natural number then there exists an
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element poi W such that f is equal to the sequence p, p, p, ...
of Tw rnoael. So, s(f), sucessor off, is equal of sequence
s(p), s(p), s(p), ..• of Tw rnoael (s(p) being the sucessor of p)
ana, consequently, s(f) is lirnitea.
Ana an unlirnitea natural nurnber is, far exarnple, g such that far
all elernent n of w·, g(n)=n (that is, gis equal to the sequence
s ( 4>w) , s ( s ( 4>w) ) , s ( s ( s ( 4>w) ) ) , • • • of Tw rnoael) .
Appenaix of first part.
A "filter" on a non ernpty set E is, by aefinition, a subset F of
P(E), such that F is not the ernpty set, the ernpty set is notan
elernent of F, «if A, B are elernents of F then AílB is an elernent
of F» ana if A is an elernent of F, Bis a subset of E ana A is
a subset of B then Bis a elernent of F.
So, if F is a filter on a set E ana A is an elernent of F then cA
(cornplernentary of A with respect to E) is notan elernent of F.
An "ultrafilter" on a non ernpty set E is a filter U such that if
F is a filter on E ana UCF then U=F.
It can be easily provea that if U is an ultrafilter on a set E,
A, B are subset of E, AUB is an elernent of U ana A is notan
elernent of U then Bisan elernent of U (So, if A is a subset of
E ana A is notan elernent of U then cA is an elernent of U).
Using Zorn's lernrna (valia to use in ZFC, because ZFC satisfies
the axiorn of choice), it is provea that if F is a filter on a set
E then there exists a ultrafilter U containing F.
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If W is the set of natural numbers, we say that "F is a Frechét's
filter" if and only if F is a filter on W such that A is an
element of F if and only if A is a subset of W such that •A
(complementary of A with respect to W) is a finite subset of W.
We say that "F,, is a Frechét's ultrafilter" on W if and only if
F,, is a ultrafilter on W containing the Frechét's filter on W.
Second part: Support of sorne principles far nonstandard analysis.
In this Second part we denote O,,, 1,,, 2,,, 3,, , ... to natural
numbers of T,, model (that is, to elements of W) and we denote
O, 1, 2, 3, ... to natural numbers the T" model (that is, to the
elemens of N) .
In the mentioned T,, model and T" model, we suppose defined in a
classical way, the structure of natural numbers and , with its
respective structures, the sets of the integer, rational, real
and complex numbers. 20
Definition 2.1. If f is a set of T" model then we say that f is
a "classic" set if and only if there exists a set A of T,, model
20In the T" model, apart from this classical defini tions,
there exist the corresponding no classical interpretations,
specific of this mentioned model of ZFC.
Far example, the non classical interpretations (specific of T"
model) of arder and of addition on the set R (of real numbers of
T" model), are the following ones:
If f, g are elements of R then we say that,
fsg if and only if the set of elements n of w• such that
f(n)sg(n), is an element of F,,.
if h is an element of R then f+g=h if and only if the set of
elements n of w• such that f(n)+g(n)=h(n), is an element of
F,,.
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such that the set of elements n of w· such that f(n);A, is an
element of Fw. And we say that a set of T" model is a
"nonclassic" set if and only if is nota classic set .
Theorem 2.1. If f is a natural number of T" model (is that, if f
is an element of N) then f is classic if and only if f is limited
(so, an element of Nis nqnclassic if and only if is unlimited).
Easy proof, considering Definition 1.2 and Definition 2.1.
Theorem 2 . 2. If f, g are' natural numbers, f is classic and gis
nonclassic then fsg.
Proof. If f is a classic natural number, then there exists a
natural number p of Tw model such that the set of elements n of
w· such that f(n);p is an element of Fw. And if h is a natural
number of TN model such that hsf then the set of element n of W
such that h(n)Sf(n) is an element of Fw. So, there exists ari
element q of W such that qsp and the set of element n of W such
that h(n);q, is an element of Fw. So, h is limited and,
consequently, if gis a nonclassic natural number then fsg.
