Veteran TSZers may recall an entertaining thread in which a bunch of us tried to explain the cardinality of infinite sets to Joe G:
A lesson in cardinality for Joe G
At UD, commenters daveS and kairosfocus are now engaged in a long discussion of the transfinite, spanning three threads:
An infinite past can’t save Darwin?
An infinite past?
Durston and Craig on an infinite temporal past…
The sticking point, which keeps arising in different forms, is that KF cannot wrap his head around this simple fact: There are infinitely many integers, but each of them is finite.
For example, KF writes:
DS, I note to you that if you wish to define “all” integers as finite -which then raises serious concerns on then claiming the cardinality of the set of integers is transfinite if such be applied…
The same confusion arises in the context of Hilbert’s Hotel:
KF:
Try, the manager inspects each room in turn, and has been doing so forever at a rate of one per second. When does he arrive at the front desk, 0?
daveS:
Re: your HH explanation: If the manager was in room number -100 one hundred seconds ago, he arrives at the desk now.
KF:
Yes a manager can span the finite in finite time. But the issue is to span the proposed transfinite with an inherently finite stepwise process. KF
daveS:
In the scenario I described above, the manager was in room -n n seconds ago, for each natural number n. Given any room in the hotel, I can tell you when he was there.
KF:
DS, being in room n, n seconds past does not bridge to reaching the front desk at 0 when we deal with the transfinitely remote rooms; when also the inspection process is a finite step by step process.
What KF doesn’t get is that there are no ‘transfinitely remote rooms’. Each room is only finitely remote. It’s just that there are infinitely many of them.
Any bets on when — or whether — KF will finally get it?
KF is on the verge (finally!) of seeing his error:
Now he just needs to realize that there are no natural numbers other than the ones reachable by successive +1 steps starting from 0.
The natural numbers are endless — there are infinitely many of them — yet each one is finite.
keiths,
Nonsense. The 1 that replaced the 0 already existed. You didn’t add a new element.
But thank you proving that you can’t even follow your own posts
Frankie,
No. If you step through from left to right, at some point in the process you have two ‘1’.
Richardthughes,
Richardthughes,
Richardthughes,
Again, getting technical, you have to be careful with this.
If you replace 0 with 1, then you now have the set {1,2,3} which has a different cardinality. You would have to work in reverse order — first replace the 3 with 4 — in order for that argument to work.
Neil,
No, because the substitutions are done simultaneously.
keiths:
cupcake:
LoL! Yeah like when you travel through infinity and end up back at the beginning. BWAAAAAAAHAHAHAHAHAHAHAHAHAHAHA
Perhaps Richie is ignored by everyone and that is why no one has corrected him.
How’s that working out for you at UD?
Then what you are really doing, is replacing the 0 with a 4.
Neil,
No, I’m replacing
a) the 0 with a 1,
b) the 1 with a 2,
c) the 2 with a 3,
…and so on.
Actually you are just removing the 0. Everything else remained the same.
Then you are not replacing them simultaneously.
Initially, the set is {0,1,2,3}
Replace “the 0 with a 1”.
The set is now {1,2,3}
Replace “the 1 with a 2”.
The set is now {2,3}
Replace “the 2 with a 3”.
The set is now {3}.
If you did it simultaneously, you started with {0,1,2,3}, and instanteously changed that to {1,2,3,4} which is just a change of the 0 to a 4 (the order is not relevant).
Neil, this was the original claim:
4. Take the set {0,1,2,3,…}. It has a cardinality. Add 1 to each of its elements without adding or removing any elements. You’ll obtain {1,2,3,4,…}. Has the cardinality changed?
The cardinality hasn’t changed. You are wrong about that. However, there are some technical problems with the argument that keiths was using.
Set subtraction says that the cardinality has changed. But yes there are technical problems with his argument
Set subtraction shows that the set has changed. It does not follow that the cardinality has changed.
Of course it shows that the cardinality has changed. With set subtraction the 0, which is an element, is left. That means one set has one more element than the other.
Neil,
First of all, you were the one who insisted that elements cannot change, so your reference to “a change of the 0 to a 4” is nonsensical by your own (pedantic) standard.
Second, the process of replacing each n in the set with n+1 is not the same as the process of replacing 0 with 4.
This is trivial to demonstrate. Take the set {0,1,2}. Replace each n in the set with n+1, and the result is {1,2,3}. Now start with the original set again, and replace the 0 with 4. You get {1,2,4}. Two different processes, two different results: {1,2,3} vs {1,2,4}.
