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Mirrors > Home > MPE Home > Th. List > df-no | Structured version Visualization version GIF version |
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 Gonshor, 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 Gonshor'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.) |
Ref | Expression |
---|---|
df-no | ⊢ No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csur 27489 | . 2 class No | |
2 | va | . . . . . 6 setvar 𝑎 | |
3 | 2 | cv 1532 | . . . . 5 class 𝑎 |
4 | c1o 8454 | . . . . . 6 class 1o | |
5 | c2o 8455 | . . . . . 6 class 2o | |
6 | 4, 5 | cpr 4622 | . . . . 5 class {1o, 2o} |
7 | vf | . . . . . 6 setvar 𝑓 | |
8 | 7 | cv 1532 | . . . . 5 class 𝑓 |
9 | 3, 6, 8 | wf 6529 | . . . 4 wff 𝑓:𝑎⟶{1o, 2o} |
10 | con0 6354 | . . . 4 class On | |
11 | 9, 2, 10 | wrex 3062 | . . 3 wff ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o} |
12 | 11, 7 | cab 2701 | . 2 class {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}} |
13 | 1, 12 | wceq 1533 | 1 wff No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1o, 2o}} |
Colors of variables: wff setvar class |
This definition is referenced by: elno 27495 sltso 27525 dfno2 42668 |
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