I’m wondering if the following supertask is logically impossible.
Suppose at 7 a.m. a container with infinite capacity is empty. Nobody but Quinn adds something to the container, and once something is put in the container it never comes back out. Quinn wants to put as many balls as he can in this container but does not want the container to contain infinitely many balls. So Quinn determines whether it’s possible that if after adding a ball, the container would contain infinitely many balls (if for example the container already contained infinitely many balls, the answer would be “yes”). If the answer is “yes” then Quinn does not add the ball. If the answer is “no” then Quinn adds the ball. Where n is the nth ball Quinn puts in if it meets his condition, Quinn does so at 9 a.m. – 1/n hour. Quinn puts the first ball at 8 a.m. (9 a.m. – 1/1 hour), the second ball at 8:30 (9 a.m. – 1/2 hour), the third ball at 8:40 (9 a.m. – 1/3 hour), and so forth. Since Quinn does not add a ball into the container if there’s any chance that the container would have infinitely many balls after adding the ball, how many balls does Quinn put in? It seems he couldn’t put in infinitely many, because Quinn makes sure that this won’t happen; he starts with an empty container, he’s the only one adding any balls, and he won’t add a ball if adding it would make the set of balls infinite. On the other hand, there doesn’t appear to be any ball n that Quinn wouldn’t put in the container at 9 a.m. – 1/n hour, and if that’s true Quinn puts in infinitely many balls.
So does this supertask entail a bona fide contradiction or am I missing something?
It seems a very long way around of asking whether there’s a number one below infinity.
It’s a purely theoretical question - posing it as a concrete, practical problem with balls and containers doesn’t make it less so, alas - and as such depends entirely on what you take “infinity” to be. “Infinity” doesn’t refer to anything out there, it’s a mathematical concept. It’s not a part of the real numbers that you use to count things. Or at least, not usually. It gets complicated.
So (as I understand it, using standard axioms and assumptions) it’s not a logical contradiction, but it doesn’t make sense; your example assumes some sort of countable infinity (or at least a countable infinity-1). If you really want to get into it, a maths degree would probably be a good place to start.
I may be missing the point, any one of you feel free to let me know if I am.
However, if there is a ball in existence anywhere in the World (regardless of whether or not that ball is in Quinn’s possession), doesn’t that automatically prohibit the container from housing an infinite number of balls because there exists (at least) one more ball the container could theoretically have if it is able to contain an infinite amount?
The amount of balls he puts in is unlimited, not infinite. There’s a difference.
IE assuming he’s immortal and all that sorta shit, there’s no cap on the number of balls he can put in, but they are always countable and finite.
In order for the amount of balls in the container to be infinite, that means that the number of balls in the container must also be uncountable, which is entirely different from there being too many balls to have the time to count. The fact that there is at least one ball that exists outside of the container results in the ability of a person to count that one ball. If that one ball can be counted, then we know that regardless of the number of balls in the container, that those balls can also theoretically be counted.
Therefore, the existence of one ball anywhere but inside the container renders the amount of balls in the container finite.
I came up with that because a ball is a physical object of which we have a concept and to which we apply a description. If there are certain (individual) physical objects that can meet our conception of what a ball ought be, and those objects can be counted, then we are able to count how many balls there are in a given place.
The inside of the container qualifies as a, “Given place,” and although the OP stated that when something goes into the container it never comes back out, the OP did not state that Quinn (or any other person) could not go in the container if they so chose.
Therefore, since the balls are nothing but individual physical objects to which we apply a conception to determine if they, “Fit the bill,” to be called balls, and the inside of the container qualifies as a, “Given place,” and we can count the number of balls in a, “Given Place,” and the ability to (theoretically) count the balls in a given place means that the number of balls in a given place cannot possibly be infinite we must be able to count the number of balls in the container. Therefore, the container does not contain an infinite number of balls. To suggest otherwise would be to assert that we could not count by ones.
Alternatively, if there was an independent individual situated outside of the container at such a time before the first ball went in, that person could simply count every ball that Quinn puts in…theoretically. Therefore, while the number of balls Quinn is putting in may seem limitless, it is not proven that the container can hold an infinite number of balls.
you’re right, the balls in the container aren’t infinite, but it’s not because there are countable balls outside of the container. that’s pretty irrelevant.
Okay, I just thought of an example of why your logic really isn’t working for me:
There are infinite numbers greater than 2. You can’t count all the numbers greater than 2. If you can, I’d really love to see that. I’d probably get bored by the time you hit 500 or so though.
But, in any case, there are infinite numbers greater than 2. You can still count 1 and 2, though. Just because you can count certain instances of an object (in this case numbers, in that case balls) outside of a certain category (in this case > 2, in that case inside the container) doesn’t have anything to do with whether or not the objects inside the category are infinite or not.
So, like I said, I agree that there aren’t infinite balls in there, but not because you can count the balls that aren’t in there. There’s no logical connection there.
The fact that I can count a number of balls that aren’t in there demonstrates that I can count the number of balls that are in there. Any ball that is in there was, at one time, “Out of there,” because it is required that someone puts the balls in there.
Anything that starts at an initial state of zero can never reach a state of infinite so he already gets his wish.
We must presume that Quinn has an infinity meter of some kind in order to accomplish that part of the task.
Either Quinn has the magic ability to inject a ball in zero time, or he simply will never get to the 9th hour. Anything requiring any more than zero time can never get to infinity.
He is free to add as many as he wants without even counting because his infinity meter could never get off of the zero reading where it began. On the other hand, if he could add the balls in zero time (not presumed until now), by 9AM, he would already have placed an infinite number of balls in the container as per the required “Quinn does so at 9 a.m. – 1/n hour”. Again, only if he could add the balls in zero time, could he ever even reach 9AM.
That is the point where the presumption of impossible talents leads to a contradiction in consequence. If Quinn can do the impossible, can he accomplish an illogical consequence? - Certainly - “IF” logic doesn’t apply to Quinn, then what Quinn does will not be logical either (logically speaking).
Any change in the universe taking only zero time is a logical contradiction and thus an impossibility.
Yes, you are missing the fact that one cannot propose a logically impossible situation and then logically conclude a logical consequence.
There’s no logic here. There’s no connection. I’m missing premises to make the logical connection you seem to be so easily making. Could you try to make a syllogism out of it perhaps? I really need those missing premises, cuz you’re basically speaking Chinese to me right now without them.
No, because adding one to infinity gives you infinity. It’s not a real number. In fact, you could have two containers, each containing infinitely many balls. And one left over in your hand.
Faust is an expert on taking what I say, breaking it down, clarifying it, and then expressing it in one sentence.
The number of balls is going to be expressed in real numbers no matter what you do, because there are individual balls, therefore, because we can theoretically count to any given number (creating numbers if necessary), the number of balls in the container cannot be infinite. The number of balls is theoretically countable no matter what you do.
If I put 10,000,000 people in Room 1, the owner would probably fire me.
I understand that adding one to infinity gives you infinity, but you cannot have infinity with respect to a defined physical object (of which there is known to be more than one) because any defined physical object can be counted.
I might be inclined to agree with this, but once again this has nothing to do with what your original claim was: that if u can count one ball outside of the container then you can count all the balls inside the container. This and that have nothing in common. There’s no shared train of logic. No connection.