Inifinities.....

...come in all shapes and sizes.

We all know what infinity is, don't we? Or do we?

Sure we do! It's where something goes on forever, never stopping, like the number of numbers. There is no highest number because they go on forever, right? And the decimal portion of Pi goes on forever, does it not? It begins 3.1415926... and never stops and never repeats.

We are so used to this concept of "infinity" that we are almost blase about it.

Allow enlightenment to take place.....

Suppose there is a hotel with an inifinite number of rooms, all occupied by guests.

What does the owner do when a new guest arrives? Why, he puts guest 1 in room 2, guest 2 in room 3, guest 3 in room 4....and so on up to infinity, leaving room 1 empty.

What does the owner do when an infinite number of new guests arrive? This one requires a little more thought.....

Simple, really. He puts guest 1 in room 2, guest 2 in room 4, guest 3 in room 6 and so on up to infinity, leaving an infinite number of "holes" ie vacant odd-numbered rooms. All guests are now occupying rooms. Does that mean the owner somehow now has 2 times an inifinite number of rooms? NO! he still has an infinite number of rooms with a guest in each!

It gets even stranger. Georg Cantor proved that there is more than one "type" of infinity. This guy was involved in set theory, the theory of certain classes of objects, such as the set of cars, the set of brown cars, the set of brown cars with at least 2 wheels, etc.

Sometimes sets can be inside each other, sometimes not. For instance, the set of Green Cars is different from the set of Brown cars, but above the third set "the set of brown cars with at least 2 wheels" is contained inside set 2 "the set of brown cars" AND set 1 "the set of cars".

Cantor took this a little further and argued that sets sometimes can contain themselves. For instance, the set of cars is not a car itself, but the set of all sets IS a set, so it is contained within itself. These are called "self-swallowing" sets. Another example is the set of ALL things that are not brown cars.

So, You have ordinary sets eg "the set of brown cars" which are not members of themselves, and self-swallowing sets eg "the set of all sets".

The question arises: Is the set of all ordinary sets ordinary or self-swallowing?

Inspection shows it is neither. Try it! Assume it is an ordinary set.

So, the set of all ordinary sets belongs to neither group.

The argument put forward above is called the "diagonal" argument for reasons which are not at all clear. Here comes Cantor's same argument, but in the form he proposed.

To begin, suppose we wish to count all numbers. Not just integers such as 1, 2, 3 etc, but ALL numbers, including those with decimals. Well, one of them is pi = 3.1415926.... Another is e = 2.718281828....... and ALL the others in between and around , such 9.80665, and 5.5, 100.99999 and 2.9979. Suppose we list them all in order. For simplicity, let's just consider all those less than 1.

It would look something like

0.1111111111111111....>>>> to infinity
0.2222222222222222....>>>> to infinity
0.3333333333333333....>>>> to infinity....

and suppose our table lists them ALL, one after the other. Obviously there are an infinite number. There is a proof that there is at least one number NOT in our table of the infinitely many "numbers".

It goes like this:

Take the first number, 0.1111111111111.....and change the first column to a "2" creating 0.2111111111111....

Take the second number, 0.222222222222.... and change the SECOND column to a "3" creating 0.2322222222222....

Do the same with the third number. So we will have

0.2111111111111111....>>>> to infinity
0.2322222222222222....>>>> to infinity
0.3343333333333333....>>>> to infinity....

where I have changed the third column in the third number to a "4". Now look at the diagonal. It says 0.234.......to infinity.

It is clear that this number is NOT in our list, as the first column is different from number 1, the second is different for number 2, and so on up to infinity. For any "n", the digit in column n is different from the nth digit of the number "n", so our "made-up" number, derived from adding 1 to the diagonal is absolutely NOT in the list.

But didn't we say that our list was of ALL the numbers all the way to infinity?

The reasoning is that there MUST be another "kind" of inifinity somehow "bigger" than the one we had. In fact, there is not only "one" different kind of infinity, but an infinite many of them, each, in some abstract sense, "bigger" than the last.