Theorem 2.3. If f is a classic non empty set (of T" model) then
there exists a classic set g (of T" model) such that g , is element
off.
Proof. If f is a classic non empty set then there exists a non
empty set A of Tw model such that the set of elements n of w·
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such that f(n)=A, is an element of Fw. Then, there exists a set
B (of Tw model ) such that Bisan element of A and, consequently,
if g is a set of T" model such that for all element n of W' ,
g(n)=B then gis a classic set and gis and element off.
Theorem 2.4. If f is a real number of T" model such that there
exists a classic natural number p such that l f l ~P then there
exists an only classic real number g such that for a l l class i c
natural number q~Ow, l f-g l ~(l / q).
Proof. If p is a classic natural number then there exists a
natural number h of Tw model (that is, an element of W) such that
the set of elements n of w· such that p(n) =h , i s an element of
Fw. And if f is a real number of T" model such that l f l ~P then
the set of elements n of w· such that -h~f(n)~h, is an element of
Fw. So, if a 1=-h and b1=h then the set of elements n of w· such
that a 1Sf(n)Sbi, is an element of Fw. So (since Fw is an
ul trafil ter), the set of elements n of w> such that ei ther
a 1 sf ( n) s ( a 1 +b1 ) / 2w is an element of Fw or the set of elements n of
w· such that (a1+b1 )/2wsf(n)Sb1 , is an element of Fw. Continuing
this way, we define two sequences a 1 , a 2 , a 3 , ••• and
b3, .•• , of real numbers of Tw model such that
a 1sa2sa3s ...... sb3Sb2Sb1 and there exists one and only one real
number r ( of Tw model) such that for all element n of w•, ansrsbn.
Therefore, if gis a set of T" model such that for all element n
of W-, g ( n) =r then ( since for all natural number p~O of Tw model ,
the set of elements n of w• such that -(lw/ 2/)sf(n)-rs(lw/2/), is
an element of Fw) gis a classic real number such that for all
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classi~ natural number q~Ow, lf-gls(l/q).
- Theorem 2.5. If f is a set of TN model then f is classic if and
only if there exists a formula a(x) with a free variable x, with
no other free variables, wi t h no nonclassic constants21 and such
that the statements 31xa(x) 22 and a(f) are true.
Proof. If a(x) is a formula of TN model, with a free variable x,
with no other free variables and with no nonclassic constants and
for each element n of w•, a,,(x) is the component n of a(x), then
there exists a element p of w• such that the set of elements n of
w• such that a,,(x) is (identical to) ~(x) is an element of Fw.
So, if the statement ( of TN model) 31xa( x) is true then the
statement (of Tw model) 31x~(x) is true and, consequently, there
exists one and only one set A (of Tw model) such that ~(A) is
true. Therefore, if f is a set of TN model such that for all
element n of W', f(n)=A then f is classic and a(f) is true.
If, reciprocally, we suppose that f is a classic set ( of TN
model) then x=f is a formula (of TN model) with a free variable
x, with no other free variables, with no nonclassic constants
( since f is a el as sic constant) and such that the statement
31x(x=f) and f=f are true.
21We call "classic constants" to the signs which denote
classic sets and "nonclassic constants" to the signs which denote
nonclassic sets.
22Bearing in mind that the statement 3 1a(x) is equivalent of
statement 3x( a(x)A'v't( a( t) = t=x)).
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Third part: Sorne principles far nonstandard analysis.
We suppose all what follows in a model T" of ZFC in which sorne
principles far nonstandard analysis (compatible with the axioms
of the classic ZFC) are proposed and, asan example, there appear
sorne non standard defini tions and theorems of sorne basic concepts
of mathematical analysis on sequences or standard functions. 23
But, what it not possible at this point (because it would make
the paper too long) is to make a study of the properties
corresponding to the befare mentioned concepts which we define.
