𝙒𝙚 𝙣𝙚𝙚𝙙 𝙩𝙝𝙚 𝙏𝙍𝘼𝙉𝙎𝙋𝘼𝙍𝙀𝙉𝘾𝙔

By definition,
1
=
𝑆
(
0
)
1=S(0).
According to our recursive definition of addition:
1
+
1
=
1
+
𝑆
(
0
)
1+1=1+S(0)
Using the rule
𝑛
+
𝑆
(
𝑚
)
=
𝑆
(
𝑛
+
𝑚
)
n+S(m)=S(n+m), we can rewrite this as:
1
+
𝑆
(
0
)
=
𝑆
(
1
+
0
)
1+S(0)=S(1+0)
By definition of addition,
1
+
0
=
1
1+0=1.
So, substituting back, we get:
1
+
𝑆
(
0
)
=
𝑆
(
1
)
=
2
1+S(0)=S(1)=2
where
2
2 is defined as
𝑆
(
𝑆
(
0
)
)
S(S(0)).
Conclusion
Therefore, by the formal definitions of addition and the natural numbers, we have shown that
1
+
1
=
2
1+1=2.