Some Math Paradoxes

A Logical Paradox – The British Air Force Bombing Paradox

The British Airforce announces that it will bomb the city sometime during this week; but at an unpredictable date.

Why is this paradoxical. Suppose, they announce this on a Monday. Now, they plan to bomb sometime between this Monday and Sunday. However, it cannot be Sunday – because then it wouldn’t be unexpected. It would be expected since the bombing didn’t happen on any other day.

So, it cannot be Sunday. How about Saturday? Again, since it CANNOT be Sunday, one is limited to Monday through Saturday. However, if the bombing hasn’t happened till Friday, then everyone will expect it on Saturday. So, it cannot be  unexpected? So, Saturday is out as well.

So, it cannot be Saturday or Sunday.

What about Friday? Same logic applies; if it hasn’t happened till Thursday, it cannot happen on Friday – since that would be ‘expected’.

So, essentially, the bombing cannot happen on any day of the week – if we believe the premise that it is to be truly UNEXPECTED.

The Logical Flaw in the Bombing Paradox

Based on pure logic, there isn’t really a flaw in the argument above. However, logic is not applicable to someone else’s thoughts! Logic is only applicable in the realm where everyone is on the same page about the current facts – and no one is in the dark.

A Counting Paradox – How many REAL numbers are there?

How many real numbers are there between 0 and 1? Can they be mapped to the integers – is there a 1-1 correspondence?

Counting the Rationals First (before we get to the Reals)

It is easy to put the rational numbers in a 1-1 correspondence with the integers.

The problem with the Reals

In order to ‘count’ the reals. it is enough to try and just count the reals between 0 and 0.999..

Now, in order to assign an integer to each real number, we need a FIRST real number after 0. What is that supposed to be? 0.000000001, 0.000000000000001,….. ? There is no clear answers. In fact, if you look from the 0 side, there isn’t a FIRST number to pick. However, you can start from the other side- and pick 0.9999 as your FIRST number.

Then, you assign ‘1’ to 0.9999. Now to get the next real number, again you have an infinite number of choices. But say you pick 0.999999999..8 as the next real number. Now you assign ‘2’ to it. And so on. So – you are able to ORDER these somewhat – using a little bit of hand-waving. This means that it is possible to list all these reals as a sequence  – r1, r2, r3….

Now that you have a sequence, it is easy to see how you can ‘construct’ a real that is NOT in this sequence.

To begin with, assume that this sequence contains ALL the reals between 0 and 0.9999.

Now, look closely at a particular piece of this sequence – say from

0.123456789

0.237329923

0.2332323232

0.2433445455

……

Supposing you construct a new decimal by ‘subtracting’ 1 from the first digit of the first number, from the second digit of the second number….and so on..

Then, you end up with:

0.0222…….

This new number is different from ALL the numbers in our sequence – it differs from the first one in the first decimal place, the second one in the second decimal place…and so on.

So – our original assumption that our sequence contains all the reals cannot be true. There are more reals between 0 and 0.99999 than can be labeled with integers.

Summary

This was just a brief intro to some logical paradoxes and counting paradoxes in math. If you have any interesting ones, feel free to comment on this post.