dividing an infinite quantity

[tpogge]

What happens if one divides an infinite quantity by a finite number >1 (e.g. 2) an infinite number of times?

Two intuitions:

1) after each division, the remaining quantity is still infinitely large, thus even an infinite number of divisions cannot reduce it below infinity.

2) Eliminate from the set of natural numbers every other member repeatedly. After the first step, only every other number is left. After the second step, only every fourth is left. After the 3rd step only every 8th. And so on. If one does this forever, no natural number will escape elimination. So the process converges toward zero.

Which intuition is correct? And why is the other one deceptive?

 

[Zermelo]

What happens if one divides an infinite quantity by a finite number >1 (e.g. 2) an infinite number of times?

Two intuitions:

1) after each division, the remaining quantity is still infinitely large, thus even an infinite number of divisions cannot reduce it below infinity.

2) Eliminate from the set of natural numbers every other member repeatedly. After the first step, only every other number is left. After the second step, only every fourth is left. After the 3rd step only every 8th. And so on. If one does this forever, no natural number will escape elimination. So the process converges toward zero.

Which intuition is correct? And why is the other one deceptive?

Depends on what you mean by “infinite quantity”. Do you know what infinite cardinal numbers are, are you talking about them? Or is this an idea from calculus and you are thinking about limits and functions that diverge to infinity?

 

[tpogge]

I was actually thinking of infinitely large sets, such as the set of all natural numbers.

 

[darkyadoo]

An infinite quantity is not defined.
the word infinite is used for limits and cardinals and there exists a definition for each one.

If you don't use the definitions, you risk thinking for an infinite time!

 

[HallsofIvy]

I was actually thinking of infinitely large sets, such as the set of all natural numbers.

Then what do you mean dividing a set by a number???

 

[tpogge]

Divide one set into two, eliminating one part and retaining the other. Each step eliminates infinitely many members & retains infinitely many. This is repeated infinitely many times.

 

[Steven G]

Divide one set into two, eliminating one part and retaining the other. Each step eliminates infinitely many members & retains infinitely many. This is repeated infinitely many times.

Please stop saying divide sets.

Let A = {1,2,3, 4}. What the heck does A divided by 2 mean?

Let B = {0, 1, 2, 3, 4, ....}.
Step 1: remove every other number. Left with {0, 2, 4, 6, 8, ...}
Step 2: Remove every other number. Left with {0, 4, 8, 16,....}
Step 3: Remove every other number. Left with {0, 8, 16, 24, 32, 40, ...}
...
Please tell me in which step 0 gets removed

After each step you still have an infinite number of numbers!

 

[Zermelo]

Divide one set into two, eliminating one part and retaining the other. Each step eliminates infinitely many members & retains infinitely many. This is repeated infinitely many times.

The area you’re talking about is very slippery, weird and somewhat controversial, and it’s called Cardinal Numbers. Investigating this kind of topic requires a lot of firm knowledge of set theory. Cardinal numbers are used to extend the idea of how many elements are there in a set to infinite sets like N. We can even define multiplication and division of those numbers, but that may not be what you’re looking for. I would reccomend reading about them, and looking at Jomos response for this particular topic

 

[tpogge]

Thank you, Zermelo.
Let me explain why I find Jomo's answer unhelpful.
His final sentence - "After each step you still have an infinite number of numbers!" - is my first intuition.
In order to trust it, I need to see how my second intuition is wrong.
Jomo's argument here fails. His elimination is constructed never to eliminate 0. This (a) shows at best that one element survives, not that infinitely many do; and (b) may well be avoidable by designing the elimination differently.
Using Jomo's set B, here is how I would set up the elimination:
Step 1: remove every other number beginning with the first (0). Left with {1, 3, 5, 7, 9, ...}
Step 2: remove every other number beginning with the first (1). Left with {3, 7, 11, 15, 19, ...}
Step 3: remove every other number beginning with the first (3). Left with {7, 15, 23, 31, 39, ...}
Step 4: remove every other number beginning with the first (7). Left with {15, 31, 47, 63, 79, ...}
and so on.
Doing this forever, we will be eliminating every element of B. This is my second intuition. And I have yet to see what's wrong with it.