The fact that the two processes give the same result in one particular case — when the set being operated on is {0,1,2,3} — does not mean that the processes are identical. It’s as silly as arguing that 2x and x^2 are the same function, because they give the same answer when x=2.
Neil,
You’re making the same mistake as Frankie, which ought to concern you. Like you, Frankie is assuming that if the results are the same, the process must be the same. Regarding my conversion of the set {0,1,2,3…} to {1,2,3,4…} by replacing every n with n+1, he writes:
He’s wrong for the same reason that you are wrong. The fact that the results are the same in one particular case does not mean that the processes are the same.
Again, this is trivial to demonstrate. Take the set {0,2,4,6…} and replace each n in the set with n+1. The result is {1,3,5,7…}. Now take the original set and remove the 0, as Frankie suggests. The result is {2,4,6,8…}. Two different processes, two different results.
keiths,
The mistakes are all yours, keiths
That is your bullshit opinion and you can’t support it
Actually you are just removing the 0. Everything else remained the same.
That happens to be true.
That is what you did. I didn’t have to suggest it
Technical, not pedantic.
We can change a 0 to a 4 in the set by removing the 0 and replacing it with the 4. It is, of course, a different set after that replacement.
You can change the numeral “1” to the numeral “4” without removal or replacement, because numerals are just marks (material representations). But you cannot change the number 1 to the number 4 without removal and replacement. But if the number 4 is already in the set, you cannot add it to the set as a duplicate, since it is already there. There can be many numerals “4” but there is only one number 4.
I am not being pedantic about change vs. replace. Rather, I am being technical about numbers being singular objects, rather than representations or marks.
This may be technical. But the way that we talk about numbers requires it.
No, I’m not making a mistake.
Numbers are not numerals or marks. Sets are not buckets (or containers).
The language of changing a 0 to a 1 might be good as an intuition pump, but it is not technically correct.
Neil:
Sure you are, and I’ve explained exactly what your error is:
Which axioms of ZF (or ZFC) license the process that you claim to be using?
Neil,
Heh. Nice try at changing the subject, but I’m not biting. Your question is irrelevant. You already asserted that my process — replacing n with n+1 for each n in a set — is the same as your process — replacing 0 with 4:
I’ve shown that your claim is incorrect:
Do you have a counterargument to offer?
Sigh.
Start with {0,1,2}
Replace 0 with 1. That gives {1,1,2}, which is just a badly described {1,2}.
Replace 1 with 2. Now you have {2}
Replace 2 with 3. Now you have {3}.
Yes, I understand that you fail to get this. But that’s your misunderstanding.
You can say “do it all simultaneously” and then ask me to look at the process. But if it all happens simultaneously, there isn’t a process. There’s only a final result.
You are treating this as if you have billiard balls in a barrel, with numerals painted on them. But it doesn’t work that way with actual numbers.
Yes, I understand that you were using an informal intuitive description. That’s fine. I don’t object to that. However, Frankie rejected your intuitive description and made a technicallly correct objection. You said that he was wrong. I was correcting the record on that.
Neil,
I’ve specified that the steps happen simultaneously, yet you bizarrely continue to take them as happening sequentially. Why?
Good grief, Neil. The fact that the steps happen at the same time doesn’t mean that they don’t happen. Where do you think the result comes from?
Replacing n with n+1 for all n in a set is a distinct process from replacing 0 with 4. You got confused because the two processes have the same result when the set in question happens to be {0,1,2,3}. You weren’t thinking like a mathematician, so you failed to notice that the two processes gave different results for other starting sets such as {0,1,2}.
Yes, it’s an embarrassing mistake, but why prolong your embarrassment? Why not just acknowledge the mistake, resolve to think more carefully next time, and move on?
Neil,
The process of replacing each number n in a set with n+1 does not depend on “treating this as if you have billiard balls in a barrel, with numerals painted on them.”
At UD, daveS sums up KF’s position:
Meanwhile, Virgil is in trouble again:
Virgil notpologizes:
Self-control is a far-off dream for Virgil.
Good grief yourself.
You seem to be thinking of this as if you had a physical barrel of billiard balls that you can change. But mathematics is abstract. Unless the axioms give an account of changing, then what you have is one set replaced by another set. There isn’t a process unless a process is defined. Loose talk of changing simultaneously is not a definition.
Unless you can define “replace” in accordance with the axioms, then the talk of replacement is just extra-axiomatic talk and of zero relevance. What you have is a different set than you started with. The process of replacement might be going on in your mind, but is not available to anybody who doesn’t read your mind. It’s made up bullshit.
