Explain why 1 + 1 equals 2 in the most complex way possible

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

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