E
lf|Er
SvsrBus
Rules oF THE Rono: Axtotuettc
1.2
E
lh ti
Axiomatic Systems and Their Properties
knoun
384-CA' 322 B'c') Aristotle is
of ancient times' At the age of sevm-
tIus
,ft m:--tr
-Fr
ft
il,-
-m= :
1t1f nilirp:
ri
--
Tb
n-tr:=Erl
tsrrr'
n:Stqr*rFx t
--t--t nsED-
1||:nS
-tl=L-E
for
J
fl:O=id
f,.lts ;IETEL' ':': - ;l
The icr.--i s-
aII of Greek mathematics'
:aEunderimportant that we have a basic
in the develoPment of all of modern
--lf
t ! E: -air:-i:\=+ i
n: :c-':f Er-f3 u-I
irz<
CL-:
=
!.fl
Drcr?
td:=.-J
/r;!'Tt-
3nt Jetn
tr{-f l\]r}EIZIt
mu-r sup'Pi1 i.r
.-1
trirc- -e
riit*G'. *?riEFDLi t0
tc d':e- ge ir-rtf,: ''hd
E trt-ri --:'-tleriq-. cot
Iue =.remens. '-eik
-Inr&s'
r::airmnu'
f
urnrnr"ize- :n1 -q:ic=l
:=!EtE
lL1
tbe F€Jtern
t
rr
ani-rmatir- :)l
tEorr-\ anJ tbecrern
f ;frm-rce ;n
E rryC-i :rr-rr :.f :tr-czl
E:E
\5j
--L1
reT!-e-= E
: FfiE.l;t\€tsi
m -r=E={-i
ltirm:-::rad
-idefinidons, or
it
must stoP at some
AxtoMATIc Sysrtus nuo
Tugtn pRopERTTES g
Tabte 1.2.1
The Axiomatic Nlethod
1n)/ axiomatic system
must contain a set of technical
terms that are de_
-:berately chosen _J
as undefined
u'sulrsrr r..rn,
rsrms and
uJ
u, subject to the
are
interpretation
tf the rcader.
- {t
other technicar terms
of the system
are ultimatcly defined
by means
of the undefined terms' These
terms ur"irr. definitions
of the
system.
-'
The axiomatic system
contains a set of starements,
dealing with undc_
hned terms and definitions,
ttrat arc-c;;r;; ," remain
unprovcd. ,l.hese
are the axioms of the
system.
:
r\ll
other statemcnts of-the
systcm must bc rogicar
"
conscquences of thc
derived' tot"m'n
tT;"#:
rs-
r;[?:
.:I:"il;,testrnsaristying,
*. -r.J,n",h."*il;'
;,i.
o*-
so the collection of definirions
musr end at
:il,;il;;"^.:,::,:;:1,::r'::;7,;:,,:,":,;::^:i;1T1,:*;, j$::
and definirions caniow
be combined inro the
n-'.,1n;|f,;:l;.::i:,
stare_
our axromatic systcm.
For these
ematical value, we m
th.o."_. in i.
nowneedadditionar*:Fl{.i'.."X,:',,f+fi1i;l,i,J[:f
or
_
;li*l j:
^ott
proof' As before, we ftrrm
o.rruin'orr",ateme:ts that
leads us to the concru_
sron that' to avoid circularity,
un. * r"r.
these statements mrs.
unproved' These statements,
remain
called c xir,tns or",.
and lbrm the fundamenral ,,rrurhs,,*
.postLtlules,,L3 must be assume.
;il;;axiomatic system.
To summarize, any logical
d*.top*"it
of an axiomatic system
therefore confbrm
must
to
r
he
altern
..or.r.ii.i',n
Table 1.2.
lb
I
.
illustrale an axiomatic
system and the relarinn.hi^- .._^
^ e d rerm
nr
s, u,
io
rr, lna
tn"
o."rr,
*i ffi ;.#j:: il: iiltffiJ:*;ll.
fil'ffiff*il
':;:i.'y:,:ff:;::,;::ii::t
un,u..sat truth
I'i*'l-"1:'ricalrv''1hcw'rd
oppi,"our. ,"i:i#:rtff#,tr:-subjecr arca,
rrurh of rhese o*,:'o..
