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Definition df-no 25511
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 25508 . 2  class  No
2 va . . . . . 6  set  a
32cv 1648 . . . . 5  class  a
4 c1o 6676 . . . . . 6  class  1o
5 c2o 6677 . . . . . 6  class  2o
64, 5cpr 3775 . . . . 5  class  { 1o ,  2o }
7 vf . . . . . 6  set  f
87cv 1648 . . . . 5  class  f
93, 6, 8wf 5409 . . . 4  wff  f : a --> { 1o ,  2o }
10 con0 4541 . . . 4  class  On
119, 2, 10wrex 2667 . . 3  wff  E. a  e.  On  f : a --> { 1o ,  2o }
1211, 7cab 2390 . 2  class  { f  |  E. a  e.  On  f : a --> { 1o ,  2o } }
131, 12wceq 1649 1  wff  No  =  { f  |  E. a  e.  On  f : a --> { 1o ,  2o } }
Colors of variables: wff set class
This definition is referenced by:  elno  25514  sltso  25537
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