You are confusing infinity with the process of going through an infinite list of items (e.g. the process of watering an infinite number of flowers.)
You do understand there is such a thing as the process of going through a finite list of items just as well? If you do, why don’t you consider number (2) to be a process just as well?
I consider the number 2 to be a process that doubles the length of any line segment via multiplication.
With this point of view, the imaginary unit i is an operator that rotates a line segment one quarter turn counterclockwise in the plane. Do it twice in a row and that’s the same as the operator -1, which preserves the length of a line segment but flips its orientation 180 degrees. So (i^2 = -1) is a natural and obvious truth about the quarter-turn process; and no longer some deep mystery that confuses people.
Numbers are operators, or processes if you like.
Infinity is not a number, endless is not a value. What you just said is meaningless, which leads me to believe we’re not speaking the same language.
I understand what the words “reduce” and “subtract” mean… but what I’m suggesting is that they are category errors because “infinite” is not a number nor a quantity such that it can be reduced. In fact it’s relationship with quantities is to describe the quality of being immune to reduction, regardless of how much is taken.
So you might as well talk to me about square circles.
If I own a restaurant that can seat and serve endless guests, then no matter how many guests sat down I could never say I had a reduced capacity to seat and serve guests…
If my capacity is limitless… it does not get reduced by any given number of guests… I will remain equally capable regardless of the number of guests.
If this rings false to you then we have a semantic disagreement… which means it has nothing to do with rational argument.
We just have to agree to speak the same language or learn to understand each other’s language.
But at this point I have no idea what you think infinity means or even implies.
I have to admit I’ve not read this entire thread… so pardon if I’m not up to speed with what you’re talking about
You are using terms that I’m not familiar with and the structure of the argument itself is a non-sequitur.
If A is B then B must be C… is missing a premise, like say all A’s are C’s or something.
Consequently I don’t understand your question… you’re going to have to present this argument to me more comprehensively for me to grasp it.
The unit circle is a square in the taxicab metric. People should stop using this example because there are in fact square circles. There’s a picture of a square circle here:
So you agree that a statement such as “We removed one apple from an infinite number of apples” is a contradictory one?
And you agree that the statement “The number of people in the universe is infinite” implies that no person can cease to exist?
I don’t think so. Its relationship with the reals (what you consider to be quantities) is that it’s greater than them. Infinity is a number greater than every real number.
(By the way, you’re still contradicting yourself.)
No and No to both of those questions.
That means you can’t say whether or not a line is finite in it’s length until you settle on a measurement for it, such that you can ascribe it a number according to that arbitrary metric.
Let’s say we draw a finite line as measured in the metric system to be 1 meter long. It could in theory qualify as infinite in length if the arbitrary unit of length we decided to operate by was equivalent of 1/infinite of a meter say. Let’s call this unit infinitesemeters… that means the line that we’ve just drawn is infinite in length as measured in infinitesemeters, but only 1 meter long in the metric system.
I like my definition better…
I’ll just have to take your word for it, I guess.
Alright.
Earlier you said that you understand what the word “remove” means. I take this to mean that you agree that to remove one thing from a group of things is to alter that group of things and to alter it in a specific way, namely, by decreasing the number of things it consists of. Can you confirm this?
If this is the case, if you agree that to remove one thing from a group of things is to decrease the number of things in that group, then it follows that the statement “We removed an apple from an infinite number of apples” implies that we decreased the number of apples in that group. And yet, you disagree with this. Why?
Let me make that argument explicit:
To remove one thing from a group of things is to decrease the number of things in that group.
We removed an apple (one thing) from a group of apples consisting of an infinite number of apples (a group of things).
Therefore, we decreased the number of apples in that group.
What part of this argument do you disagree with?
On the other hand, if you think that the statement “An infinite number of apples” describes a group of apples the size of which (in terms of how many apples it consists of) cannot be changed, then it follows that the statement “We removed an apple from an infinite number of apples” is a contradictory statement because it is saying that we decreased the size of a group whose size cannot be changed (decreased or increased.)
Step 3…
I don’t consider “infinite” to be a number like you do… “infinite” is a property, not a number.
Under normal circumstances, within a finite set, removing a unit from the set does decrese the number of units remaining in the set.
In an infinite set, removing a unit uniquely does NOT decrese the number of units in the set. THAT is the property that qualifies the set as INFINITE.
In an infinite set of positive whole numbers {1,2,3,4…} if we removed the number 1 from the set… we still have an equal number of members remaining in the set, demonstrable by lining the two new sets up like so
{1, 2, 3, 4…} before
{2, 3, 4, 5…} after
We can start the second set at 10545815236 and it would still have every bit the same membership as a count that started at 1…
It is completely irrelevant where you begin your count, none of that in any way impacts the size of an infinite set.
Adding or subtracting another infinite set is how we can meaningfully measure relative size of infinite sets… let’s say we add the infinite set of whole negative numbers.
That can be meaningfully called a larger set, because it consists of two infinite sets.
So we might say, in that respect, If we removed THE INFINITE SET of every even number, we would have reduced the size of the infinite set…
This larger or smaller talk is not me counting total members, this is me counting the number of infinite sets…
But unfortunately that does not seem to be what you’re saying… you are suggesting if we took one singular member away, we would have a smaller infinite set…
Which to be fair makes sense in light of your definition of infinite as a number… I just don’t care to adopt your definition.
Man,
Mad Man P!
Wtf already answered you on this and you still repeated it!!!
Look at the set:
1, -1, 2, -2, 3, -3… etc…
1:1 correspondence!
So you’re saying that the conclusion does not logically follow from the premises? In other words, that the argument is logically invalid? But how?
