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ORDINAL REAL  NUMBER 3. The techniques of transfinite real , surreal, 

      ordinal real, numbers ; unification .

 

By Dr. Constantine E. Kyritsis

Department of Mathematics and Computer Science

University of Portsmouth

 Software Laboratory

Electrical and Computer Engineering

National Technical University of Athens

http://www.softlab.ntua.gr/~kyritsis

kyritsis@softlab.ntua.gr

 

 

 

 

 

§ 0  Introduction . In this third paper on ordinal real numbers, it is proved that the three  different techniques and Hierarchies of transfinite real number-fields, of surreal numbers, and of ordinal real numbers , give by inductive limit ,or union, the same class of numbers ,already known as the class No .It can be characterized, simply, as the smallest (linearly ordered field  which is a ) class and contains every linearly ordered set- field as a subfield . Ôhis class ,and also the category of linearly ordered set-fields, we call    " the linearly ordered Newton-Leibniz realm of numbers". It is obvious that without the set theory of G. Cantor as it is formalized, for instance, by Zermelo-Frankel  and a correct thinking about the infinite, this "realm of  numbers" would   not be definable.

 Nevertheless we could follow a still different approach. If in the completion  of rational numbers to real numbers we ramify the equivalence relation of convergent sequences to others that include  not only where the sequences  converge (if they converge at the same point) but also how fast (if they converge in the same way e.g finally equal ,an attribute related also to computer algorithms complexity)  ,then we get non-linearly ordered topological fields that contain ordinal numbers (certainly up to ùa , a=   ùù) and are closer to practical applications. This approach does not involve the random variables at all, but involves directly  sequences of rational numbers as "Newtonian fluxions". This creation can be considered as a model of such linearly ordered fields (up-to-characteristic  ùa , a=   ùù) ,when these linearly ordered fields are defined axiomatically. This gives also a construction of the real numbers with a set which is countable. This does not contradict that all models of the real numbers are isomorphic as the field- isomorphism is not in this case also an Î-isomorphism so the Cauchy-real numbers and such a model still have different cardinality.

If we want to define in this way all the Ordinal real numbers , then it is still possible but then this would give also a device for a model of all the ZFC-set theory! And such a model is indeed possible: By taking again sequences of non-decreasing (in the inclusion) finite sets of ZFC , and requiring that  any property ,relation or operation if it is to hold for this set-sequence it must hold finally for each term of the sequence and finite set. In other words we take a minimality for every set of it, relative to the axiom of infinite. It is easy to prove that the (absolute) cardinality of such a model is at most 2ù  , that is at most the cardinality of the continuum. We could conceive  such a model as the  way a computer with its algorithms , data bases tables etc would represents sets of ZFC in a logically consistent way. Needles to say that a similar model of ZFC set theory could be given within Euclidean geometry! (e.g. after the Hilbert's axiom system, and in addition accepting only the natural numbers). There is no contradiction with the 2nd-incompletness theorem of Gödel as the argument to prove that it is a model of ZFC-set theory is already outside ZFC-set theory (as are also the arguments of Gödel ,or of Lowenheim-Skolem  that gives a countable model of ZFC-set theory).

 

 

 

 

 

