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Definition df-no 31550
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 31547 . 2 class No
2 va . . . . . 6 setvar 𝑎
32cv 1479 . . . . 5 class 𝑎
4 c1o 7513 . . . . . 6 class 1𝑜
5 c2o 7514 . . . . . 6 class 2𝑜
64, 5cpr 4157 . . . . 5 class {1𝑜, 2𝑜}
7 vf . . . . . 6 setvar 𝑓
87cv 1479 . . . . 5 class 𝑓
93, 6, 8wf 5853 . . . 4 wff 𝑓:𝑎⟶{1𝑜, 2𝑜}
10 con0 5692 . . . 4 class On
119, 2, 10wrex 2909 . . 3 wff 𝑎 ∈ On 𝑓:𝑎⟶{1𝑜, 2𝑜}
1211, 7cab 2607 . 2 class {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1𝑜, 2𝑜}}
131, 12wceq 1480 1 wff No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1𝑜, 2𝑜}}
Colors of variables: wff setvar class
This definition is referenced by:  elno  31553  sltso  31582
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