In formal arithmetic, especially in systems like Peano arithmetic or set-theoretic constructions of natural numbers, numbers and operations are built from logical foundations. Let's consider the construction of natural numbers using set theory, where:
0
=
∅
0=∅ (the empty set)
1
=
{
0
}
=
{
∅
}
1={0}={∅}
2
=
{
0
,
1
}
=
{
∅
,
{
∅
}
}
2={0,1}={∅,{∅}}
Addition is then defined recursively:
For natural numbers
𝑎
a and
𝑏
b,
𝑎
+
0
=
𝑎
𝑎
+
𝑆
(
𝑏
)
=
𝑆
(
𝑎
+
𝑏
)
a+0=a
a+S(b)=S(a+b)
where
𝑆
(
𝑏
)
S(b) denotes the successor of
𝑏
b, or
𝑏
+
1
b+1.
Using this, we can prove:
1
+
1
=
𝑆
(
1
)
=
2
1+1=S(1)=2
This recursive structure shows how addition is not just intuitive, but rigorously built up from logic and definitions. In Principia Mathematica, Whitehead and Russell famously worked through the logical underpinnings of this very equation, taking over 300 pages to derive:
1
+
1
=
2
1+1=2
as Theorem ✸54.43. Thus, what seems self-evident is actually the end result of a structured, layered mathematical system—demonstrating that "1 + 1 = 2" is not only a basic truth, but a profound one anchored in the foundations of logic, arithmetic, and human understanding.
Caption this Meme
