Imagine the following perpetual motor machine: a small conveyor belt running in a loop. Written on the belt is a single line of digits, which is continuous round the loop. When the conveyor belt reaches the top you can see some digits. What are they?

Of course, my 'infinitely long number' does not reflect the real complexity of the concept of infinity.

The fact that I suggest we can see digits in infinity implies that it either can be used to state the number of objects or some other unique value - ie.,
7 refers to 7 objects, 7th position etc. not to any other number of objects or position. Clearly, infinity is not like those numbers - as there seem to be sets of different sizes that include an infinite number of numbers (ie., set of rational numbers, intergers, odd numbers, primes etc.).

My understanding of infinity is that it refers to any set that has an uncountable number of members, ie., when you think you have counted them all there's always another exemplar that can be found.

However, infinity also implies the greatest magnitude (in contrast to infinitessimals) and from this perspective, I think we can say something about what we would see on my never-ending loop.

Welcome to my webpage Brian

A FASCINATING QUESTION BRIAN - below is my answer right now - I think your question needs a deeper answer but don't know what it is. I'd also like you to comment more & to see what other kids say.
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MY COMMENTS

IS SUCH A CONCEPT NECESSARY ? Could we understand the world without it? Let us look at some examples.

COUNTING

Imagine if everything in the universe could be lined up and counted. You would get a certain number. When you had counted all the things have you reached the end of numbers?

Of course not, because adding on a number is open-ended, ie., you can always add more. The effect of the adding on process is obvious even with the very small numbers, 0 1 2 .....9 and the number of digits increases from units, tens, hundreds ......... To me, this open-endedness is one necessary concept of infinity. Without it, numbers would necessarily be tied to objects & hence we couldn't understand mathematics as a thing in itself

SETS

Hope you know what a set is - like the different suits in cards. As noted above, counting is never-ending hence numbers are infinite. But also is the set of even numbers, primes, multiples of a million etc. Clearly such sets have a different number of members (ie., there are 2x more even numbers than odd & even numbers). This is only strange if you think of infinity as a number like 3 or 5.

It makes sense though if you think of infinity as the notion that new members of each set can always be generated (ie., the greatest prime number can never be found) and therefore, the members of an infinite set can never be counted.

PHYSICS

IN simple terms I think there are two concepts of infinity in physics (I'm sure I'll be corrected here !).

First, imagine being inside a ball where you don't know where the boundaries are. You travel outwards trying to find the inside surface of the ball but never do. Is the ball of infinite size ?

Second, imagine travelling round the circumference of a circle. Where is the beginning of the circle? And it's end ? If you had to start at the beginning and stop at the end such a trip would be impossible? Imagine a computer program that controlled a robot train to go round a circular track. How would the program tell it to stop (without creating an external criterion, like putting a station beside the circle)?

The above two notions I think underlie the notions of space being infinite.

UNSTOPPABLE COMPUTER PROGRAM

Imagine a computer program that sole job is to +1 to some starting number,
ie., 1 2 3 4 5 ......... Such a program would run forever. In computer science theory there are many unstoppable programs that define the limits of what a computers can do. (Can't think of any off hand.) (In contrast superficially with minds, but minds can find external factors to stop a mental program.)
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There is many other examples of how the notion of infinity is necessary. However, I'll finish with a simple point. It seems to me that the notion of infinity occurs because we have imaginations and that we can think logically and rationally about abstract relationships. In the absence of such abilities, we would be like no different from the apes or primitive man, who understanding of the world seem limited to what he can experience directly through the senses and through associating experiences as related only if they occur at very close together in time or/and space.

In a previous note I said that since there is no way (without external criteria) for specifying the beginning and end of a circle then a robot travelling round the circle would travel forever unless stopped by some external force (ie., it's battery ran out).

I know think this may not be [quite] correct. Imagine a robot that leaves slime behind it to mark the circular path it is travelling around. The program could be programmed to stop when it reaches slime. Therefore, for an unmarked circle there would be a marked start and finish point if the traveller left a trail behind them as they travelled, I don't think this works for a premarked circle as the trail-making substance would be exterior to the marked circle.

Christopher I look forward to seeing your hyperbolic algebraic infinity-expressive equation. I also challenge you to answer what digit[s] would be seen on my perpetual motion machine I described above.

Below are three books that have informed me about infinity.

Darling, D. (2004). The Universal Book of Mathematics. Wiley.

Kaplan, R & Kaplan, E. (2003). The Art Of The Infinite. Penquin.

Wallace, D.F. (2003). Everything and More: A compact history of infinity. Phoenix.

Also use it as a key-word on the net.