I came across this elementary introduction to Gödel's theorem on incompleteness in math. It contained this statement:

[C]onsider a system that consists of the axioms of arithmetic excluding the axiom of mathematical induction. In this system, it would be impossible to prove the following statement (among others), even though it is true:

1 + 2 + 3 + 4 + 5 + ... +

n=n(n+1)/2

Oh really? I put it to you, the left side of the equality can't even be defined without mathematical induction.

Let's be clearer. I'll give my axioms defining the natural numbers (positive integers) **N**:

**N**contains**one**.- There exists a injective successor function
*s*:**N**-->**N**-{**one**}. (An injective function is defined: if*s*(*m*) =*s*(*n*) then*m*=*n*. An injective function is also known as "one-to-one".) - Suppose we have a theorem (or claim) for the natural numbers (say T
_{n}). We prove T_{one}, and then prove if T_{n}then T_{s(n)}(equivalently, T_{n}implies T_{s(n)}). Then T_{n}is guaranteed true for all natural numbers*n*.

**N**-{**one**} means **N** without **one**. I use that (and the injective requirement) to ensure that every number is followed by a new and different number. The final axiom is the axiom of mathematical induction -- or the special case starting with **one**. Remove it, and what can happen? Here are some sets that will satisfy the beginning and middle axioms:

- The positive integers: 1, 2, 3, ...
- The positive integers and half-integers: 0.5, 1, 1.5, 2, 2.5, 3, 3.5 ...
- The following numbers: 0.2, 1, 1.2, 2, 2.2, 3, 3.2 ...
- The rational numbers excluding zero and the negative integers
- The real numbers excluding zero and the negative integers

Now consider any of the sets which include 13.5. How do we even define 1 + 2 + 3 + 4 + 5 + ... + 13.5? Clearly the theorem, valid for positive integers (okay, possibly zero as well) is invalid in general on sets satisfying the other axioms. The induction axiom excludes all but the first example.

In my definition of the natural numbers, the induction axiom is part of the definition. Remove it, and we have so many other possibilities satisfying the other axioms that statements proved using induction in general aren't true.

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