)ms rs not at issue_just
the reader,s wiltingnes,
as true.
to u.."pt tt,".
r+The
10
-LE:ra{ff
SYsrEMs
Rules oF THE Ronp: AxIoMATIC
L L if,rE
ExamPle 1.2.1
A simple abstract axiomatic system
and the relation "belongs to'"15
Undefined terms: Fe's, Fo's'
lI-qSC',ip-'=t
istinct Fe's in this sYstem'
ng to exactly one Fo'
same Fo'
to
ntain at least one Fe that belongs
both'
Fe-Fo Theorem
1'
one Fe'
Two distinct Fo's contain exactly
we
Fo's conain at least one Fe'
no more than one
Fe' For
this
Fo
ow
cts
we are done.
Fe-Fo Theorem
2'
re r:f,
rriEEr:rE[Dr
:EErrc
There are exactly three Fo's'
isrxn se &;rsaJl
rEi-.lGEc +sEn- F''+r t
tsprncs
f.re
Srnce 6ese trrm< r
--d. treaJ€rrnat r-hclse
Dfng -l tmfufnal term in a 1
olthar +sem- [f. itx s
?m-.
E -Jrr€ct-
statefileoLi- se
c
htrlrr
lEmie I'-l rle can desipate 6
b edoms becorne d€ folk
G,
l. There are er.arlY $ru
rr- 1 Tuo distinct PeoPle bt
lr- 3. \ot all peoPle belong
rr- { -\nt' Cuo distinct coml
-
to both.
I-:6e people be Bob. Ted. and
u'B.ir and Ted), Finance tTaJ
{E-[ js€ Frg. 1.2.1). We can see d
d *ments. and therefore this ir
i
QrJ3
Fe'Fo Theorem
3'
that belon7 to
Each Fo has exactly two Fe's
2
that each Fo has at le
exactly one. Axiom I
Proof. BY Theorem
il'
Fo's' Now Axiom
f
Fr*--rFrle 1.2. I we can designare
G'rcs. and the relation "belongs tc
tr
IFoP:""*:']:i:T
from contamlng
two Fe's-
u
t
in::me the following:
-lritm 1. There are exactll'the
-frbm 2. Any two books are or
-frbm 3. Not dl books are on I
-fri{m 4. Any two distinct shel
*nce Axioms 2 and 4 are not r model for the system.
Smce the theorems of anY arior
in the Exercises'
"a Fo contains a Fe'inology "a Fe is on a Fo" or
gs to" the Fo'
ft rtioms of the system, their talid
L undefined terms. One of the mai
lem is that ail theorems of the ry
nbefore,
the theorems proved in
d
AXIO]VIATIC SYSTEN{S AND THEIR PROPERTIES 11
In the following section we will investigate the consequences of giving
:ome type of meaning or interpretation to the undefined terms of our axiomatic
s\ stem.
\Iodels
In the previous section we discussed the axiomatic method and an exirmple
of
l abstract axiomatic system. Each axiomatic svstem contains a number of
Eramplc 1,2.2
,r Example 7'2.1 we can designatc
the Fe's as people and the Fo's as commrt.:es. and the axioms bccome the lbllowing:
Axiom 1.
Axiom 2.
Axiom 3.
Axiom 4.
There arc exactly three peoplc.
Two distinct people belong to cxactly one committee.
Not all pcoplc belong to the samc committee.
Any trvo distinct committees contain one pcrson who beronas
to borh
Lct the people be Bob, Ted, and carol, and the committees be Entertain_
rent (Bob and red), Finance (Ted and caror), and Relieshments (Bob and
carol) (sce Fig. 1.2. l). we can see that as a collection these axioms
..corare
..ct" statcments, and thercfbre this interpretation is an example of a model.
Eramplc 1.2.3
Erample l-2.1 rve can dcsignatc the Fe's as books, the Fo's as horizontar
.he l'es, and the rclation "belongs to" as "is on" (see
Fig. ) ,2.2); the axroms
:len becomc thc lbllowing:
'r
Axiom 1. There are exactly threc books.
Axiom 2. Aoy two books are on exactly one shcli-.
Axiom 3. Not all books are on thc same shelf.
Axiom 4. Any two distinct shelves contain one book that is on both.
