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Definition df-no 32073
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 32070 . 2 class No
2 va . . . . . 6 setvar 𝑎
32cv 1619 . . . . 5 class 𝑎
4 c1o 7710 . . . . . 6 class 1𝑜
5 c2o 7711 . . . . . 6 class 2𝑜
64, 5cpr 4311 . . . . 5 class {1𝑜, 2𝑜}
7 vf . . . . . 6 setvar 𝑓
87cv 1619 . . . . 5 class 𝑓
93, 6, 8wf 6033 . . . 4 wff 𝑓:𝑎⟶{1𝑜, 2𝑜}
10 con0 5872 . . . 4 class On
119, 2, 10wrex 3039 . . 3 wff 𝑎 ∈ On 𝑓:𝑎⟶{1𝑜, 2𝑜}
1211, 7cab 2734 . 2 class {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1𝑜, 2𝑜}}
131, 12wceq 1620 1 wff No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1𝑜, 2𝑜}}
Colors of variables: wff setvar class
This definition is referenced by:  elno  32076  sltso  32104
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