Human ecology

Two proofs of 1+1=2

Proof I: From Ernest (1998, p.35)

Notation:
a(x,y)=x+y
s(x)="the increment of x"

A1 a(x,s(y))=s(a(x,y)) AXIOM x+sy=s(x+y)
T2 a(1,s(y))=s(a(1,y)) FROM A1 1+sy=s(1+y)
T3 a(1,s(0))=s(a(1,0)) FROM T2 1+s0=s(1+0)
A4 a(x,0)=x AXIOM x+0=x
T5 a(1,0)=1 FROM A4 1+0=1
T6 a(1,s(0))=s(1) FROM T3 AND T5 1+s0=s1
D7 s(0)=1 DEFINITION s0=1
T8 a(1,1)=s(1) FROM T6 AND D7 1+1=s1
D9 s(1)=2 DEFINITION s1=2
T10 a(1,1)=2 FROM T8 AND D9 1+1=2

Expression tree of this proof structure


Expression trees of the structure of these statements

x+sy=s(x+y)
1+sy=s(1+y) x+0=x
1+s0=s(1+0) 1+0=1
1+s0=s1 s0=1
1+1=s1 s1=2
1+1=2

Alternative Proof II

Notation:
a(x,y)=x+y
i(x)="the increment of x"
d(x)="the decrement of x"

A1 a(x,y)=a(i(x),d(y)) AXIOM
T2 a(1,y)=a(i(1),d(y)) FROM A1
T3 a(1,1)=a(i(1),d(1)) FROM T2
D4 i(1)=2 DEFINITION
T5 a(1,1)=a(2,d(1)) FROM T3 AND D4
D6 i(0)=1 DEFINITION
T7 a(1,1)=a(2,d(i(0))) FROM T5 AND D6
A8 d(i(x))=x AXIOM
T9 d(i(0))=0 AXIOM
T10 a(1,1)=a(2,0) FROM T7 AND D9
A11 a(x,0)=x AXIOM
T12 a(2,0)=2 FROM A11
T13 a(1,1)=2 FROM T10 AND T12



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Created 16/8/99