 

[Steven G]

In my opinion, and I by no means a set theorist, I constructed sets using your method and 0 never got excluded. You method, according to you, removes all numbers. How can this be?

 

[tpogge]

Jomo. You added the element 0 to the set of natural numbers. My method of elimination was geared to the set of natural numbers and (as you showed) never reaches the element 0. But my method can easily be adapted to the enlarged set (as I showed) so it eventually removes every element.
So, paradoxically, the remainder set has infinitely many elements at each step but, after infinitely many steps (if one may say this), is empty.

 

[Dr.Peterson]

What happens if one divides an infinite quantity by a finite number >1 (e.g. 2) an infinite number of times?

Two intuitions:

1) after each division, the remaining quantity is still infinitely large, thus even an infinite number of divisions cannot reduce it below infinity.

2) Eliminate from the set of natural numbers every other member repeatedly. After the first step, only every other number is left. After the second step, only every fourth is left. After the 3rd step only every 8th. And so on. If one does this forever, no natural number will escape elimination. So the process converges toward zero.

Which intuition is correct? And why is the other one deceptive?

This is a familiar kind of paradox; the infinite tends to do that to us! For a particularly interesting variation on yours, consider

Ross–Littlewood paradox - Wikipedia

en.wikipedia.org

Also,

Hilbert's paradox of the Grand Hotel - Wikipedia

en.wikipedia.org

 

[Steven G]

Jomo. You added the element 0 to the set of natural numbers. My method of elimination was geared to the set of natural numbers and (as you showed) never reaches the element 0. But my method can easily be adapted to the enlarged set (as I showed) so it eventually removes every element.
So, paradoxically, the remainder set has infinitely many elements at each step but, after infinitely many steps (if one may say this), is empty.

You said start with an infinite set. Fine, I'll use {1, 2, 3, 4, 5,...}
Step 1: remove 2, 4, 6, 8, ..., left with {1, 3, 5, 7, 9, 11, ...}
Step 2: remove, 3, 7, 11, 15, ...., left with {1, 5, 9, 13, ...}
...
1 is never removed.

As Dr Peterson suggested, read about the infinite hotel.
It should also make it clear that my set starting with 0 or 1 does not matter!

 

[Steven G]

Jomo. You added the element 0 to the set of natural numbers. My method of elimination was geared to the set of natural numbers and (as you showed) never reaches the element 0. But my method can easily be adapted to the enlarged set (as I showed) so it eventually removes every element.
So, paradoxically, the remainder set has infinitely many elements at each step but, after infinitely many steps (if one may say this), is empty.

the remainder set has infinitely many elements at each step but, after infinitely many steps (if one may say this), is empty.
I showed two (really one) counter examples. Comment on that.

 

[Steven G]

Suppose you have an infinite number of marbles numbered 1, 2, 3, .... in a HUGE container (#1).
You also have another currently emptied HUGE container (#2).
Step 1a) Take balls numbered 1-10 from C1 (container #1) and put them into C2
Step 1b) Remove marble labeled #10 from C2 (throw the marble away)
Step 2a) Take balls numbered 11-20 from C1 (container #1) and put them into C2
Step 2b) Remove marble labeled #20 from C2 (throw the marble away)
Step 3a) Take balls numbered 21-30 from C1 (container #1) and put them into C2
Step 3b) Remove marble labeled #30 from C2 (throw the marble away)
Step 4a) Take balls numbered 31-40 from C1 (container #1) and put them into C2
Step 4b) Remove marble labeled #40 from C2 (throw the marble away)
continue this process.
In the end how many balls will be left in C2?

Suppose the steps above were replaced with
Step 1a) Take balls numbered 1-10 from C1 (container #1) and put them into C2
Step 1b) Remove marble labeled #1 from C2 (throw the marble away)
Step 2a) Take balls numbered 11-20 from C1 (container #1) and put them into C2
Step 2b) Remove marble labeled #2 from C2 (throw the marble away)
Step 3a) Take balls numbered 21-30 from C1 (container #1) and put them into C2
Step 3b) Remove marble labeled #3 from C2 (throw the marble away)
Step 4a) Take balls numbered 31-40 from C1 (container #1) and put them into C2
Step 4b) Remove marble labeled #4 from C2 (throw the marble away)
continue this process.
In the end how many balls will be left in C2?

Infinity is different!