Sincc Axioms 2 and 4 arc not .,correct" statements, this intemretation
is
:lot a model for the system.
r_'a- v_rj
tz
RuLES oF THE
Rono: AxloMATlc Svsrelts
Refreshmenl
Entertainment
Eirr, rtr
L-t;
f =-g----- -=::.=<:-€ - - - F- :r==:-:- F:-ft
'"trilrffinrH ::1.--_:::i 3\_-i--. -i!-J
Finance
3:r*,?:l ffle*l,S#l'
r,nun.e, and Rerreshments
Figure 1.2.1
1 =:rrrnE - ] -:. -S --=t-=--' ;g;1
{Gi Ift l:q -:-i:--;.
l,&.r=:- 1-.; tf '^If,.-hF.:s-i L1-e -l
llf llt :rL-:. j--3T. ;e -:13 : :*iia€i
I r tr-|: - '-: . a: ;;t;'-- 1;1-'; :-l
tcs=Sl{-:_: alrlels'5e ca-t _i3i: -'i
-- :
-t :E--!---E- -: .-i: n.rl:i:ri::taiel3:
r-tl
lbe
:i'
ffiJri
SOO1
trf EE -t:rit-_:€ 1i n6t ,tlr a\.-r inielr:r-. .lrt! i.Ot ensUre i-S Jt13i
E I :B:Er :: the a\iL1matrJ ii:-J
G+fr. :i:=er morlels $h--^-:-.3:
z-:e-- ---alng modeis. :: :r l-q
llr;p._- f.a rall closer in_iFe-:.r_-El iarr:- -r3 mean dat theie :r-'
r fi:-r::.lnon of each st .rl .ii
E.r: - undefrned terms in.'re
.--=i-r-r\=!.-:. in the second rD.f
ErE E: -id to be isomorPhic. .r-.
5. ;:F;=r.itm.
tilI*e
1:.-l
l;n<rk rte ariomatic s)'stem in E
& L-:-T{T-i of H1 annispon on C-;
4 \:--r-fet on Nantucket I-'la::d.
l:. i.l.3t. This model is ir'n
5r-de lJ.5
- =e Fe's of our axiomatic slit
- : P. 0. R), and let the Fo's h
n j hrs modei is isomorPhic tu'I
E-:Ce -irTespondence:
P € Bob iP.Q:
O +> Ted {P. R;
R <+ Carol {0. Rl
I:rs
T-1
Figure 1.2.2
model is also isomorPhi;
illustrate the fact that ever'
interpretations of the unc
f==nt
AXIOMATIC SYSTEMS AND THEIR PRopeRTTgs
13
ation in
s: Each
loms are not
n found
the ax-
satisfied.
stud.y of axiomatic systems. Suppose
that
of a
r which we are unsure of the existence
ether the statement is a theorem or not).
By
investigating
models we can gain insight into the correctness
of the .sratement.
In particular, if one moder exists in wriich
the statement is incorrect, we can be
assured that no proof of the statement exists.
The reader is cautionecr, however,
that the reverse is not always true. Simply
because a statement is correct in
may not
to create
different
sentially
between
the interpretation of each set of undefinecl
terms such that any relatronship
between the undefined terms ir.r one model
is preserved, under that one-ro-one
correspondence, in the second model,
Two
moders that are essentiauy the
same are said to be isomorprtfu:, and
the one-to-one correspondence is
cailed
an isomorphisnt.
Example 1.2.4
Example 1.2.5
Let the Fe's of our axiomatic system
be rnterpretecl as the letters in the set
s : {P, Q, R}, and ret theFo's be interprefed
as all the two-erement srbsets
of 'l' This model is isomorphic to the
moder
in Exampie r.2.2 under rhe one-
to-one correspondence:
P
0
R
+ Bob {p, e} <-> EntertainmentCommittee
e Ted {p, R) e Re freshments Committee
<-> Carol
{e,
R}
<-> Finance Committee
This model is also isomoryhic to the
model in Example 1.2.4. Why?
To i'ustrate the fact that every one-to-one
correspondence between two
different interpretations of the uniefined
te-.,s of some axiomatic system is
14
RuLES oF THE Roe.o:
Axlounrtc
$:'-i
SYSTEMS
U
lr.:'
Yawth
m lE =-=:r- --i <-;- lt
- ==:r-::-. :rr:.-ic
l[rr*s
of -fsiomak Sp
-.- n:s =N-r=-r .s f-s ,ti
.t:'rR--?r{-._
er_r:.s
-\ sr
::N rfiJG-< a
=r.
-!-!
\\llt-1i] r'r'r.
i1
---r-3I
rea 1a-. irrd rur$sr rudr
-iltrssE:L--. ,-; :-n -rtit-rrl.38r- S\Sl
- .fl:--sti rn --re la-.t:sr---lol
l{:r;e-: rial'he sp'arared i-nu
t8r=
ffi.