Also we suppose all what follows, that O, 1, 2, 3, ... are the
natural numbers, Nis the set of natural numbers and R is the set
of real numbers, of mentioned TN model (and that ~).
Definition 3.1. If A is a set then we say that "a(x) is a formula
defining A" if and only if a(x) is with a free variable x and
with no other free variables and the statements 31xa(x) and a(A)
are true.
Definition 3.2. If A is a set (whichever), we say that,
"A is standard" if and only if there exists a formula
23Such concepts are, far the sequences of real numbers,
those of regular sequence and limit. And, for real functions of
one real variable, those of limit, continuity ata point, uniform
continuity, derivative at a point, derivative function and
Riemann integral.
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defining A, with no nonstandard constants.
"A is nonstandard" if and only if A is not standard. 24
Definition 3.3. If a is a formula (whichever) then we say that
"a is standard" if and only if a is with no nonstandard
constants. 25
Principle l . There exists a nonstandard natural number (that is,
there exists a natural number that is a nonstandard set). 26
Principle 2. If f, g are natural numbers, f is standard and gis
nonstandard then fsg."
Principle 3. If A is a non empty standard set then there exists
a standard set which an element of A. 28
Definition 3 . 4 . If risa real number then we say that,
24In this Third part we call "standard sets" and
"nonstandard sets" to the set that in Second part we call,
respectively, "classic set" and "nonclassic sets". And (in this
Third part) we call "standard constans" and "nonstandard
constants" to the constants that in Second part, we call,
respectively, "classic constants" and "nonclassic constants".
2 5So, if a is a formula wi th no constants then a is a
standard formula. And if A is a set such that there exists a
formula defining A with no constants then A is a . standard set.
26The Principle 1 is supported in Theorem 1.14, Theorem 2.1 ,
Theorem 2.5, Definition 3.1 and Definition 3.2.
27The Pr i nciple 2 is supported in Theorem 2.2, Theorem 2.5,
Definition 3.1 and Definition 3.2.
28The Principle 3 is supported in Theorem 2.3, Theorem 2.5,
Definition 3 . 1 and Definition 3.2.
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"r is limited" if and only if there exists a standard natural
number p such that lrl~p.
"r is unlimited" if and only if r is not limited.
"r is infinitesimal" if and only if either r=O or 1/r is
unlimited.
"r is appreciable" if and only if r is limited and is not
infinitesimal.
Principle 4. If risa limited real number then there exists an
only standard real number ºr, that we call "standard part of r",
such that r-ºr is infinitesimal.n
Theorem 3.1. If E is a set then «E is standard if and only if
there exists a standard formula ~(t) with only a free variable
t and such that E={tl~(t)}».
Proof. If E is a standard set then tEE is a standard formula with
only a free variable t and such that E={tltEE}.
Reciprocally, if E is a set and ~(t) is a standard formula with
only a free variable t and such that E={tl~(t)} then
Vt(tEx <:e:>~(~)) is a standard formula with only a free variable
x and is a formula defining E. So, E is standard.
Theorem 3.2. O is a standard natural number, 1 is a standard
natural number and if n, pare standard natural numbers then,
n+p, n.p are standard.
29The Principle 4 is supported in Theorem 2.4, Theorem 2.5,
Definition 3.1, Definition 3.2 and Definition 3.4.
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if r~o then nP is standard.
Poof. Vt(t~) is a standard formula (since is with no constants)
defining O (since 0=~ and the statements 31xVt(t$x) and Vt(tj!:<l>)
are true) and, consequently, O is standard.
Vt(tEx e:, t=O) is a standard formula defining 1 (since O is a
standard constant and 1={0}). So, 1 is standard.
And if a(x) is a standard formula defining n (with free variable
x and with no variables t, u) and ~(t) is a standard formula
defining p (with free variable t and with no variables x, u)
then,
3x3t((u=x+t)Aa(x)A~(t)) is a standard formula defining n+p
(with free variable u) and 3x3t((u=x.t)Aa(x)A~(t)) is a
standard formula defining n.p (with free variable u).