 It  is  not directly apparent that  so different  techniques  and  ideas  would have  such an  underlying   unity.  It  is, also ,  surprising  that, although    the    Hessenberg     operations    were     very early known in the theory of ordinal  numbers, (at least since 1906, see  [ Gleyzal A. 1937])  no one went far  enough to define through them, fields in a way similar to  the way that the real numbers are defined from the natural numbers.  Although  G.Cantor,  himself  was  conceiving   the ordinals as a natural continuation of  the  natural   numbers (see [Frankell A. A. 1953 ] introduction pp 3 ) ,as  it  is  kmown,  he    rejected the attempts to define  infinitesimals   through  them  . (see [Frankell A. A. 1953] ch ii § 7.7 pp   120). We could speculate that un underlying  reason  for  this, might  be  that,  his   set-theory  was  already strongly  attacked  and  was   facing  the  danger of final rejection ,and these were good  enough reasons to  avoid  the additional  charge  that  his  theory   "  opened  the  door"   to infinitesimals . In spite of this, there  are many who might  consider that although the present results are coming now, nevertheless it is too late and ,they might speculate ,for this long    delay    (more  than  eighty years) and diversion   of ideas and  technique,  nevertheless  on the   same   subject,    we    could      suspect      systematic obstructions, that    came    outside    the    mathematics. Nevertheless, there are others who consider that it is too  early for  such  a development ,and especially for an analysis  on   such numbers. It  seems that it has never been published   any   "partially ordered Newton-Leibniz realm of  numbers" (in    other words a category of transcendental extensions of  the real    numbers ,that are partially ordered   fields and complete in the order topology )  with reasonably   "good" properties  for a  classification.      In this  paper we use the surreal numbers, as they are definable in the Zermelo-Frankel set theory, through the binary trees, directly as a class, and not as union of some set fields.(The original technique of J.H.Conway).I met J.H.Conway during 1992 at Philadelphia in the USA, I talked to him about the new developments in this area of research and I gave to him the present work but as he told me he had more than a decade that for the last time he had active interest in the subject. I is somehow necessary to make use of classes instead of sets; since, for the kind of "induction" that the J.H.Conway uses, we prove that it  is reduced to the usual transfinite induction on the height of the elements of the trees; but in their union as a class and not for each one of them separately as a set; in the latter case in  which  the  trees  are  sets  the induction  fails .The key-point is to prove that for every cut that J.H.Conway uses it does really exist a unique element of the trees of least height . "simplest number" as it is used to be called ).  This is  a  very crucial point, for the whole technique of the surreal numbers,  to  work,  and it   seems   that   it   has   been obscured,  by  not paying sufficient  attention to it

The author has initially included also the non-standard real numbers in the classification. As they are also linearly ordered fields and the present classification is of all linearly ordered fields it was natural to include them. There were experts in non-standard analysis that were glad about it. Nevertheless there were experts that insisted that according to the initial definition of A.Robinson and not of later definitions, it was not claimed that the non-standard real numbers were sets inside Zermelo-Frankel system. Only if Zermelo-Frankel system was used to model meta-mathematics also the they would be also sets. This was nevertheless different as such sets would models of meta-mathematical entities different than the sets that are models of mathematical and not meta-mathematical entities. Because of their arguments and in spite the fact that this made some other researchers of non-standard mathematics unhappy, the author prefers in this first publication about ordinal real numbers not to include the non-standard real numbers in the unification. Any definition nevertheless that has the non-standard real numbers as ordinary sets of Zermelo-Frankel set theory, would naturally lead to a straightforward proof that such fields are always subfields of some field of ordinal real numbers! The author has already produced pages with this proof that is based on the premise that I mentioned.

§ 2.  The   surreal numbers .

     In this paragraph we define the class No of surreal numbers inside the ZF-set theory.We use the binary  trees  (see  [ Conway J.H. 1976] appendix to part zero pp 65  and   [Kuratwski K.-Mostowski A. 1968]  Ch  IX  §1,  §2).The crucial point is to prove that for the cuts defined by J.H.Conway in these trees it does really exist a unique element strictly greater than all the elements of the left section and strictly smaller than  all  the  elements  of  the  right section  (the "simplest number" ).Through this the Conway-induction us reduced to the usual transfinite induction on the height of the elements of the tree .As we shall see this works for the union of all trees as a class but fails for each one set-tree .For the definition  of the tree, binary tree, height, levels of the tree ,Hî-set see   [Kuratwski K.-Mostowski A. 1968]  Ch  IX  §1, §2 Theorem 2, . The binary tree of height á we denote by Dá . More precisely we are  interested for the trees of the next definition.     

Definition 1. Let á be an ordinal . We define = {x|x  Da such that there  is â<á such that for  the  element x  as  a zero-one sequence x = {xî|î<a} holds that xâ = 1 and xî = 0 for  î>â}.

We call the set  the open full-binary tree of height á.

