2009-10-24

The countability of real numbers

Natural numbers (symbol N) are positive whole integers starting with either 0 or 1. It is obvious that there are an infinite number of natural numbers, and that they are countable. By starting with the lowest natural number and counting long enough, you will eventually reach any other chosen natural number.

Integers (symbol Z) are countable as well. The set of integers contains zero and all natural numbers as well as their negatives. To count them all, just start with 0, then 1 and -1, then 2 and -2, etc.

Rational numbers (symbol Q) are those numbers that are expressible as fractions, e.g. 1/2. Rational numbers are also countable. For an idea of how to count them, imagine a two-dimensional grid that has integers on each axis, and contains infinitely many horizontal and vertical lines, one for each integer on the respective axis. To count all rational numbers, start at (0,0). Then, count all intersections on the border of the square between (-1,-1) and (1,1). Then, count all intersections on the border of the larger square between (-2,-2) and (2,2). And so on.

Algebraic numbers are those numbers that are roots of a polynomial. Algebraic numbers include e.g. the square root of 2, which is the root if (x^2 - 2 = 0). Algebraic numbers aren't necessarily rational, but they are also countable. They are countable because their quantity is limited by the quantity of polynomials, and polynomials are countable. (A polynomial is determined by its coefficients, and the coefficients can be written down as a number.)

Finally, we arrive at real numbers. Real numbers (symbol R) are an expansion of rational (and algebraic) numbers that is intended to address a certain deficiency. The deficiency is that there exist convergent series of rational/algebraic numbers which appear to converge to something, but that something is not a rational (or algebraic) number. A classic example of such a convergence is Pi.

The idea behind real numbers is to have a set which, in addition to rational numbers, also contains the converging points of all converging series that can be constructed within the set. This makes the set "complete".

The traditional idea is to envision a real number as an infinitely long series of zeroes and ones (i.e. the number itself expressed in binary, with a decimal point some place).

If one imagines a real number like that, then it intuitively follows that a set of all-real-numbers is the set of all such infinite-series-of-0-and-1.

Cantor's diagonal argument can then be used to show that a set defined that way is uncountable. If you have an infinite set of unique infinite-series-of-0-and-1, such that each series in the set is marked with an index; then you can construct another infinite-series-of-0-and-1 which differs from all the other series that have an index. So therefore, the indexing is not complete. The set contains more series than can be counted.

Well yes, but: if we are constructing the set of real numbers as the set of rational numbers + the converging points of all converging series, then aren't the converging series countable? Every converging series must be the result of some algorithm. But algorithms are expressible as code, and code is expressible as numbers. So the number of possible algorithms is countable. So the number of all converging series must also be countable. If we construct the set of real numbers this way, then, it is a union of two countable sets - so it must itself be countable.

Does it not follow, then, that the standard envisioning of real numbers as infinite-series-of-0-and-1 must include an infinite number of them that are neither rational, nor the result of any limit? If so, what is the use of all these extra, unidentifiable "real numbers"?

It sure seems that, all those real numbers that we can actually express, or use in any way - are countable.

Or is the number of converging series uncountable? If so - how?

2009-10-19

Somali pirates, Italian mafia, and radioactive waste

Looks like the story about Somali pirates has some twists and turns. Johann Hari:
As soon as the government was gone, mysterious European ships started appearing off the coast of Somalia, dumping vast barrels into the ocean. The coastal population began to sicken. At first they suffered strange rashes, nausea and malformed babies. Then, after the 2005 tsunami, hundreds of the dumped and leaking barrels washed up on shore. People began to suffer from radiation sickness, and more than 300 died. Ahmedou Ould-Abdallah, the UN envoy to Somalia, tells me: "Somebody is dumping nuclear material here. There is also lead, and heavy metals such as cadmium and mercury - you name it." Much of it can be traced back to European hospitals and factories, who seem to be passing it on to the Italian mafia to "dispose" of cheaply.
In other news, the Italian mafia is reported to have sunk as many as 30 ships of toxic waste off the coast of Italy.