This result is true and has had many implications for math, especially for that man Godel again, who said that not everything is proveable within the system. This is a paraphrase of that theorem.

Age-old Debate

WARNING! VERY SUBJECTIVE POST! Comments welcome!

Tonight members of the UK Parliament will vote on a new Law, or rather a change in an existing one.

At the time of writing, the maximum legal term allowed for an abortion is 24 weeks. If the embryo ("foetus" to abortionists, "unborn child" for anti-abortionists) is older than 24 weeks, it is illegal to perform an abortion. Tonight it will be decided whether to lower this to 20 weeks.

The reasoning behind this is cited as improvements in embryo survival rate in premature children. Although this data refers to embryos born after 24 weeks, it is decided tonight whether it may apply to embryos between 20 and 24 weeks.

The Catholic Church's stand on this is as it has always been: "Abortion is murder". Other bodies of people object on moral ground. The question remains: After how many weeks does it become wrong, if ever or never?

Facts aside, there is another viewpoint other than "Is it right or wrong, and if wrong, after how many weeks?".

It is this:

For nine months, the embryo resides in another person's body, feeding off that body as the "host" ingests food for it to grow and survive upon.

It may sound a callous way of putting it, but essentially these are the facts as to what is happening.

I can even think of a name for the embryo during those stages.

A parasite.

What do you do with parasites?

Squish them .

Just For Laughs Part One

This is the sort of thing that makes me laugh.......

A Sequence of Events

A sequence is a succession of numbers that follow on from one another.

The simplest is

1, 2, 3, 4, 5, 6.....

Where all you do to get the next is add 1 to the last.

2, 4, 6, 8, 10....

is similar, in that you add 2 to the last one.

1, 3, 5, 7, 9....

is also similar, adding 2, but starting at a different number.

So much for addition.

How about 1, 2, 2, 4, 8, 16, 32....

This one is where you multiply the previous two to get the next.

Or even,

1, 2, 3, 5, 8, 13, 21, ......

where you add the previous 2 to get the next.

And so it goes on.

So, a common puzzle is to guess what the next one is....

Try this..

1, 4, 9, 16, 25....?

The answer is of course 36, each being the square of the integers.

This one..

1, 8, 27, 64....

The answer is 125, since they are the cubes of integers.

What about this one?

3, 3, 5, 4, 4, 3, 5, ?

That's harder - the answer is 5, the numbers being the "number of letters" in the integers. ONE = 3, TWO = 3, THREE = 5 etc

NOW, try THIS one...

1, 1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, 11, ?

What is next? First prize is a can of worms

The Most Remarkable Formula in All Math.......

.....involves three of the most famous constants of all, all encompassed in five symbols.

The constants are:

"pi", the number equal to the circumference of a circle divided by it's diameter, numerically, 3.1415926.....

"e", perhaps less-known, but maybe equally widely used. It is the base of natural logarithms and is the limit of ((1 +1/n)**(n)) as "n" gets nearer to infinity:





"i", the square root of -1, the "imaginary number",

-1 itself,

and an equals "=".

In visual form it is:









In order to get to this amazing result, an understanding of calculus, complex numbers, and trigonometry is needed.

Now When I'm Smiling I Know Why.......

Perfect Day, Perfect Night,

Perfect Song, Mr. Right...?

Loose Elastic

One of the funniest events ever to happen to me occurred in one of my Korean classrooms. My job was to teach these kids (aged 8-15) English in English.

The problem with Korea is that almost the entire population wants to speak English but can't. But they are getting there (gradually). They employ native speakers of English to be present in their English Language, er "Schools", and although technically my mother tongue is NOT English, INHA University managed to get me clearance to teach as a special case.

It's very easy. However, because of the way the system works, I never knew what classes I was to teach next, nor the age range.

So I got into the habit of asking each student what their names were before I started to teach, writing them down in my notebook, along with their position in the classroom (with 300 students, it was sometimes very difficult to recall names. These Koreans ALL look alike!)

