which is the bigger set

[mic-d]

If you made a group with every whole number and another group with only every even whole number, which group would contain the most numbers?

[Avery]

both the same.

The Hotel INFINITY

At the Hotel Infinity things are different. Suppose that you turn up at the check-in counter of the
Hotel Infinity only to find that the infinite number of rooms (numbered 1, 2, 3, 4, . . . and so on,
forever) are all occupied. The receptionist is perplexed – the Hotel is full – but the manager is unperturbed.

No problem, he says: move the guest in room 1 to room 2, the guest in room 2 to room
3, and so on, forever. This leaves room 1 vacant for you to take and everyone still has a room!

You are so pleased with this service that you return to the Hotel Infinity on the next occasion
that you are in town, this time with an infinite number of friends for the ultimate reunion. Again,
this popular hotel is full, but again, the manager is unperturbed. We can easily accommodate an unexpected
party of infinity, he explains to the nervous receptionist. And so he does, by moving the
guest in room 1 to room 2, the guest in room 2 to room 4, the guest in room 3 to room 6, and so
on, forever.

This leaves all the odd-numbered rooms empty. There are an infinite number of them
and they are free to accommodate you and your infinitely numerous companions without leaving
anyone out in the cold. Needless to say room service is a little slow at times to some of the highnumbered
rooms.

[mic-d]

Correct. Now if our universe is not infinite, would it be able to contain an infinite set? Even the concept of an infinite set, such as whole numbers? Just a philosophical question...

[Avery]

There can never be an infinite set of numbers.
Should you ever think that you have found one, let me know, I will add 1 to it.


‘There is no smallest among the small and no largest among the large but always
something still smaller and something still larger.’

Anaxagoras (500–428 BC)


‘The infinite has a potential existence . . . There will not be an actual infinite’

Aristotle


‘The infinite turns out to be the opposite of what they say. The infinite is not that of which
nothing is outside, but that of which there is always something outside. That of which
nothing is outside is complete and whole. By contrast, that of which something, whatever
it might be, is absent is not everlasting.’

Aristotle (again)

[_fly_]

If you made a group with every whole number and another group with only every even whole number, which group would contain the most numbers?

And I'm guessing you worded this wrong.
you didn't say the first group is all the odd, you didn't say take the even ones out of the first to put in the second, you didn't say you couldn't have duplicate numbers (u.e only one set). first group could have 1234 and the second 2 and 4.

[mic-d]

Perhaps no largest among the large, but I think it's been suggested that there is perhaps a smallest possible dimension named after somebody's theory, something called a Planck distance?? from memory...

[mic-d]

And I'm guessing you worded this wrong.
you didn't say the first group is all the odd, you didn't say take the even ones out of the first to put in the second, you didn't say you couldn't have duplicate numbers (u.e only one set). first group could have 1234 and the second 2 and 4.

Nope I wrote what I meant, but to say it another way... I said the first group contains every whole number (odds and evens), but the second set contains only every even number, or in other words, all the odd numbers have been dropped. Both groups still contain the same number of er, numbers...

[Avery]

Perhaps no largest among the large, but I think it's been suggested that there is perhaps a smallest possible dimension named after somebody's theory, something called a Planck distance?? from memory...

Planck's length.


Consider, if you start on a journey of 1 kilometer, before you get there, you must first complete a distance of 500 meters (half the length of your journey). Then you must complete a length of 250 meters, then 125 meters ,etc. etc..

Eventually you will have only 1 Planck's length to travel. Before you can complete that, you will have to travel one half Planck's length.

You will never get there...

[mic-d]

Planck's length.


Consider, if you start on a journey of 1 kilometer, before you get there, you must first complete a distance of 500 meters (half the length of your journey). Then you must complete a length of 250 meters, then 125 meters ,etc. etc..

Eventually you will have only 1 Planck's length to travel. Before you can complete that, you will have to travel one half Planck's length.

You will never get there...

I understand that, intuitively it is reasonable. My caution would be whether it is possible to rely on intuition, based on our macroscopic experience, when we reach these incredibly small dimensions.

[Avery]

I understand that, intuitively it is reasonable. My caution would be whether it is possible to rely on intuition, based on our macroscopic experience, when we reach these incredibly small dimensions.

It's a bit like cutting a critical tenon on your table saw. Leave it a bit long at first and creep upon it with a few cuts - just dial the fence in a little more... and a bit more... and then

ah well. there's always another piece of timber.

Although ,I don't think my table saw fence goes down to Planck's length, just a plancks length.

[rwbuild]

Planck's length.


Consider, if you start on a journey of 1 kilometer, before you get there, you must first complete a distance of 500 meters (half the length of your journey). Then you must complete a length of 250 meters, then 125 meters ,etc. etc..

Eventually you will have only 1 Planck's length to travel. Before you can complete that, you will have to travel one half Planck's length.

You will never get there...

Your analogy is incorrect, otherwise we would never arrive home, at work, fly from one country to another, etc, conversely, having arrived, you then return doubling the distance as you travel, thus, you travel half a planck, then a full planck, etc, 1 kilometer then two kilometer, etc.

So in effect, whilst you keep shortening the distance to the destination, you are also increasing the distance from departure, each being relative to your need or point of view. Both are finite, however, if you were to set off in a space craft and keep traveling away from earth, how far could you go . We will never ever know

[Avery]

Your analogy is incorrect, otherwise we would never arrive home, at work, fly from one country to another, etc, conversely, having arrived, you then return doubling the distance as you travel, thus, you travel half a planck, then a full planck, etc, 1 kilometer then two kilometer, etc.

So in effect, whilst you keep shortening the distance to the destination, you are also increasing the distance from departure, each being relative to your need or point of view.

Of course.

This is known as Zeno's (450 BC (or BCE if you like)) paradox.

Zeno was a disciple of a local philosopher called Parmenides who held that the
Universe was just one thing and that it consisted of just a single thing. This one thing was timeless
and changeless. This led to the immediate conclusion that nothing really moves because this would
require more than one thing, or state, to exist: one before the movement occurred and another after
it had finished. All the movement we see, claimed Parmenides, was just an illusion on the surface
of things. Deep down, the Universe was a single changeless reality.

This idea was not readily accepted by the locals who could see motion everywhere they looked. In order to support his teacher Zeno came up with 4 ideas that showed motion was not all it was cracked up to be. His second paradox involved a race between Achilles and a tortoise. It showed that Achilles could never actually catch and pass the tortoise. I think this paradox was later modified and used in another story.


‘The will is infinite and the execution confined. The desire is boundless and the act
is a slave to limit.’

William Shakespeare, Troilus and Cressida

Which I think was a play about two Japanese cars, which of course, are renowned for going on forever.



"Both are finite, however, if you were to set off in a space craft and keep traveling away from earth, how far could you go . We will never ever know "


But just suppose you went as far as you could go in your spaceship, and when you got there you threw a rock over the boundary...

‘Thus there are properties common to all things, and the knowledge of them opens the
mind to the greatest wonders of nature. The principal one includes the two infinities which
are to be found in all things, infinite largeness and infinite smallness.’
Blaise Pascal

[gilamonster]

I just read all these posts and now I'm infinitely tired!