**untheorem**:

Given a

room with any (

finite)

number of

people in it, if

1 of them is a

redhead, then all of them are.

This is patently false, of course. Nonetheless we have the following "proof" by induction.

**unproof**:

We'll denote the number of people in the room by *n*.

- base case:
*n*=1.

This case is clear: it says that for a room with one person in it, if one of them (and there's only one) is a redhead, then all of them are redheads.
- induction case: we shall suppose the untheorem true for
*n*, and prove it for *n*+1.
Consider a room with *n*+1 people in it, one of them a redhead. Ask one of the non-redheads to step outside the room for a moment. Then we are left with a room with *n* people in it, and one of them is (still) a redhead. By the induction hypothesis, all *n* people are redheads!

Now ask the (only) non-redhead who stepped out to return to the room; ask some other person to step out, and again we're left with *n* people in a room, one of them (in fact, all but one of them!) a redhead. By the hypothesis, we deduce that all *n* are redheads, in particular the remaining person. Returning the redhead who stepped outside to the room, we see that all *n*+1 people are redheads.

By the

principle of mathematical

induction, the

theorem is proved (or rather, the

untheorem is

unproven).

What's wrong here??

**Notes about comments:**

Both prole and DaVinciLe0 are giving wrong arguments. prole is mistaken about the proof: every time we ask someone who is *not* a redhead to step out, thus fulfilling the induction hypothesis. DaVinciLe0 says something about not being able to prove by induction anything regarding *n* people. Apparently he does not believe (a finite number of) people can be counted, since the primary way of telling if a set can be placed in a bijection with an initial subset of the naturals is to count it!
This is a common problem with this unproof: it's false, but for much simpler reasons than what people try to make up!