It is your embarrassment, not mine.
I fully acknowledge that keiths has made a fundamental mistake here. Moreover, I acknowledge that keiths doesn’t understand his mistake and will probably continue to make a big fuss about it.
I’m just walking away, as I usually do when keiths gets onto the wrong side of an argument.
Neil,
Okay, I see you’ve chosen to prolong your embarrassment. As you wish.
No, as I’ve already explained:
Neil:
Whether you take 0 as being changed to 1 within a set, replaced by 1 within the set, or mapping to 1 in a new set makes no difference for the purposes of this discussion. Success in abstract thinking requires an understanding of which details matter and which can be abstracted away. What matters is that you start with the set {0,1,2,3}, you apply the specified process, and you end up with {1,2,3,4}.
I defined the process. For each n in the set, you change it to (or equivalently, replace it with or map it to) n+1. That’s different from changing 0 to 4 (or equivalently, replacing 0 with 4 or mapping 0 to 4), as I demonstrated.
You made a silly mistake. As I said earlier, it’s like concluding that 2x and x^2 are the same function because they have the same value when x=2.
You were the one who introduced replacement into the conversation:
And you have been discussing replacement throughout the thread. For example:
By your own standard, then, you have been spouting “made-up bullshit”.
How does that help your case?
You’re tripping over your own shoelaces, Neil.
LOL.
Virgil:
ellazimm patiently explains Virgil’s error:
keiths,
Except Virgil wasn’t talking about mathematical ideas at that time. He was talking about ellazimm’s continued lies about the utility of the concept being debated. All ellazimm has done is bluff and lie. But I understand why that would appeal to you.
keiths,
LoL! The people who spew lies and bluffs have lost self-control. But I can understand why you would agree with that tactic and disagree with anyone exposing it.
KF:
daveS:
Will it finally sink in?
daveS, to KF:
It’s remarkable that Mr. “Principles of Right Reason” doesn’t see the need to justify his intuitions.
A commenter at AtBC brings up an interesting problem for Virgil’s cardinality “theory”, although Virgil is banned now at UD so I won’t post this there. But this is nice.
1. Virgil argues as follows: Let A = the naturals, B = the evens, and C = the odds. Then, since A – B = C, this proves that B has fewer elements than A – presumably 1/2 as many.
2. But, on the other hand, start with A = {1,2,3,…} and multiply each element by 2 to get set D = {2,4,6,…}. Set D has the same number of elements as A because all we did was double each element. However D is just all the evens, so this proves the evens have the same number of elements as N
Therefore, arguments 1 and 2 contradict each other, one saying that the cardinality of E is less than N and one saying the cardinality of E is the same as N. Both can’t be right, so there must be a flaw in one of the two arguments.
Wonder what Virgil would say to this?
aleta,
That’s the homework I gave to Joe/Virgil/Frankie back in 2013:
Also:
Joe’s response was pretty much what you’d expect:
And More stupidity from keiths
aleta,
That doesn’t follow at all. By doubling each element you have also removed elements.
keiths,
LoL! We have already addressed your points, keiths. Merely repeating what we have refuted is a sign of desperation. In #4 you did remove the 0
That doesn’t follow at all. You just make shit up and post it as fact. Mapping converts one set into the other. It just shows the sets are countable.
Pathetic
Frankie:
You, Virgil, and Joe?
No you haven’t, Frankie. 1 became 2, 2 became 4, 3 became 6. You haven’t removed elements, you’ve changed the value of the elements.
Let’s make it simple: let A = {1,2,3,… 1 billion} and double each element to get B = {2,4,6,… 2 billion}. Set A has 1 billion elements and set B has 1 billion elements. The number of elements didn’t change.
We are legion. Ask Zachriel
Of course elements were removed as they are no longer there. All of the odds have been removed. And all the evens already existed. And yes finite sets are not the same as infinite sets, are they?
Frankie, to Aleta:
I explained this to you three years ago, “Frankie”:
keiths,
LoL! Once you add 1 to the 0 the 0 is gone.
By removing the 0 from the first set
Lol! It is impossible to remove all of the elements from an infinite set
The result is even more amusing in this case:
JoeMath says that the cardinality of the result is only 1/17 of the cardinality of the original set.
Where did all of those elements go, Joe?
keiths,
You are removing elements, again
Who cares? They are definitely no longer in the set
keiths thinks that you can remove all of the elements in an infinite set. How can anyone take him seriously when it comes to math and infinity?