Jarvis the type of guy to explain why1+1=2

When mathematicians want to be absolutely rigorous about numbers and arithmetic, they don't just say "it's obvious from counting." They build the system from a set of very basic, self-evident truths called axioms. The Peano Axioms (named after Giuseppe Peano, who published them in 1889) are the most common set of axioms for defining the natural numbers (0, 1, 2, 3, ... or 1, 2, 3, ... depending on the convention, but often including 0).

These axioms are like the "rules of the game" for numbers. From these simple rules, you can logically derive all of arithmetic (addition, subtraction, multiplication, etc.) and all the properties of numbers.

Let's break them down and then show how 1+1=2 is a direct logical consequence.

The Five Peano Axioms (Common Formulation, starting with 0):

Axiom of Zero: 0 is a natural number.

This simply establishes the existence of a starting point for our number system.

Axiom of Successor: Every natural number n has a successor, denoted S(n).

This is crucial. It means after any number, there's always a "next" number.

We can think of S(n) as "n + 1". It's the operation that gets you to the next integer.

For example: S(0) is the next number after 0. S(S(0)) is the number after S(0), and so on.

Axiom of No Predecessor for Zero: 0 is not the successor of any natural number.

This means 0 is truly the "first" number in the sequence. You can't get to 0 by taking the successor of some other natural number.

Axiom of Unique Successor: If S(n) = S(m), then n = m.

This ensures that each natural number has a unique successor. If two numbers have the same successor, then they must have been the same number to begin with. This prevents "loops" or numbers having multiple "next" numbers.

Axiom of Induction: If a property P is true for 0, and if for every natural number n, if P is true for n then it is also true for S(n), then P is true for all natural numbers.

This is the most complex axiom and is fundamental for proving statements about all natural numbers (like mathematical induction). It ensures that if you can establish a base case and a step-by-step progression, the property holds for the entire infinite set of natural numbers.

Defining Numbers from Axioms:

Based on these axioms, we define numbers:

1 is defined as the successor of 0: 1 = S(0)

2 is defined as the successor of 1: 2 = S(1)

3 is defined as the successor of 2: 3 = S(2), and so on.

Defining Addition from Axioms:

Addition (a+b) is also defined recursively using the successor function:

Base Case: a+0=a

Adding zero to any number doesn't change the number. This establishes the starting point for addition.

Recursive Step: a+S(b)=S(a+b)

This is the critical rule for adding numbers other than zero. It states that adding the successor of a number (b) to a is the same as finding a+b first, and then taking the successor of that result.

Think of it as: "Adding (b+1) to 'a' is the same as adding 'b' to 'a', and then adding 1 (taking the successor)."

Deriving 1+1=2 Using Peano Axioms and Definitions:

Now, let's rigorously prove 1+1=2 step-by-step:

Our goal is to show that 1+1 is equivalent to the number we defined as 2.

Start with the expression: 1+1

Apply Definition of 1: We know that 1=S(0) (by definition, from the axioms).

So, we can rewrite the expression as: 1+S(0)

Apply the Recursive Definition of Addition: We use the rule a+S(b)=S(a+b).

In our current expression, let a=1 and b=0.

Applying the rule, 1+S(0) becomes S(1+0).

Apply the Base Case Definition of Addition: We use the rule a+0=a.

Here, 1+0=1.

Substitute this back into our expression: S(1+0) becomes S(1).

Apply Definition of 2: We know that 2=S(1) (by definition, from the axioms).

Therefore, S(1) is equal to 2.

Conclusion:

By starting with the Peano Axioms and the definitions derived from them, we have logically demonstrated that 1+1 is equal to 2. This illustrates how basic arithmetic is built upon a very small set of foundational truths and definitions, making mathematics incredibly consistent and powerful.