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Definition df-no 27713
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 27710 . 2  class  No
2 va . . . . . 6  setvar  a
32cv 1363 . . . . 5  class  a
4 c1o 6909 . . . . . 6  class  1o
5 c2o 6910 . . . . . 6  class  2o
64, 5cpr 3876 . . . . 5  class  { 1o ,  2o }
7 vf . . . . . 6  setvar  f
87cv 1363 . . . . 5  class  f
93, 6, 8wf 5411 . . . 4  wff  f : a --> { 1o ,  2o }
10 con0 4715 . . . 4  class  On
119, 2, 10wrex 2714 . . 3  wff  E. a  e.  On  f : a --> { 1o ,  2o }
1211, 7cab 2427 . 2  class  { f  |  E. a  e.  On  f : a --> { 1o ,  2o } }
131, 12wceq 1364 1  wff  No  =  { f  |  E. a  e.  On  f : a --> { 1o ,  2o } }
Colors of variables: wff setvar class
This definition is referenced by:  elno  27716  sltso  27739
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