Answer:
By constructing arithmetics from Peano’s axioms (or equivalent).
Define 1 as suc0 .
Define 2 as suc1 .
Define addition as: ∀∈ℕ,+0= and ∀,∈ℕ,+suc=(+) .
Prove that suc=+1 . ( +1=+suc0=suc(+0)=suc ).
Therefore, 1+1=suc1=2 .
Then, prove that in any system which include a subset of inductive number which is compatible with Peano numbers, it is indeed compatible and the definitions of addition, 1, and 2 hold.
Second system:
Defining (natural) numbers as finite cardinals, and defining addition of two numbers , : + , as the cardinality of the union of two disjoint sets of cardinality and respectively.
We could define 1 as the cardinality of set {{}} , and 2 as the cardinality of set {{},{{}}} .
First I would prove that cardinality is an equivalence relationship.
Then I could prove that sets {{}} and {{{}}} are disjoint, each has cardinality 1 and the union has cardinality 2, which would fix my definitions.
Third system:
Use any other set of definitions and work from it. What should I define as 1? What should I define as 2? How I define addition?
For example, let’s have a field (a set with a commutative, associative, operation with identity property called addition, and a second commutativee, associative, operation with identity property that distributes the first one called multiplication) with total order which is closed by addition and multiplication. Let’s define 0 as the identity element of addition and 1 as the identity element of multiplication. Then find a way to define 2 differently than 1+1 , then prove that 2 is 1+1 . The tricky part is to use a coherent intuitive definition of 2 that is not 1+1
Step-by-step explanation:
