January 05, 2017 Thursday
Bedtime Story
Hilbert's Grand Hotel is Really Grand and Magnanimous
For time being, I shall not go into the birthday problem and leave
it to your curiosity.
The point that I wish to establish here is that in general, the
pigeonhole principle is not applicable to infinities (though in particular
cases it has been used) and hence to the Hilbert’s paradox of the Grand Hotel.
Let us review once again what Hilbert proposed.
Consider a hypothetical hotel with countably infinite number of
rooms, all of which are occupied.
Would it be possible to add any new guest in it?
Hilbert answer was yes.
All one needs to do to lodge in another guest is to shift the
guest in room 1 to room 2, the one in room 2 to room 3 and so on.
In this manner, not only one guest could be accommodated but
countably infinite number of patrons could be taken in.
Alas, mathematicians never let things be simple.
Their imaginations are vivid and let it run wild.
They tend to think extreme, creating their own examples and ask of
themselves “What happens if…?”
In fact it is even possible (though not likely) that a person is a
dreamer and can think like a true mathematician yet quite poor or average in
keeping his accounts or solve mathematical problems.
Not happy with countably infinite, Hilbert proposed what if
multiple tourist coaches came into that hotel each coach ferrying in countably
infinite number of passengers?
What then?
Can the Grand Hotel still accommodate all of them?
Remarkably enough, it is possible!
And how?
All you need to do is identify each and every person using two
numbers, one being the seat number (n) and one being the coach number (c).
And as long as you can pair each one specific combination of n and
c with a natural number, you would arrive at the positive answer.
Using the pairing function and using the axiom of choice, it can
be done!
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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