In one of the classes, I had a new student. A little girl, aged maybe eight, very small in stature and quite angelic looking. She sat on the front row, right beneath my lectern. She had very big eyes couched in innocence. So, I started asking names. "My name is Danny", My name is Emma", "My name is Sandra", all of them announced rather shyly, and I wrote the names down.

Then the new student. I bent my head down and said quietly, so as not to embarrass her. "What is YOUR name?" I barely whispered.

Looking straight ahead, taking a huge deep breath, and with a voice Mr Richter would have been proud of, she bellowed

"MY NAME IS L U C Y !!!!!"

to the astonishment of all present.

I almost dropped my marker pen.

A H S S A !

It just does not get any better than this!

This song is still very popular in Korea. The guys who sing it are almost National Heroes. Everyone knows the Turtles!. The song impresses me simply because they ALL seem to be able to sing. My upbringing is such that I am not English anymore. I am more Korean than anything else - even Japanese.

In this wonderful happy song, there is "Hana - Dul - Se - Ne" which means 1-2-3-4, and "Ahssa!" (Yahoo!) One of my students was named Hannah and I made fun of her because it sounds like "1"!!

Suffice it to say, this video just about sums it all up for me right now. The translation is clear to see, if translation is needed!

There IS more to life than McFly or Leona Lewis!

Fuzzy Logic

The previous post attracted a comment which incensed me greatly. The comment was about probability and quantum mechanics, and made dubious claims.

What the reader failed to state at any time was this:

"Suppose normal everyday objects obeyed the Laws of Quantum Mechanics".

And the person went on to describe a "marble-in-box" puzzle where the marble could have been been anywhere.

IT DOESN'T MATTER!!!! In normal everyday life, there is NO quantum mechanics!!! Quantum Mechanics was invented to explain normal everyday life when it got very small!

To those people who want to apply these rules to everything they see, let them. They will fail. There is a video out called "What the (bleep) do we know?" that claims everything is just an illusion because of Quantum Mechanics. Is it a vision by scientists, known and respected throughout the community? NO! it is a video made by doctors (neuro surgeons), philosophers, psychiartrists and other thinkers who claim they know Quantum Mechanics better than those who actually use those laws.

The best example of what I call to be the "Quantum Fuzziness" is Shrodinger's Cat (see the post "Wanted - Dead or alive"). He invented the Cat just to show everyone how silly it is to take Quantum Mechanics beyond the realm it was invented for.

Sadly, some people still believe that when they go to the store to buy an apple, there is a distinct probability that they will instead, buy a 14-toed giant sluglike marsupial Mexican Bandit instead.

On the planet Oohhuhuhooompha.

In subjunctive tents.

Find the Queen..revisited.

This question first appeared on this blog a couple of years back.

Check out my 10th July 2005 post:

"Five Clues - One for each 10 of 50"

The question was this...

You have three cards, (two aces and a queen), all face down. You are asked to find the queen, and you are asked to pick one card only. Then, the magician looks at them all, and promptly turns face up an ace. What is the probability that your card is the queen?

The answer is a third, since you made the choice when all three were face down.

NOW consider the same scenario...

You have three cards, (two aces and a queen), all face down. You are asked to find the queen, and you are asked to pick one card only. Then, the magician looks at them all, and promptly turns face up, an ace.

Your delicate finger is still on the card you chose before any were revealed, and YOU KNOW the probability is 1/3 that yours is the queen.

So, you have 3 cards, two face down, and an ace face up. You have tiny finger on your face down card, knowing that it is 1/3 that it is the one queen.

Now, lift you finger up, really carefully. You have 3 cards, one face up, an ace, and two face down, the other ace and the queen. You are then asked to make your choice again, so you put your same finger down, ever more carefully, ON THE SAME CARD YOU CHOSE BEFORE.

NOW, what is the probability that your card is the queen?

The astonishing answer is that it is now 1/2.

So....what changed between your lifting your finger off the card, to replacing it back, to cause this change in probability?

Moreover, is the answer different if you had chosen the other card instead?

Comments welcome.