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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 Goshnor, and is based on the conception of a
"sign
expansion" of a surreal number. We define the surreals as
ordinal-indexed sequences of 1_{o} and
2_{o}, 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.) |
Ref | Expression |
---|---|
df-no | ⊢ No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1_{o}, 2_{o}}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csur 32332 | . 2 class No | |
2 | va | . . . . . 6 setvar 𝑎 | |
3 | 2 | cv 1657 | . . . . 5 class 𝑎 |
4 | c1o 7819 | . . . . . 6 class 1_{o} | |
5 | c2o 7820 | . . . . . 6 class 2_{o} | |
6 | 4, 5 | cpr 4399 | . . . . 5 class {1_{o}, 2_{o}} |
7 | vf | . . . . . 6 setvar 𝑓 | |
8 | 7 | cv 1657 | . . . . 5 class 𝑓 |
9 | 3, 6, 8 | wf 6119 | . . . 4 wff 𝑓:𝑎⟶{1_{o}, 2_{o}} |
10 | con0 5963 | . . . 4 class On | |
11 | 9, 2, 10 | wrex 3118 | . . 3 wff ∃𝑎 ∈ On 𝑓:𝑎⟶{1_{o}, 2_{o}} |
12 | 11, 7 | cab 2811 | . 2 class {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1_{o}, 2_{o}}} |
13 | 1, 12 | wceq 1658 | 1 wff No = {𝑓 ∣ ∃𝑎 ∈ On 𝑓:𝑎⟶{1_{o}, 2_{o}}} |
Colors of variables: wff setvar class |
This definition is referenced by: elno 32338 sltso 32366 |
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