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?
How can you have a thread if you keep adding posts?
keiths,
Except there isn’t any such thing as an infinite Turing machine tape. So if your point is that people have to pull things out of their ass to try to win then you did good.
How can you have a point if you never make one?
http://www.merriam-webster.com/dictionary/infinite
1 : extending indefinitely : endless
Nothing is “already infinite”. Only someone unaware that infinity is a journey would say such a thing. However your sets that keep growing and will never stop growing, fit the definition of infinite sets
Frankie:
Let’s suppose your choo-choo train leaves the station at exactly 3 o’clock, on a journey that will never end. It is carrying an empty set on board. (I’m trying to keep this as concrete as possible, for your sake.) One minute later, the choo-choo train passes a marker with the number ‘1’ painted on it. The conductor adds the number 1 to the set S. Every minute thereafter, the choo-choo train passes another marker with the next higher integer on it, and the conductor adds that number to the set.
What does the set S look like at 3:07:10? What is the size of the set? Is the set finite or infinite?
keiths,
Nice non-sequitur. Try again:
However your sets that keep growing and will never stop growing, fit the definition of infinite sets
Not my fault that you have difficulties with definitions and still cannot grasp that infinity is a journey
Why not answer the questions, Frankie?
What does the set S look like at 3:07:10? What is the size of the set? Is the set finite or infinite?
ETA: It’s as if you’re embarrassed to claim that a set containing seven elements is an infinite set.
*guffaw*
However your sets that keep growing and will never stop growing, fit the definition of infinite sets
Do you agree or disagree? I know that you are embarrassed to answer because you already have one foot in your mouth and don’t want to add the other
Please explain what that has to do with anything I have said.
Frankie/Joe/Virgil:
How soon they forget:
And “choo-choo math” was born.
Frankie,
I disagree, of course. A set containing seven elements is not an infinite set.
Jesus Christ, Joey.
Right, it has nothing to do with anything I have said.
keiths,
Wow. keiths proves that he is incapable of following along.
However your sets that keep growing and will never stop growing, fit the definition of infinite sets
Do you agree or disagree?
That has nothing to do with a set containing seven elements. keiths is afraid to answer my questions because they prove he doesn’t know jack about infinity.
And your ignorance was exposed.
Frankie:
I just gave you an example of a set with seven elements that keeps growing and will never stop growing.
You claim that my set “fits the definition” of an infinite set. That is what is known as “a gobsmackingly stupid statement.”
KF is still hung up on the idea of a “far zone”:
LoL! It the set keeps growing then obviously it will have more than seven elements. IOW yours is what is known as “a gobsmackingly stupid statement.”
Again you don’t know jack about infinity and you can’t even understand the definition of infinite.
Here it is again:
http://www.merriam-webster.com/dictionary/infinite
1 : extending indefinitely : endless
Nothing is “already infinite”. Only someone unaware that infinity is a journey would say such a thing. However your sets that keep growing and will never stop growing, fit the definition of infinite sets.
Joe:
I see. They aren’t infinite, but they “fit the definition of infinite sets”.
Keep going, Joey, you’re doing great.
The funny thing is that Joe is wrong even according to JoeMath:
By that same logic, the counts are finite — always and forever. Therefore the sets are finite.
keiths,
What? Obviously if they fit the definition they are infinite. But thank you for continuing your “gobsmackingly stupid statements”.
That doesn’t follow at all. But it is another one of your “gobsmackingly stupid statements”.
(if saying that is good enough for keiths…)
Joe:
Of course it doesn’t follow. It’s JoeMath.
Again:
Here is another question not one of my detractors can’t answer:
If Cantor was wrong about countable infinite sets having the same cardinality, what would be affected or would there be no effect at all?
It doesn’t follow from what was quoted. Obviously you have other issues
You don’t understand math and you are far from understanding what you quote-mine.
I did a comparison of CantorMath and JoeMath somewhere in the old thread. I’ll dig it up later and post it here.
keithsmath, where the infinite is really just the finite that keeps going forever. 😛
I did comparison of keithsassmath with reality. keithsassmath lost.
What do you think it is?
Could you be more coherent, please. What “it” are you referring to?
How’s the search for the largest number coming, Joe? Where are they looking?
You should publish this fine work Frankie! Show the mathematical world what’s what!
I’m sure you’ll have no problem getting your thoughts through peer review.
What if I have a set that I intend to grow to infinity, past seven members, and label it as such. But then I forget to extend the set to infinity and only get to ten instead.