Consequently, n+p and n.p are standard.
Similar proof to the preceding.
Theorem 3.3. N and R are standard sets and the class of standard
natural number is nota set (of T" model).
Proof. By virtue of axiom of infini ty, there is a standard
formula defining N 30 and, consequently, Nis standard.
By virtue of classic construction of real numbers, there is a
standard formula defining R and, consequently, R is standard.
By virtue of Thorem 3.1, O is standard and if nis a standard
30So, for example, the formula (~Ex)AVt(tEx = s(t)Ex)A
AVu(((uCx)A((~Eu)A'v'y(yEu = s(y)Eu))) = u=x) (with only free
variable x and wi th only standard constant ~) is a standard
formula defining N.
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natural number then n+l is standard. And, by virtue of
Principle 1, there exists a nonstandard natural number. Then,
considering the classic recurrence principle, the class of
standard natural number is nota set (of TN model ) .
Theorem 3.4. If pis a natural number then pis (a real number )
limited if and only if pis standard.
Easy proof considering Principle 3.2 and Definition 3.4.
Theorem 3.5. If risa real number then,
r is unlimited if and only if there exists a nonstandard
natural number p such that p< l rl.
r is infinitesimal if and only if there exists a nonstandard
natural number such that lr l <l/p.
r is appreciable if and only if there exists a standard
natural number p~O such that 1/p< l rl<p.
Easy proof considering Definition 3.4, Theorem 3.2 and recurrence
principle.
Theorem 3.6. If risa real number then,
if r is limited then -r is limited.
if r is limited and sis a limited real number then r+s, r.s
are limited.
if r is infinitesimal then res limited.
if r is infinitesimal then -r is infinitesimal.
if r is infinitesimal and sisan infinitesimal real number
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then r+s is infinitesimal.
if r is infinitesimal and sis limited real number then r.s
is infinitesimal.
if r is appreciable then 1 / r is appreciable.
if r is appreciable and sisan appreciable natural number
then r.s is appreciable.
if r is standard then either r=O or r is appreciable.
if r is standard and sis a standard real number then r+s, r.s
are standard.
if r is standard then «-r is standard and if r-o then 1 / r is
standard».
Easy proof considering Defini tion 3 .1, Defini tion 3. 2, Defini tion
3.4, Principle 2, Principle 4, Theorem 3.2 and Theorem 3.5.
Defini tion 3. 5. I f r, s are real numbers we say tath "r is almost
s" if and only if r-s is infinitesimal.
Definition 3.6. If r 1 , r 2 , r 3 , ••• is a standard sequence of real
numbers then we say that "r1 , r 2 , r 3 , ••• is regular" if and only
if, for all unlimited natural numbers n, m, rn is almost rm.
Definition 3.7. If r 1 , r 2 , r 3 , ••• is a standard sequence of real
numbers then we say that "r1 , r 2 , r 3 , ••• is convergent to r",
"r=limnrn" and "r is limit of r 1 , r 2 , r 3 , ••• " if and only if r is
a standard real number and for all ilimited natural number n, rn
is almost r.
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Definition 3 .8. If f is a standard function frorn an open real set
o into R and sis a standard elernent of adherence of D then we
say that r•lirax_. f(x) and "r is the limit off at point s'', if
and only if risa standard real number and for all element u of
D such that u is almost s and u~s. f (u) is almost r.
Definition 3.9. If f is a standard func tion from an open real set
D into R and sis a standard element of D then we say that " f is
continuous at point s" if and only if there exists
l imx-.Í ( X ) • f ( S ) •
Definition 3.10. If f is a standard function from an open real
set D into R then we say that "f is uniformly continuous" if and
only if, if r, s are points of D and r is almost s then f(r) is
almost f( s ) .
Definition 3.11. If f is a standard real function from an open
real set D into R and sis a standard element of D then we say
"derivative of f at point s" to limxx-s( ( f(x)-f (s)) / (x-s)), if
there exists this limit.