Nantucket
Sound
Er:uiL1n-. u.f tbe unjeEneJ
E. =i- z,f-ii. an'l rli dc_;r4--'
-- -Ai :rre talten frort si---rr,e I
r _ - : :_i tn e1ample Oi a .--.-11-;
s.[ ]bsrsr model.
&
a--= a concrete modei ha-r I
-i.---J-? consistencv of tlLi
lm,:r:--: deduced trom the aritl
Nantucket lsland
E
Figure 1.2.3
i
rr-rrld, uhich u'e acccpr j
ae ihall see. horverer. thar
LZ a\ s possible. since ir n:a1
=e-
r
the following example' in which the
not necessarily an isomorphism' consider
preserved'
rJationsnip bltween the undeflned terms is not
Example 1.2.6
: lx' )' z]' and let
Let the Fo's be the elements of the set T
(tx, yi, {x, z}'and {-v' z) arethe
representedUytnepairof Fo'sthatshareit.
one-to-one correspondence
Fe's of our model.) Now consider the following
this model and the model in Example l'2'5:
each Fe be
between
{x,y} <+P lP,Q} +z
{y, z} <+ Q {Q, Rl <> }
{x,z} <+R {P'R} ex
.2.5 share the Fe R' while
hare the Fe [x, Y] and not
Therefore, the illustrated
very important role
that the creation of models will play a
and determining
models
in creating
in our future study of geometry' Practice
relationships can be found in the Exercises'
We
witl soon
iro.o.ptti.
see
Lr
=r :r:rr.ple.
should an arion:ari
q-e i.l.l require an infinite nur
*:r_:5e
impossible since. in th
=t*';rion of "things" that u'or:ld
:x.nir>h a model using concept
::ei.rency we are willing to a_.s
fE -u axiomatic system is co!
:ur :r.odel is consistent. \\'hen d
:srl
the relative consistenc\ oi
are said tobe relativelt c
il'=ms
Tbe next two propenies of ;
a*ndence and completeness. h
E :roperty of consistency. Thi
l:csistency prope(y, we do not
r
-:perties to be useful (worthl o;
.l=.-us for us to employ systems tl
-\n individual axiom is said rr
from the other axioms in r
r=ced
be independent if each of irs a
AxtoHr.trrc SysrErvrs .{ND THErR pRopERTrEs
\ll"t,"t
15
In thc next section we shall briefly investigate several important properties
.rat can be exhibited by axiomatic systems.
Properties of Axiomatic Systems
- re most important and rnost fundamental propefiy of an axiomatic system
. cortsistenc-v. A set of axioms is said to be consistent if it is impossible to
-r:duce from these axioms a thcorem that contradicts any axiom or previously
:ror'ed thcorcrn. without this property, an ariomatic system has no mathe:atical value. and furthcr study of its properties is useless. To establish thc
:r-'nsistcncy of an axiomatic system. rve rvill makc usc of modcls similar to
rrcsc discusscd in the last section.
Models may be sepiratcd into trvo categories: (l\ c'ctncrete moclels, where
rrrterpretations of the undefined terms are objects or rclations adaptccl from
.:e rcal world, and (2) abstract modcls. whcre interpretations of the unde:ncd terms are taken from some other axiomatic system. The model in Exam-
:le 1.2.4 is an cxarnplc
. in uhslracl motlel.
clf a corrcretc mcldcl, ll,hereas the clne in Exarnnle I .2.5
when a concrete model has been procluced, u'c
'-|,c absolute
consistcncy of our axiomatic systern.
claim to have established
otherwise, contraclictory
-hcorems deduccd from the axioms ri'ould havc contradictory counl-erparts in
,he rcal world, which wc acccpt as in-rpossible.
we shall sec, horvcver, that the establishrncnt of absolute consistcncy is
nich the
hFebe
are the
r)ndcnce
rot always possible, since it may not be possible to establish
while
and not
,ustrated
rant role
srmlnlng
concrete nrodcl.
''iould be impossible since, in thc real u'orld, Lhere does not cxist an infinite
-'ollcction of "things" that u'oulc] servc as an interpreLation. In such cases we
r'stablish a rnodcl using conccpts from somc other axiomatic svstenr whose
consistency wc arc willing to assume (i.e.. the rL-irl numbers). In other worcls,
-he new axiomatic systen is consistcnt if rhe one in ri'hich we have cnosen
our urodel is consistcnt. when this situation arises, we claim to have cstab-ished thc relativ'e consistency of our ixiomatic s1,stcm, and the two axiomatic
s)'stcms arc said
R^
a
ior example, should an axiomatic system sirnilar tcl the clne cliscussed in Exinple | .2. I require irn infinite number of Fe's. the creation of a concrete rnodel
t<t
be relaLively con,sistent.
