Sunday, February 17, 2013

Misconceptions you may have

If the Universe is infinite, there must be a copy of me out there.

False. An infinite universe doesn't mean that everything necessarily repeats.

The incorrect assumption here is that all repetitions must occur. If the universe is infinite but this is the only populated planet (all others contain no life, or at least no human life), then clearly there is no copy of you.

What is true is that if there are only a limited number of configurations per space unit (say, all possible quantum states), then at least one such configuration must repeat infinitely in an infinite universe. However, it is possible that the state of total vacuum repeats infinitely much, so that outside a finite region of the universe (our observable one), there is only empty space. Universe is infinite, but most of it is empty and there is only one copy of you.

Whether these scenarios are plausible or not is another question, but logic and the laws of physics (as we understand them today) doesn't force multiple copies of you just because the universe is infinite.

If something has non-zero probability of occurring, it will eventually happen for sure.


Well, if the event has fixed non-zero probability (it doesn't change with time), then the event is bound to happen eventually.

If, however, the event has a time-dependent probability (it changes with time, as is usually the case in real life), this is false, even if the probabilities are always non-zero. The trick is that if the time-dependent probability is lowered fast enough, there is not necessarily a 100% chance of it happening at any point in time. Consider the event "World War breaks out", with probability     of breaking out in year k (it may happen more than once, but the events across years do not affect each other for the sake of simplicity). For the first year, the probability is .  In the second year, the probability drops to , then to   in the third year, and so on. The probability that World War never breaks out is then ,  so the probability that World War will break out (at least once) is 63%. Not for sure at all.

Everything that has a beginning must have an end.

False. The natural numbers have a beginning at 0, but, as there is no largest number, there is no end:
beginning ---> 0,1,2,3,4,5,6,.... ---> just keeps counting, no end.
Time itself may have a beginning in the big bang, but not necessarily an end (even when the universe reaches its maximal entropy state, quantum fluctuations ensure that change still occurs).

There's only one size of infinity.

False. There's an infinity of infinities. You can't count infinities, but you can pair one value from each of the two infinities, and see if there's anything left in either afterwards. This is analogous to determining whether there are more passengers or seats in an airplane by asking everything to set down, and check whether some people are still standing, or some seats available after the process. You don't count, you only compare relatively.

In this sense the set of natural numbers is smaller than the set of real numbers. That's two different infinities. To keep going, one shows that the power set is always larger than the original (infinite) set. The power set is the set of all (unordered) permutations of the original set. For example, the power set of {1,2} is { {}, {1}, {2}, {1,2} }. Hence an infinite hierarchy of sets.

Another example: when you enumerate numbers as 0,1,2,3,4,... you will eventually reach the first infinite number (ordinal), but logically speaking, there's nothing preventing you from continuing by doing infinity+1, infinity+2,... until you reach 2*infinity. By the same logic, then, you can have infinity*infinity, infinity^infinity, and so on.

Do these "higher" infinities exist in the real world? Maybe not. But in that case, the real numbers are not real at all, since they belong to a higher infinity...

See ordinal and cardinal numbers for more information.

Computers only deal with numbers. They can't know that , as we do.

False. This is based on the assumption that the only way a computer can compute the equality is by first squaring pi, and then taking the square root of that number.

Now, pi has an infinite non-repeating decimal sequence (it is an irrational number; in fact transcendental), so it is not even possible for a computer to store all the decimals of pi. This reasoning  is correct; what is incorrect is the assumption that the computer is limited to only performing such calculations when determining the equality.

Just like humans, the computer can store the algebraic rewrite rules
 ,    ,    ,  ,  and 

 from which it can easily deduce the equality:  

In other words, the computer can do it in just the same way as humans do it. And just like the computer can't store the infinite decimal sequence of pi and square that, neither can we.

All numbers have a unique representation. Thus 0.99999... = 1 is not true.

False. When you write 0.999..., the ellipsis is intended to mean "the 9s continue forever". This is thus a compact way to write 0.9 + 0.09 + 0.009 + ... This infinite series equals 1, so 0.99999... = 1 is in fact true. (Mathematically, the sentence asserts that the partial sums converge to 1, which is true.)

The number 1 can be represented in many other (decimal) ways: 1.0, 1.00, 1.000, and so on. If we only use infinite decimals, there are only two ways to represent each number. For example, the number 12.345 can be represented as 12.3450000... or as 12.3449999...

If we can compute an infinity of things in finite time, we can break any problem.

False. Assume for instance we have a hypercomputer blackbox that decides whether any given algorithm (not using the hypercomputer) terminates on all inputs or not (it solves the Halting problem). Such a hypercomputer cannot actually decide whether machines equipped with its own blackbox functionality terminates on all inputs, and thus, it cannot decide this particular "hyper" halting problem.

See arithmetical hierarchy for more information.