We also remind that if for the height á, holds that À(á) is a cofinal to á regular aleph: À(á) = Àcf(á)= Àî the open full-binary tree is an Hî set,  (see [Kuratwski K.-Mostowski A. 1968] ChIX §2 Theorem 2,  the proof works  also   for   trees  where À(á)= Àcf(á))

Lemma  2.  For  every  pair  of  subsets   L,   R   of   the open-full-binary tree  of height the  ordinal  á, such that À(á) is a regular aleph, and holds that: for every lL, rR, l < r, and À(L), À(R) < À(á),  there  is  exactly  one element x0 of least height in  such that l <  x0  <  r  for every  l  L, r  R.

Proof. Let D(L) = {x|x   such that there exists l  L with x £ l} and I(R) = {x|x   such that there exists r  R with r  x} that is D(L), I(R) are the decreasing  and  increasing lower and upper half subsets of  determined by L, R, in the linear ordering of  as a tree (see [Kuratwski K.-Mostowski A. 1968] Ch IX §1  Lemma A). Let the set M = {x|x e  and for every Ë l  D(L), r  I(R) it holds that l < x < r}. By the Hî property of   it  holds that M ¹ Æ . Let A = {â|â is an ordinal number such that there is x  M with x   Tâ where  Tâ is  the  â-level  of  Dá  in other words there is x  M of height â}. Let á0 =  min  A.  Let Dá0(L) Iá0(R) the subsets of D(L) R(L) of elements  of  height less than á0, and let Má0 Í M the subset of M that consists of elements of height á0. Suppose that the set Má0  contains  two elements x, y with e.g. x £ y. We will prove that Má0 contains only one element.

          Let x'={xâ|â<á0} that is that part of the á0-sequence  x with terms of indifes less than á0. And the  same  also  with y' = {yâ|â < á0}. Then there  is  lx  or  rx  and  ly  or  ry respectively in Dá0(L), Iá0(R) such that they are equal  with x', y'. If x=rx then, if the á0-term of x is 0 or 1, in  both cases x > rx, contradiction. Hence there is no  such  rx  and also such ry. Then lx=x' ly=y' and lx£ly. The á0-term  of  x and y might be 0 or 1. The  only  possible  cases  are  {x  = (lx,0), y = (ly,0)}, {x = (lx,0), y = (ly,1} {x = (lx,1), (ly,1)}, {x = (lx,1), y = (ly,0} where with  the  parenthesis we symbolize the á0- sequence which is the elements x, y. Let us suppose that x ¹ y and ,  the  part of the á0-sequence with terms with indices less than ä,  with ä £ á . Let the least value of ä, be denoted  by  ä0  such  that . If holds that  because x<y. In the sequent, let z=(Dâ(x)=Dâ(y) â< ä0, 1). Then x < z £ y. If ä0 = a0 then x=y because Xá0 = Yá0 = 1. Then ä0 < á0 and also z < y and x < z<  y and  the height of z is ä0 < a0 contradiction.  Hence  x=  y,  and  Má0 contains only one element.  It also holds that if  we  restrict to Dc(L), Ic(R) where  (and L, R have height <c), then a0 £  c by the Hî-property of  if c  is also such that À(á)= Àcf(á)      q.e.d.

Definition 3. The open full binary tree  of height á, such that À(á) is a cofinal to á, regular aleph , I  call  regular open  full- binary tree.

The property of the previous lemma of a regular open full-binary  tree  I call Hî-leveled Dedekind completness.

We remark that the class of regular alephs is unbounded (see [Kuratwski K.-Mostowski A. 1968] p. 275 relation 5 ) Thus the class of ordinals á  such that À(á)= Àcf(á) is unbounded.

The next definition is the definition of the class of surreal numbers in the ZF-set theory and it depends as we mentioned on the lemma 2 .As it is seen  ,in  the  hypotheses  of  the lemma 2 the cardinality of halfs of the cut is bounded by À(á). If it is to include all possible cuts of the tree  then the lemma 2 will give the element xï in some tree  , of  sufficient  greater  height,thus   outside   the original  tree  Dá. This  is  why  we  mentioned  that   the definition of surreal numbers (with the original technique of J.H.Conway ) does not apply to the trees  separately .

