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Definition df-no 33150
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 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}}
Distinct variable group:   𝑓,𝑎

Detailed syntax breakdown of Definition df-no
StepHypRef Expression
1 csur 33147 . 2 class No
2 va . . . . . 6 setvar 𝑎
32cv 1532 . . . . 5 class 𝑎
4 c1o 8094 . . . . . 6 class 1o
5 c2o 8095 . . . . . 6 class 2o
64, 5cpr 4568 . . . . 5 class {1o, 2o}
7 vf . . . . . 6 setvar 𝑓
87cv 1532 . . . . 5 class 𝑓
93, 6, 8wf 6350 . . . 4 wff 𝑓:𝑎⟶{1o, 2o}
10 con0 6190 . . . 4 class On
119, 2, 10wrex 3139 . . 3 wff 𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}
1211, 7cab 2799 . 2 class {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}}
131, 12wceq 1533 1 wff No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}}
Colors of variables: wff setvar class
This definition is referenced by:  elno  33153  sltso  33181
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