Mathematical Induction is Wrong

Mathematical Induction is a greatly misunderstood subject. According to its Wikipedia page, it is a "mathematical proof technique".

Bullshit.

In order to criticise something, one should first understand it properly. So here goes.

Proof by induction supposedly works for statements about a well-ordered set. This means there is some order, continuation to the elements.

Many sequences pertaining to the natural numbers can be apparently proved by mathematical induction. Let's see the principle behind the beloved mathematical induction.

The technique works by first proving what's called the base case, which proves the proposition for the lowest element in the ordered set. This is usually the first element.

For example, let's say we want to prove this proposition:

where n is a natural number, for every natural number.

It would, of course, be impractical to prove every single natural number, so that's where induction should come in.

The base case would be proving that if n = 0, the proposition would make sense.

Since the equation holds, the proposition makes sense for the first element, 0. This is known as the base case.

The inductive step is essentially proving that as long as there is any natural number that satisfies the proposition, the next highest natural number will definitely satisfy the proposition. This step implies continuation, without a fixed point. Any random element will satisfy this. Hence every natural number will satisfy the proposition.

This is done like this:

We first assume that the proposition is true for some natural number. (It already is true for 0 but let's see what happens when we make it more general.)

In other words, we assume that this is true for some k.

Then we examine the case with k+1:

Basically we're adding (k+1) to both sides, then seeing whether it can be reduced to a similar form in terms of k+1.

In this case, the manipulation goes like this:

As we can see, if we replace k+1 with something simpler, like a,

we get this:

This is the same as we have seen previously. The inductive step proves that if one case is true, the subsequent cases must be true.

Since we have already proven the first base case, mathematical induction allows us to carry out a leap of faith and say that every natural number satisfies the proposition stated.

If any of this is hard to grasp, it's not your fault, because mathematical induction is wrong. Just because these two conditions hold, it does not mean that the resulting proposition is true for all elements.

Imagine jumping off a platform suspended 1cm away from the ground. Then you jump off another platform suspended 2cm away from the ground. In both cases, you would be unharmed. But does that mean that jumping off from any height will leave you unharmed?

Another example involves removing grains of sand from a heap of sand. If removing one grain of sand from a heap leaves a heap, and removing another still leaves a heap of sand, then a single grain of sand (or no sand at all) must still be called a heap of sand. This is called the Sorites paradox.

There are so many other examples that prove that mathematical induction is wrong. But it has already been so well established as a technique for proving. Many people simply don't realise that what they're doing is fundamentally wrong and illogical, attempting to transcend "everything" with just a few simple steps. They are deluded into thinking that math is just so simple. They will probably kill me if I don't say Happy April Fools.