Definition 4. Let U=No be  the union of all regular  open full-binary trees. It is a class (after axiom A2.(see [Cohn P.M. 1965] p1-36)) Operations may be defined in this linearly ordered  class according to the formulae of Lemma 2 in [Kyritsis C.1991 Alt. or Free etc.)] II, that hold for every linearly ordered field that is:

     1.   let á be an ordinal with À(á)= Àcf(á) and L,R subsets of such that   for every l Î L, r Î R holds that l < r. Then there exists a regular aleph â such that  and À(L), À(R)< À(â). Then there is by lemma 2 a unique element  of least height  such that l < x0 < r for every l Î L, r Î R, we  denote  this element by {L|R} and we write x0=  {L|R}.  We  note  that  although  ,  it holds that  and á <â.

     2.   If and we  denote the height of x, y by  h(x), h(y) and by L(x), L(y), R(x), R(y) the sets

Then  the operations are  defined through  simultaneous two-variable  transfinite induction in the form of the lemma 2,3 in [ Kyritsis C. 1991 Free etc.], for  the  heights of the trees  where for the initial segments of ordinals  we substitute the corresponding trees of No  (For  every ordinal â<á such  that N(â)=Ncf(â) corresponds a tree ).  Thus  the function of operation is defined not  on  w(á)2  but  on 2. For the addition, the next  rule  is  used  x+y={L(x)+y È x+L(y)| x+R(y)} È R(x)+y}.

     3.   The opposite  is  defined  by:

          -x = {-R(x)|-L(x)}

     4.   Multiplication  is defined by

          x.y={L(x).y+xL(y)-L(x).L(y)ÈR(x).y+xR(y)-R(x)R(y)½

|L(x).y+x.R(y)-L(x).R(y) È R(x).y+x.L(x)-R(x).L(x)}.

          This definition presupposes the definition of addition.

     5.   Inverse is defined by

 As it is proved in [Conway J.H. 1976] Ch0, 1 the set No is a linearly ordered c-field. The characteristic of No is easily proved to  be   Ù1, we call this c-field, c-field of surreal numbers.  According to Definition 3 No is an Hî -leveled Dedekind  complete  field.

§ 3  The unification .

     In this paragraph  we  prove  that  all  the  three  different techniques and hierarchies of transfinite real ,of surreal ,of ordinal real numbers  give by inductive limit or union the same class of  numbers  .We  have  already  proved  that CR=Ù1R=C*R. (see corollary 10) and it remains to prove No=CR.

Lemma 5 . It holds that CR=Ù1R=C*RÍ No.

Proof  .Let  an  open  full  binary   tree    of   height the principal ordinal a .Then  Í No  ,and  the  field-inherited operations in the initial  segment  W(á)  are  the  Hessenberg operations (see [Conway J.H. 1976] ch 2 § ""containment of the ordinals "note pp 28 and also [Kyritsis C.1991 Alt] the characterisation theorem ).If  á was not a principal ordinal, the W(á) would not be closed to the Hessenberg operations .Thus the  Ná, Zá, Qá are contained in No ,since what it is used to define them from  W(á)  is  only the field  operations  .The  Qá is  a  field   and   from   the fact that No is  closed  to  extensions of  its  set-subfields (see[ Conway J.H. 1976] ch  4  theorem  28  )we  deduce  that  the  field  of ordinal real  numbers  Rá is  contained  in   No, for   every principal ordinal number á .Thus ÈRá=Ù1RÍNo .Q.E.D.

Lemma 6 . For every regular open full binary  tree  , it holds  that  Í Râ, for  some  sufficiently  big  principal ordinal number â . (With the inclusion is meant that the restriction of ordering of Rá in the tree, coincides with the ordering of the tree).

Proof . We shall prove it by transfinite induction .It holds  for the trees of finite height. The transfinite induction shall be on the transfinite sequence of all ordinal numbers such that À(á)= Àcf(á) and À(á) is a regular aleph. Let us suppose that it holds for all such ordinal numbers of W(á), and À(á)= Àcf(á) and À(á) is a regular aleph .Then :  where â(á) is a principal ordinal with . q.e.d.