'l-hc next tu'o properties
of axiomatic systems that we will discuss, inJcpendence and completeness, have a nature that is distinctly different from
rhe property of consistcncy. 1-his dift'erence lies in the fact that, unlike
the
consrstency property, rve do not require that a.riomatic s;,srems possess
these
propertics to be useful (u'orthy of study). In fact, later it may prove advantageous for us to employ systens that do not posscss these properties.
An individual axiom is said tobe indeperdent jf it cannot be logically clcduced from the other axioms in the svstem. Thc entire set of axioms is
said
to be independcnt if cach of its axioms is indepenclcnt. It should bc clear to
:;
L6
RuI-es oF THE Roeo:
Axlovhttc
-
SYSTEMS
n :t -:
thereaderthatanaxiomaticSystemshouldnotbeinvalidatedsimplybecause
f
:4
-
\---:.:r- ;: ----
..:
--.-
itsaxiomsarenotindependent.Theworstthatcanbesaidisthatanynonindependent system has some redundan
gene
coul-d also appear as theorems' In
f-rrrcbe
independent axiom set, srnce anY
nu*t.t of assumPtions' It is ofte
ioms that is not independent' For examp
most
depend
often a
in
metry class
1
useful
an
and the difficulty of its proof, using
ng UetrinA
and
fJ
b rs;ceric
'T,
arily pedagogical' is that
arly in the development'
E:-l= i.-::--:=r - :::
I
-',3:-.
r:i-
{' l-,-.e: ,\:; ti:I
-
._e
ments'
-_-.:jgl
';e
:.-.
=:-1
,
F:,'=:r
-
';
--
I
:icienneC ler:.:
-$iom l, TL*=:
-lxiom
-lxiom
-lcl\i
ITISJI\-
t. .\rr\ ti
je=
3. E;r '
r'S :,ie -'::
l: :r:-.;:: Erercise s 6 throu:: i
6 kore that anr l\\o '.'s
-. Pove that not ail .r's
& kore
guaranteed that our set of axioms
iom set. Whereas ina"p"na"nte
^y
F,-.:
_a_ _1iri.-:r-
_
since only correct
undefinedterms.WesayttratanaxionSetiSofsufficientsizeorcompleteifit
isimpossibletoaddanadditionalconsistentandindependentaxiomwithout
!:::.
l-.-:
mathematical maturity'
of an axiom, we will again make use
To demonstrate the independence
which one axiom is incorrect and the
of models. ey p,oOu"1ng u *oO"f in
toi""t' we ensure the independence-of the first axiom'
remaining axioms
from correct stare^,"
statements may be logically deduced
in numtiat our,chosen axioms are sufficient
large, completeness gti"t^"tt"t
of
collection
our
statement that.arises concerning
ber to prove o, oi*pr;u"
l:-;: -\ F-.:-:.-::.-t
J::-.e: i}-e:e:\r:--< ' !
:1
maybeincludedusana*iom,thusmakingitavailabletostudentswithless
concerned with the size of the axThe property of completeness rs aiso
was not too
:- -
E
axiom set' may preclude
In such cases the theorem
its use for all but an ex'perienced iathematician.
ExamPle 1.2.7
and
elements in the set lA' B'C' D)
Let the Fe's in Example l'2'l be the
Dj'clearly'
cl' lA' D\' and {B' c'
the Fo's be the subseis lA' Bl' tA'
it contains four Fe's, but careful
Axiom 1 is inconecti" itir.noi.t since
4are all conect' Therefore' Axiom
The reader is encouraged to find
nce t'rf Axioms 2 through 4'
\tetbod
9.
"
that there eus::
kcve rhat ror an)' -, .
lro other distincr '.'s :
ltld€ls
10. \-erifv that the a\rorai
11. \-erit'v that A.xiom; l
adding additional undefined terms'
Testing for comPleteness is anot
f2.
ments and explain u b,
\'erifi'that the mole-
in Example 1.2.1.
a one-to-one ci
the models in Exampi
rerit'v I'our result.
L1. Devise
then the set
ness can be s
scoPe of this discussion'to
Founcategoricalness' see HowardEves'
Pleteness and
:-: ;rl Fundam-enml Coriczi:.
r't- .:: ^60-i62.
---
..