There is a possibility that you do not truly understand (but merely think that you understand) what the first premise in my argument means.
This means that given ANY group of things (i.e. regardless of how many things and what kind of things there are in a group) to remove one thing from that group is to decrease the number of things in that group. In other words, to remove one thing from a group consisting of an infinite number of things means to decrease that number of things in that group. THe word “remove” does not mean ONE thing when it comes to finite groups and ANOTHER when it comes to infinite groups.
Thus, in order to remain logically consistent, you have to accept one of the following two statements:
the word “remove” means ONE thing when it comes to finite groups and ANOTHER when it comes to infinite groups
the statement “We removed an element from an infinite group of elements” is a contradictory one
I don’t see a third way out.
You can also line them up any other way which is an indication that the choice is an arbitrary one (i.e. it’s not something that is logically derived from prior statements.) Here’s an example:
1 → -
2 → 2
3 → -
4 → 3
5 → -
6 → 4
7 → -
8 → 5
9 → -
…
On the other hand, you DID state that you produced the second set by making a copy of the first set and REMOVING one element from that copy which implies the number of elements in the second set is less than the number of elements in the first set.
I think you already do.
The word REMOVE does not change… but having the CONSEQUENCE of removing a member result in being REDUCED, does.
Perhaps it’s less obvious than I thought, so I’ll try to present this to you differently.
Group A is the infinite set of positive whole numbers… and Group B is the same infinite set after we remove y amount of members from it.
The premise is this: For every member x, in group A, there is a member x+y in group B.
FOR EVERY SINGLE MEMBER…
Now you can insist the membership count is different between group A and B… but I would have no idea wtf that means.
ok… are you talking about the sum of the values within the set being equal to 0?
Like (1)+(-1)+(2)+(-2) etc?
If so, I honestly don’t see how that’s relevant to anything we’re talking about here…
I don’t really understand what you’re trying to say here.
My entire point is that it logically follows that Group B has fewer elements than Group A.
To remove a thing from a group of things is by definition to decrease the number of things in that group of things. This has nothing to do with what the word “infinity” means and everything to do with what the word “remove” means. To say otherwise is to contradict yourself. (Unless, of course, you defined the word “remove” in an unusual way.)
So you, Magnus, are saying that infinite sets can increase or decrease?
But that doesn’t make them infinite then… those are not infinite sets… cam objects realistically be infinite?
Correct.
Why not? What does the word “infinite” mean to you?
Magnus,
I’m going to play with your logic a bit here, “free styling it”
Let’s say that I have this:
1.) 1/3
2.) 4/3
I have 2 infinities - obviously twice as many numbers as only having one infinity.
But let’s look at this closer with how you’re arguing.
You’re always stating that you can add and subtract TO an infinity!!
So… using this logic, I can tack on 4/3rds AFTER! 1/3!!
To just make 1 infinity!! Thus!! Using your logic, there’s only one order of infinity!
Understand?
Instead of adding it to the end, as your absurd logic implies, we can simply add them together and get 5/3!
Still only ONE infinity!
An infinite set has the property that you can remove an element without changing its cardinality. The actual definition of an infinite set is that a set is infinite if it can be bijected to one of its proper subsets. That’s the definition. Why you don’t accept standard mathematical definitions I don’t know.
If you start with the integers and throw out all the negative numbers, you’re left with the naturals. The naturals are a proper subset of the integers. They’re “smaller” in the sense of being a proper subset. Yet the two sets have the same cardinality, since they can be put into bijective correspondence. That doesn’t mean they have the “same size,” it only means that they have the same cardinality. It’s a definition.
What an odd definition of “remove”… to remove a thing is defined by the act, not by the consequence of that act.
The same way punching someone is not defined by whether or not you hurt them… that’s just what tends to happen.
The act of removing something, does not have to result in there being fewer things, for it to qualify as removing…
But let’s say it does mean that… using your defintion I would then say you cannot “remove” anything from an infinite set.
You can “dispose” of an element in the set, but that will not reduce the number of elements remaining in the set.
Unless you have an alternative definition of dispose as well…
To punch someone is to strike them with a fist. You may or may not hurt them. (It’s not implied by the word.)
But to remove a thing from a group of things is to decrease the number of things in that group. That’s what the word logically implies. How can you remove a thing from a group of things WITHOUT decreasing the number of things in that group? It makes no sense. (Unless, of course, you’ve defined the word “remove” in a strange way.)
Let’s go back to my argument:
It appears that your disagreement lies with the first premise.
And it also appears that you agree with the following statement:
That’s the strange definition of the word “remove”.
The problem is that both words imply the same thing: reduction in the number of things in the group.
Where I agree with you is that you can say that we cannot remove anything from an infinite set. That makes sense. However, that sort of statement has a different problem. Namely, it assumes the word “infinity” is defined in such a way that a statement such as “The number of people in the universe is infinite” implies that the number of people in the world cannot change. I disagree with that.
But I’m not the one with alternative definitions. It is actually you who are redefining common terms in order to defend the popular opinion (because that’s the only way one can defend the popular opinion.) Everyone knows that to destroy something is to reduce that thing to nothing (i.e. to change from having one thing to having zero things.) Thus, if the number of people in the universe is infinite, and you kill one person, you are DECREASING the total number of people in the universe. The fact that there’s no single word with which to call the resulting number of people doesn’t mean the number of people in the universe is the same as before.
Actually not. If you remove a single item from a finite set, you reduce the cardinality AND your new set is a proper subset of the original.
If you remove one element from an infinite set, the new set is indeed a proper subset of the original; but the new set and the old set have the same cardinality.
You’re just equivocating the word “number”; and, for reasons I don’t understand, refusing to engage with the standard mathematical definitions.