From the previous lemma we get that È = No Í Ù1R ,thus :

The unification theorem 7

It holds that the classes of transfinite real numbers CR , of surreal numbers No, of ordinal real numbers  Ù1R ,coincide ,and it is the smallest class (and linearly ordered  c-field ) ,that contains all linearly ordered set-fields as subfields.

We  can  have  obviously analogous  statements  for  the  other classes  of numbers (complex , quaternion e.t.c.). After the previous theorem, the binary arithnetisation of the order-types, stated  in  [ Kyritsis C. 1991]  II  ,theorem  11, is   directly provable. We remark that because the levels of the open full binary trees have the property that any upper (lower bounded set has supremum (infimum ) ,(see [Kuratowski K. -Mostowski A 1968] ch ix §1, § 2 theorem 2 ),and after the Hilbert and fundamental (Cauchy) completness of the ordinal real numbers,  and remark  after definition  13  and ù-normal form according to [ Frankel A.A. 1953] ch 3 theorem 21 ,and  after corollary 21 in [Kyritsis C. 1991] ,II , we also get:

Theorem 8 . The class of numbers CR=Ù1R=No has leveled formal  power  series   representation, leveled Hilbert completeness, leveled fundamental (Cauchy) completeness, leveled Hî Dedekind completeness ,leveled supremum completeness and representation with ù-normal forms.

Bibliography

[N. L. Alling  1987]                              Foundations  of  analysis  over surreal number fields North-Holland  Math.Studies V. 141 1987 .

[ Artin  E.  Schreier   O.1927]           Algebraishe   konstruktion reellerkorper, Abh. Math. Sem.Univ. Hamburg 5 (1927) pp 85-99  .

[ Artin E. - Tate J.1967 ]                     Class  Field   Theory  Benjamin 1967.

[Baer   Reinold,1970]                         Dichte,Archimedizitat   und Starrheit geordneter Korper, Mathematische Annalen 188 pp 165-205, 1970.

[Baker Alan1975 ]                              Trancendental Number Theory Cambridge University Press, 1975.

[   Bourbaki N.1952]                           Elemente de Mathematique algebre,  chapitre III Hermann Paris 1948, chapitre VI Hermann Paris 1952.

[Brown Ron 1971 ]                              Real places and Ordered Fields Rocky Mountain Journal of Mathematics , Vol 1 ,pp 633-636, 1971.

[Brown R.,Craven T.C.,                      Ordered   fields   satisfying   Rolle's   Theorem.

      Pelling M.J.1986]                          Illinois Journal of Mathematics Vol 30, n 1 Spring 1986 pp 66-78.

[Clliford   A. H.1954]                           Note  on  Hahn's  theorem  on ordered  abelian groups. Proc. of the Amer. Math. Soc. 5 (1954) pp 860-863.

[Cohen   L.W.-   Goffman   C.1949]  Theory  of  transfinite Convergence.Transact. of the Amer. Math. Soc. 66  (1949) pp 65-74.

[Cohn P.M.1965]                                Universal  Algebva Harper - Row 1965  .

[Conway J.H.1976]                             On numbers and  games  Academic press 1976 .

[Cuesta  Dutardi  N.1954]                  Algebra Ordinal Rev. Acad. Cientis Madrid 4 (1954) pp 103-145 .

[Dugundji J.1966]                               Topology, Allyn and  Bacon  inc. 1966.

[Ehreshmann Ch.1956]                      Categories et structure Dunod  1956

[Ehrlich P.1988]                                  An alternative construction of Conway's Ordered Field No  ,Algebra Universalis 25 (1988) pp 7-16 .

[Ehrlich P.]                                           The Dedekind completion of No ,submitted to Algebra Universalis.

[Endler  O.1972]                                 Valuation  Theory,   Springer 1972.

[Erdos P.-Gillman L.-                          An isomorphism theorem for  real  closed  fields

     Henrkiksen  M..1955]                    Ann.  of Math.(2) 61 (1955)pp 542-554.

[Frankel A.A.1953]                             Abstract  set  Theory.  North  - Holland 1953.

[Fuchs L.1963 ]                                   Partially ordered algebraic systems Pergamon   Oxford 1963.

