Nasty, British and Short

If you’ve done high school math, you probably know that there are a lot more real numbers than rational numbers. (Technically, the rationals are countable – you can pair them off with whole numbers – and the real numbers are uncountable – there are too many of them to pair off).

So it’s no surprise that between any two rational numbers, you can always find an irrational number. (Proof: if a and b are two rational numbers, a + (b – a) / pi is irrational).

The surprise is that there is a rational number between any two irrational numbers. (Proof: suppose a and b are irrational with a < b. If b – a > 1, then there is an integer between them and we’re done. Otherwse, we have 0 < b – a < 1, and so there is some integer N such that b – a > 1/N. Therefore Nb – Na > 1, so there is an integer k between Na and Nb. Then k/N is a rational number between a and b.)

So although one set is much larger than the other, there is a kind of interleaving between them that you wouldn’t intuitively expect. Similarly, there is no interleaving between the integers and the rationals, even though they are the same size.

Conclusion: infinite sets behave quite differently from finite ones. Just because you can interleave members of two infinite sets, it doesn’t mean that the sets are the same size as it does for finite ones. That said, being able to place members into a one-to-one correspondence is a valid test of relative size.

As we all know, the President carries the nuclear missile codes with him at all times.  President Carter’s codes once ended up at the dry cleaners by mistake, and the FBI accidentally took President Reagan’s codes after he was shot.

The history of nuclear codes is rather interesting.  After the Cuban Missile Crisis, Defense Secretary McNamara vowed to personally see to it that all long range missiles were protected by an eight digit code.  The protection came from a device called a Permissive Action Link (PAL).  One weapons designer stated the security requirement as “Bypassing a PAL should be about as complex as performing a tonsillectomy while entering the patient from the wrong end.”

But a security system is only as strong as its weakest link, and for 15 years until 1977, the codes were set to 00000000 so as not to interfere with the ability to launch the missiles.  Anyone with an operations manual could have done it.

Nowadays, as well as actually having a code, PALs are designed to intentionally misfire the warhead if they detect tampering, so as to render it mostly harmless.

So what kind of safeguards are in place to protect the world against nukes?  The two main requirements are called “assure” and “assure against”.  The first guarantees that missiles will be launched when appropriate authorisation is given; the second guarantees that they won’t be launched accidentally or maliciously.

Some examples of things to protect against:

• The President loses his nuclear codes, and someone else uses them to launch an attack.
• An accident happens in a nuclear base and the missiles are detonated.
• A military base is overpowered by force and the attackers get access to the missiles.
• The President and Vice President are both killed in a nuclear attack, and a retaliatory launch is impossible without their authorisation.
• A launch is authorised, but the PAL launch codes are unavailable (this is actually why the codes were set to 00000000 in 1962).

The details of how an authorization gets from the President to a launched missile aren’t really clear, but it seems to work something like this.

1. The President and the Secretary of Defence both have to authorise a launch.  The NSA provides the technology to verify the authenticity.
2. The verified authorisation goes through military chain of command to the missile locations themselves.  Likely the Presidential authorisation is used as a key at each level to produce codes for the next level down.
3. At the actual launch site, there is a safe containing the launch codes.  The safe must be opened by turning two keys together, which are far enough apart that a single person cannot do it.  Both operators must then verify their codes against the ones in the safe.  The same procedure must also happen at another launch site before the missile can be launched – presumably the safe only contains part of the PAL code and the rest of it comes from the other site.
4. Submarines have a different procedure: there are several combination-lock safes on board, and each safe contains part of the key.  The full code can be reconstructed only from a combination of keys, and additionally, no crew member has any of the safe combinations; they come as part of the launch order.

The British have a different scheme, which has some good things (a launch order has to pass through more people) and some bad things (missiles had less physical protection up until very recently).

Nearly 4 billion years ago, cyanobacteria (otherwise known as Blue Green Algae) figured out how to get energy from light, and the world has never been the same since.

The photosynthesis reaction takes in water, sunlight and carbon dioxide, and outputs oxygen and (more importantly) carbohydrates which store energy needed by the bacteria.  Plants rely on photosynthesis – and so do all animals, since they either eat plants, or eat animals that eat plants.

Back when photosynthesis first started, our planet’s atmosphere was made up of four gases: nitrogen, carbon dioxide, methane, and steam.  There was no oxygen, and the carbon dioxide trapped heat via the greenhouse effect.  But the bacteria started consuming the carbon dioxide, and a couple of billion years later, it had mostly disappeared.

As CO2 levels dropped, so did the earth’s temperature.  Ice began to form around the poles, and the more ice formed, the more sunlight got reflected back into space, until the Earth became covered in a kilometre deep layer of ice about 2 billion years ago.  The temperature from pole to equator was a chilly -40 degrees.

Cyanobacteria are very hardy, and when frozen, they just go into hibernation until a thaw happens.  There are rivers in Antarctica that have frozen and then dried out, and hibernating bacteria on the river beds have been thawed out and immediately come back to life.  So when the Earth became a giant snowball, they just waited for something to happen.

The first deep ice age lasted about ten million years.  Eventually, volcanic eruptions belched enough CO2 into the atmosphere to restart the greenhouse effect, and the first thaw happened.  The cyanobacteria then woke up and started the process all over again.  There were dozens of ice ages, each about ten million years and with thaws of about a million years in between.

Finally, about 600 million years ago, during a thaw, more complex life evolved during the cambrian explosion, which restored the CO2 balance and stopped the process.

Could this happen again?  Possibly, but not likely.  There are only two plausible scenarios: first, wait a few million years until the continents have moved around: if Antarctica travelled more towards the equator, it would be possible for a large area of ice to deflect enough sunlight to reduce surface temperature and kick off the process again.  Alternatively, a large asteroid strike could possibly block out sunlight long enough for ocean surfaces to freeze (which would have happened 65 million years ago except that the water temperature was higher then).

Something to think about on a cold day like today.