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16 Chap. i Rules of the Road AxiomaticSystems

investigation shows that Axioms2 through 4 are all correct' Therefore'

Ax-

iom 1 is independent;;;;;;;"ining three'The reader is encouraged to flnd

models that demonsrr"t" irr" i"o"p"rrld"rr""of ,q^io-* 2 through 4. o

The property of completeness is also concerned with thesize of the axiom

set. wherea, i"d"pJ;:;:;;;;;;o 't'utour set or axioms was not too

large, completeness gti'"tlJ"hat our tt'o'"" *iotttsare sufficient in number

to prove or disprove ;;;;;-"it tnil."iit"s concerningour collection of

undefined terms' W" *v'tttut an axiom '"tis oisumcient sLe ot-complete \f

it

is impossible to add iil["it""J;;;;"tt u"a na"p"ttdentaxiom without

aaoneaaaitrH:i:lfi :t*ll"Tl""'n""1:'l:'^:::':,t;ffi Aswithconsis-

tency, the failure to fi;d a newconsist""t' ;;;;";dent"axiom does not

elimi-

nate the possibility ";il ;;;""ceand trr"r"rotJitun insufficient procedure by

which to p.ou" "o*pf"i"*"'W" tu"' t'o*"i"'' rr'" ttt" isomorphism of mod-

els to demon't'ut" tio'm of complete"""' ii;i;odelsof a given axiomatic

system are isomorpil;' ;; irt" **'ot utl*is said ro be categ'orical' Ttj

nrooertv of "ut"go'r"ii""" "u"ut 'tto*"

io iryptt completeness; the proof'

il,##i, i". ;;;?; ffi;;;" "t 1ry' discussion16

In the next section' we will illustratethe properties of axiomatic systems

ina geometric context'

EXERCISES 1.2

The Axiomatic Method

To answer Exercises 1 through 4' usethe axiomatic system outlined in

Exam-

ple 1.2.1'

1. Prove: A Fo cannot contain threedistinct Fe's'

2. Prove: There exists a set of two Fo's that contains all the Fe'sof the system'

3. Prove: For every set of two distinctFe's' every Fo in the system must

contain at least

one of them.

4. Prove: A1l three Fo's cannot containthe same Fe'

5. consider the following axiom set inwhich x's, y's, and "on" are the unde{ined

terms:

Axiom 1. There exist exactly flve x's'

Axiom2.Anytwodistinctx,shaveexactlyone}onbothofthem.

Axiom 3. Eachy is on exactly two x's'

How many y's are there in the system?Prove your result'

To answer Exercises 6 through 9' usethe axiom set

_ofcomp1etenessandcategorica1ness,seeJJ.ow?:9I'"*Founda-6 Fo, u detailed discusions and Fundamentar ,"iiio.zlr'i"ii-"*r,i6

(B.jr'# rw's-rENr Publishing co'' 1990)'

pp. 160-162.

in Exercise 5.

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Sec. 1.2 Axiomatic Systems and their Properties 17

6. Prove that an)r two l"s have at most one x on hoth'

7. Prove that not all -r's arc on the same y.

8. Prove that there exist exactly four y's on each 'r 'g. prove that for an] )1 and an] )1 not on that x1 there exist exactly two other distinct

l"s on 11 that do not contain any of the i's on ,)'1'

Models

l0.VerifythattheaxiomsinExamplel.2.2are..correct,'Statements.11. Verify that Axioms 1 ancl 3 in Example I.2.3 are "correct" statements and explain

why Axioms 2 and 4 are not correct.

12. Verify that the model inExample 1.2.5 is isomorphic to the modei in Example 1,2'2,

13. Devise a one-to-one correspondence between the undefined terms in the models in

E'xamplesl.2.5andl.2.6thatisanisomorphismandverifyyourresult.14. Devise another model that is isomorphic to the one in Example \.2.Z.Find a model

that is not isomorphic. if possible.

15. Devise a model for thc axiom system described in Exercise 5'

16. Consider an inflnite set of undeflned elements s ancl the undefined relation R that

satislics thc [ollou ing axioms:

Axiom l. 7f ,r, b e S and aRb, then a + b'Axirrm2. Itia. b, c e S.aRb,ancl bRc.thenaRc

(a) Show that an interpretation with s as the set of integers and aRb as "a is lessthan b" is a model for the syslem'

(lr) Would , as the set of integers and aRb interpreted as "a is gleater than b" alsobe a model?

(c)Are

the models in parts (a) and (b) isomorphic?

(d) Would S as the set of real numbers aIKl aRb interpreted as "a is less than b"be another model?

(e) ls the model in part (i1) isomorphic to the model in part (a)?

Properties of Axiomatic SYstems

L7. Devise two additional concrete models for the axiom system in Example 1'2.1' Are

these models isomorphic? Justify your result'

18. Devise two additional abstract models for the axiom system in E'xample 1'2'1'

lg.ArethemodelsinExercise16concreteorabstract?Explainyoulanswef.20. Explain why it is not possible to devise a conclete model of the real number system'

21. Demonstfare the independence of Axioms 2 through 4 in Example 1.2.1.

22. Devise two concrete models lor the axiom set in Exercise 5' Are these modelsisomorphic? Are they isomorphic to the model found in Exercise 15? Justify your

results.