[Gillman L.-Jerison M.1960]              Rings of continuous functions.Van Nostrand Princeton 1960.

[Gleyzal A.1937]                                 Transfinite real numbers. Proc. of the Nat. Acad.of scien. 23 (1937) pp 581-587.

[Gravett K.A.H.1956]                          Ordered abelian groups. Quart. J. Math. Oxford 7 (1956) pp 57-63.

[Hahn H. 1907]                                    Uber die nichtarhimedishen Grossensysteme.S.  Ber. Akad. Wiss. Wein. Math. Natur.Wkl Abt.   IIa   116   (1907)    pp 601-655.

[Hausner M.-Wendel J.G.1952]                   Ordered Vector Spases Proc. of the Amer. Math. Soc.3 (1952) pp 977-982.

[Hessenberg G.1906]                        Grundbegriffe  der  Mengenlehre (Abh.  der  Friesschen  Schule, N.S. [1]  Heft 4) Gottingen 220 1906).

[Hilbert D.1977]                                  Grundlagen der Geometry Teubner Studienbucher 1977 .

[Hilbert D. -Ackermann W.1950]       Principles of Mathematical Logic. Chelsea Pub. Comp. N.Y. 1950.

[Kaplansky I.1942]                              Maximal fields with valuations Duke Math. J. 9 (1942)   pp 303-321.

[Krull W.1931]                                     Allgemeine Bewertungs theorie. J.reine angew. Math. 176 (1931) pp 160-196.

[Kuratowski K.1966]                           Topology  v.I   v.II   Academic Press 1966.

[Kuratowski K. -Mostowski A.1968]     Set  Theory  North  -  Holland 1968.

.

.

[ Kyritsis  C.E.1991]                           Algebraic characterisation of the Hessenberg operations in the ordinal numbers. (unpublished yet).

[ Kyritsis C.E.1991]                            Ordinal real numbers 1. The ordinal characteristic.(unpublished yet ).

[Lang S.1984]                                     Algebra .  Addison-Wesley   P.C. 1984 .

[Lam T.Y.1980]                                   The Theory of Ordered Fields Ring theory and Algebra III. Edited  by  B.R.  McDonald Dekker 1980 pp 1-268.

[Laugwitz Detler 1983]                       Ù Calculus as a Generalization of Field Extension.  An alternative approach to non-Standard analysis "Recent developments in   non-standard analysis" Lecture Notes in Math 983   Springer 1983.

[MacLane  S.1939]                             The Universality of Formal Power Series fields.Bull. of the Amer. Math. Soc. 45  (1939) pp 880-890.

[MacLane S. 1971]                             Categories for the working mathematician  Springer 1971

[Massaza ,Carla1970]                       On the completion of ordered fields . Practica (=Proceedings) of the Academy of Athens ,Vol 45 ,1970 (published 1971 )

[Monna A.F.1975]                               Analyse non-Archimedienne Springer 1970.

[Munkress J.R.1975]                          Topology. Prenctice Hall 1975.

[Nachbin  L.1976]                               Topology   and   Order.   Robert E.Krieger P.C. N.Y. 1976.

[Neubrunnova Anna1980]                  On transfinite convergence and generalised continuity. Mathematica Slovaca vol 30,1, 1980.

[Neumann B.H.1949]                          On ordered division rings. Transact. of the Amer. Math. Soc. 66 (1949) pp 202-252.   .

[Prestel Alexander1980]                    Lectures on formaly real fields .Lecture Notes 1093, Springer ,1980 .

[Robinson A.1966]                             Non-Standard analysis.North-Holland 1974 (1966).

[Robinson A.1972]                             On the real Closure of a Hardy Field. pp 427-433 in the Theory of Sets and topology. A Collection of papers in honour of  Felix Hausdorff , VEB ,1972 .

[Rolland,Raymond .1981]                  Etudes des courpure dans les groupes et corps ordonnes. Dans Geometrie Algebrique Reeles et Formes Quadradiques. Rennes 1981, Proceedings, Lecture Notes in Mathematics 959                                    Springer ,pp 386-405 .