Definition 3.12. If f is a standard real function from an open
real set D into R then we say that "f' is the derivative f unction
off" if and only if f' is a standard function from D into R such
that for all standard point s of D
f 1 ( X ) • limx-s ( ( f ( X ) - f ( S ) ) / ( X - S ) ) •
Theorem 3.7. If f is a standard function from a open real set D
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into R gis a derivative function off and h is a derivative
function off then g=h.
Proof. Then, if E={x l (xED)A(g(x)~h(x)} then E is standard (since
g and h are standard). So, if E~~ then (Principle 3) there exists
a standard element s of E and, consequently, g(s)~h(s). So (since
if sis standard then f(s)=g(s)) E=~ and, consequently, g=h.
Definition 3.13. If b, e are standard real numbers, b<c and f is
a standard function from [b,c] into R then we say that
r=J¡b,c Jf(x)dx (x being a variable) and that "r is the Riemann
integral of · f on [b,c]" if and only if r is a standard real
number such that if x 0 , x 1 , ••• , xn are n+l real numbers (n being
a natural number) such that b=x0 <x1< ... <xn=c and for all natural
number j such that 1:Sj:Sn, xJ-i is almost xJ and l;J is a real number
SUCh that XJ_1 :Sl;J:SXJ, then r is almost ÍJe{l ,..,n} f(l;)(XJ-X J-l) (that
is, J[b,cJ f( x )dx=º( f( l; 1 ). (x1-x0 )+• • • +f( l;n). (xn-xn-l))).
Remarks.
It is easy to prove that the class of real limited numbers, of
real infinitesimal numbers, of real appreciable numbers, real
standard numbers and the real unlimited numbers are not sets. And
the same happens wi th many other class whose elements are
sets. 31
31Since if, for example, there existed the set C1 of the
limited complex numbers then {xi (xEN)A(xEC1 )} would be the set of
the standard natural numbers, a set which does not exist.
1 (,<)
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If in definitions 3.6, 3.7, 3.8, 3.10 and 3.13 we replace the
terms "unlimited" and "is almost" for, respectively , the terms
"very advanced" and "very close", · we get definitions often used
by teachers of mathematics and by other professionals who, as
physicists, chemists, engineers and economists, make use of
mathematics, even thoughsuch definitions of "their" mathematical
analysis (the standard one), what they are forgiven on account
of how in tui ti ve and simple they turn out to be . And the
proverbial nuisanse of reasoning in standard analysis dueto the
"paces within limit" because of the resulting consideration of
the tradi tional "very big" natural number n0 and "very small"
real numbers E (epsilon) and ó (delta), can be avoided in
nonstandard analysis and doing so get that reasoning be "more
fluent" (the way algebra does) if, for example, the definitions
expressed are used and, in particular, the Riemann integral is
defined as the standard part of a certain fini te addi tion
(unlimited but finite).
The non standard definitions 3.6, 3.7, 3.8, 3.10 and 3.13 are
equivalent to the respective standard (or classical),
notwithstanding, to use such non standard difinitions , it is not
necessary to take into account such equivalencies, nei ther is ,
to use the referred to classical definitions.
We will not preve here the befare mentioned equivalences, but let
us say at least, that are a consequence of the above mentioned
priciples.
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REFERENCES
[l] KRIVINE, Jean-Louis: "Théorie axiomatique des ensembles".
Presses Universitaires de France. Paris. 1969 .
[2] NELSON, Edward: "Internal set theory: A new approach to
nonstandard analysis". Bulletin of the American Mathematical
Society. Volume 83. Number 6. Page 1165-1198. November 1977.
[3] ROBINSON, Abraham: "Non standard Analysis". North Holland.
1974.
Manuel Suárez Fernández
Profesor de Matemáticas
de la
Facultad de Ciencias Económicas y Empresariales.
Campus Universitario
Plaza de la Universidad, 1
02071-ALBACETE
e-mail: msuarez '@ ecem-ab. uclm. es
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