Johann Hari makes perverse conclusions - his perspective is classically ridden with misplaced Western guilt. But this makes for an interesting twist in the story. It's not just that a failed state breeds criminality; it breeds a muddled state of affairs where criminality is intertwined with vigilantism. A failed state is not one where well-minded people are absent; it is a state where well-minded people are disorganized, unable to enforce laws and counter criminals. It is a state where it is hard to maintain your principles because, in the absence of law, there are no clear guidelines on what's acceptable. One day, you use violence to chase off waste-dumping mobsters and invading fishermen. Not that long thereafter, your friends are engaging in outright piracy.

Obviously, toxic waste requires better controls. You can't just outsource its disposal to any mob-run company. They're gonna dump it either in your backyard, or someone else's backyard, or in the middle of the Atlantic. It's not that hard to manufacture papers which "prove" that the ship and its cargo were "sold" to someone in Nigeria...

2009-10-15

Gay rights worldwide

In a time when the civilized world is slowly progressing towards equal rights for gays and lesbians, it is interesting to look at this map and see what world regions have the harshest penalties for homosexuality.


Source: Wikipedia


It's striking to see how many countries have penalties as harsh as death, or life in prison. In much of the Caribbean, the penalty is 10 years. In Jamaica, it is "10 years of hard labor".

Fascinatingly, the South African Republic is an exception - same-sex civil marriage has been legal there since 2005.

2009-10-10

Jaw bone created from stem cells

 
Fantastic!

In a couple of decades, it may be standard practice to replace older people's worn out joints by growing new joints from the patients' stem cells.

Let's hope for similar breakthroughs that will allow the growing and replacement of people's organs, as well.

Now consider IBM's efforts to develop techniques for quickly and cheaply sequencing a person's DNA.

These types of breakthroughs may allow a person's DNA to be sequenced when young, and reproduced when the person is older.

The DNA in your cells gets damaged as you age, but a digital copy of your DNA takes a snapshot at the age at which it was sequenced.

Live another 50 years, and the breakthroughs that will be achieved during this time may allow us to continue living much, much longer!

2009-10-08

Religious zealotry vs. child molestation

In March 2008, an 11-year old died from undiagnosed but treatable diabetes as her parents prayed by her side without even considering taking her to a doctor.

Now, the parents have been sentenced leniently by a judge who sympathized with them.

Quotes from the article:
During the sentencing hearing, Leilani Neumann, 41, told the judge her family is loving and forgiving and has wrongly been portrayed as religious zealots.

"I do not regret trusting truly in the Lord for my daughter's health," she said. "Did we know she had a fatal illness? No. Did we act to the best of our knowledge? Yes."

Dale Neumann, 47, read from the Bible and told the judge that he loved his daughter.

"I am guilty of trusting my Lord's wisdom completely. ... Guilty of asking for heavenly intervention. Guilty of following Jesus Christ when the whole world does not understand. Guilty of obeying my God," he said.
Compare this to child molestation.

If the parents were mad in a sexual way, and believed that it's okay for them to molest their children, they would be sent away for a long, long time, and would never see their kids again.

But the parents are mad in a religious way. Religious madness is apparently much more acceptable. Even when it results in death, it does not warrant taking the children away.

The parents get a slap on the wrist, and they get to keep their kids so they can continue exposing them to religious abuse in the future.

Progress in tiny nuclear batteries

 
BBC article with a photo of the battery.

PhysOrg article with comments.

2009-10-03

Half the human population is evil

A study performed at the Faculty of Economics and Management in Magdeburg confirms my intuition that nearly half of all humans are inherently unempathic, even downright evil, but simply hide their psychopathy most of the time, because civilization expects that from them:
The Pleasure of Being Nasty

We introduce the joy-of-destruction game. Two players each receive an endowment and simultaneously decide on how much of the other player's endowment to destroy. In a treatment without fear of retaliation, money is destroyed in almost 40% of all decisions.
See Figure 1 - Destruction frequency over time. At least 25% of people will cause gratuitous harm to a random stranger if they can do so at no cost, while up to 50% will do so if they can also deny it.