[Rudin W.1960]                                   Fourier analysis on groups. Interscience Pub 1960 .

[Shilling O.F.G.1950]                          The theory of valuastions.Amer. Math. Soc. 1950.

[Schubert  H.1972 ]                            Categories Springer 1972.

[Scott, Dana 1969,]                            On Completing Ordered Fields. In Applications of Model theory to Algebra, Analysis and Probability edited by W.A.J. Luxenburg pp 274-                                 278,Holt Rinehart and Winston 1969.

[Sirkoski R. 1948]                               On an ordered algebraic field. Warsow, Towarzytwo Nankowe Warzawskie 41 (1948) pp 69-96.

[Stone A.L.1969]                                Non-Standard analysis in  topological algebra in Applications of  Model  Theory  to  Algebra, Analysis and  Probability  N.Y. (1969) PP 285-300.

[Stroyan, K.D. and                              Introduction to the  theory  of Infinitecimals

     Luxenburg W.A.J.1976]                N.Y.1976.

[Lynn A.Steen-                                    Counterexamples    in   Toplogy Springer 1970 .

     Seebach J.A. Jr.1970]             

[Viswanathan T.M.1977]                    Ordered fields and sign-changing polynomials. Journal fur reine und angewante Mathematik, 296 pp 1-9,1977.

[Van der Waerden B.L.1970]            Algebra V1  V2 Frederick  Unger Pub. Co. N.Y. 1970.

[Weil A.1967 ]                                     Basic Number  Theory,  Springer Verlag Berlin, Heidelberg  N.Y. 1967.

[Zakon E.1955]                                   Fractions  of  ordinal numbers Israel Institute of Tecnology Scient. Public. 6, 94-103 1955.

[Zariski O. -Samuel P.1958]              Commutative    Algebra   V.I.II Springer 1958.

[Zervos,S.P.1961]                              Sur les  rapportes entre la completions la cloture algebrique des corps commutatifs de caracteristique zero C.R. Acad. Sc. Paris, 5 Avril 1961 t. 282 pp 2053-2055.

[Zervos,S.P.1991]                              Simple  abstract  and concrete considerations suggested by Thue's theorem. Mathematics revisited, mathematicians remembered Vol.1                                   1991.pp 1-68.

 

 

 

 

 

 

 

 

 

 

List of special symbols

   á,â,ù           :  Small Greek letters

      Ù1         :  Capital   Greek   letter   omega   with    the subscript 1

     Fa          :  Capital   letter F  with   superscript  a. 

     N      :  Capital Aleph ,the first letter of the hebrew alphabet . In the text is used a capital script. letter n .

       : cross in a circle, point in a circle .

     Ná,Zá,Qá,Rá,:  Roman capital  letters  with subscript small Greek letters

     Cá,Há

      *X, *R et.c  :  Capital standard or roman letters with left superscript a star.

     CN,CZ,CQ,      :Capital standard letter c followed by capital letters, with possibly

     C*R                     a  left superscript  a  star

              :  Capital tstandard letter with a cap.

     Ó           :  Capital Greek letter sigma

            :  Capital standard  D with subscript a  small Greek  letter and in upper place a small zero.

 

 

 

 

 

 

 

 

 

LINEAR NUMBERS 3.

 

The technique of "free operations-fundamental completion" for the definition of a well-ordered classifying Hierarchy of Linearly ordered associative commutative number-fields. The transfinite real , the non-standard  real, the  surreal  numbers; unification .

                                             by

                                                                               Dr.Konstantine E.Kyritsis

                              Computer  Technology Institute

                                         Patra       Greece

 

                       Abstract

In this paper   is  introduced  a  forth  new  technique  of  "the enlargment real numbers",that in some sense includes the non-standard real numbers of A.Robinson.It is  proved   ,that the three different technics and   hierarchies   of   transfinite   real    numbers , of     the surreal  numbers  ,of  the  ordinal  real  numbers, give   by      inductive limit or union the same class of numbers .

 

 

 

 

 

 

 

 

 

 

 

                         Key words

linearly ordered commutative fields

transfinite real numbers

non-standard enlargment fields of the real numbers

surreal numbers

formal power series fields