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Definition df-no 29646
Description: Define the class of surreal numbers. The surreal numbers are a proper class of numbers developed by John H. Conway and introduced by Donald Knuth in 1975. They form a proper class into which all ordered fields can be embedded. The approach we take to defining them was first introduced by Hary Goshnor, and is based on the conception of a "sign expansion" of a surreal number. We define the surreals as ordinal-indexed sequences of 
1o and  2o, analagous to Goshnor's  (  -  ) and  (  +  ).

After introducing this definition, we will abstract away from it using axioms that Norman Alling developed in "Foundations of Analysis over Surreal Number Fields." This is done in an effort to be agnostic towards the exact implementation of surreals. (Contributed by Scott Fenton, 9-Jun-2011.)

Assertion
Ref Expression
df-no  |-  No  =  { f  |  E. a  e.  On  f : a --> { 1o ,  2o } }
Distinct variable group:    f, a

Detailed syntax breakdown of Definition df-no
StepHypRef Expression
1 csur 29643 . 2  class  No
2 va . . . . . 6  setvar  a
32cv 1397 . . . . 5  class  a
4 c1o 7115 . . . . . 6  class  1o
5 c2o 7116 . . . . . 6  class  2o
64, 5cpr 4018 . . . . 5  class  { 1o ,  2o }
7 vf . . . . . 6  setvar  f
87cv 1397 . . . . 5  class  f
93, 6, 8wf 5566 . . . 4  wff  f : a --> { 1o ,  2o }
10 con0 4867 . . . 4  class  On
119, 2, 10wrex 2805 . . 3  wff  E. a  e.  On  f : a --> { 1o ,  2o }
1211, 7cab 2439 . 2  class  { f  |  E. a  e.  On  f : a --> { 1o ,  2o } }
131, 12wceq 1398 1  wff  No  =  { f  |  E. a  e.  On  f : a --> { 1o ,  2o } }
Colors of variables: wff setvar class
This definition is referenced by:  elno  29649  sltso  29672
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