Was the set ever infinite? When does it become infinite? When bigNum is reached? Is that the one that there is a computer keeping track of? Do you have a link to that?
Rich,
I hear the Russians found an LKN that’s larger than ours. The President is quite worried.
‘My detractors”. Who are your supporters, Frankie? Maybe they can help us understand you.
http://math.ucr.edu/home/baez/crackpot.html
keiths,
Oh look, the monkeys are gathering
We should use the genetic algorithms inside of Lt. Data to help us! USA! USA!
Aleta attempts to nail the Jello to the wall:
KF doesn’t exactly answer, but he does emphasize the source of his confusion:
KF,
The tape is endless, but each of the cells is finitely distant.
There is no contradiction here. There is no need for any of the cells to be “endlessly distant” or part of a “far zone”.
An attempt to clarify kf’s confusion by distinguishing between the process of endlessness and the result of endlessness.
Feedback welcome: http://www.uncommondescent.com/atheism/durston-and-craig-on-an-infinite-temporal-past/#comment-597695
Aleta,
I’d have to disagree with this part:
The process is not a member of the set, but the results of the process are.
Poor Joe is trying to spin his way out of defeat at UD:
They’re ignoring him.
ETA: Correction — KF actually responded, though not very helpfully.
KF,
If you accept that the naturals are constructed upward from 0 (or 1), like this…
…and if you accept the following premise…
…then it follows that every natural number is finite.
It’s that simple.
If you disagree, then where does the argument go wrong? Do you disagree with one of the premises? With the reasoning?
When I wrote, “Endlessness as a process is what the ellipsis inside the natural numbers means. When we write {1, 2, 3, …}, the ellipsis refers to the process by which this set can be built endlessly. The ellipsis does not, however, refer to the result of the process.”
Keith disagrees with that part, saying “The process is not a member of the set, but the results of the process are.”
Yes the process is not a member of the set – the numbers generated by the process are the members of the set. So I agree with what you wrote.
But I don’t think what you wrote actually applies to what I wrote, but maybe your point, in fact, is related to kf’s issues.
Here’s an idea.
I can think of two ways to interpret the ellipsis, and this corresponds to the distinction I made to kf.
You can think of the ellipsis as standing for the rule that creates the natural numbers: given any k, there is a k + 1. This is what I mean in saying that the ellipsis stands for the process. It means “keep on going, following the pattern.”
However, and this just occurred to me in thinking about your remark, one could think of the ellipsis as standing instead for all the remaining members of the set. That is, it could represent the entire set of numbers not hitherto enumerated. With this interpretation, the ellipsis would stand for the entire infinite result of the process and not the process itself.
It may be that this ambiguity, and kf’s fixation on the ellipsis, is part of what confuses him. The ellipsis as process only produces finite numbers, and at any moment only a finite number of them. The ellipsis as product interpretation includes the entire infinite set.
So like the Gestalt faces/vase I referenced in a earlier post at UD, flipping back and forth between the two ways to interprete the ellipsis is the source of kf’s cogntivie dissonance. With one view, he only sees a finite number of elements, and with another view he sees an infinite number. However, because the process/product distinction isn’t clear, his attempt to see both meanings at once, with an emphasis on the product interpretation, leads him to his confusion.
My 2 cents.
In fact, Wikipedia says,
“The use of ellipses in mathematical proofs is often discouraged because of the potential for ambiguity. For this reason, and because the ellipsis supports no systematic rules for symbolic calculation, in recent years some authors have recommended avoiding its use in mathematics altogether.”
I don’t know whether this admonition is related to the issue we are discussing here, but it might be.
Aleta,
It doesn’t really mean “keep going, following the pattern” — it means that the results of following the pattern, in their totality, are to be regarded as part of the set. That is, the ellipsis doesn’t stand for the process, it stands for the results of the process, taken as a whole.
Suppose I write
Everything within the braces is part of the set, except that the ellipsis represents the elements that are not explicitly stated. It means that the results of continuing the pattern established by “1,2,3” — the numbers 4 through 17 — are to be regarded as part of the set S. The process itself is not an element of the set, but the numbers 4 through 17 are. The ellipsis symbolizes the latter, not the former.
Yes, and I think this is the only viable interpretation, for the reasons given above.
I think one problem is that both you and he are taking the process as something that unfolds over time, rather than as a purely mathematical process that “happens” in zero time. All of the results are already “there” — we don’t have to wait for the process to produce them.
The idea that the process is unfolding over time is what leads KF to conclude that the results are only “potentially” infinite, not “actually” infinite. They are, in fact, “actually” infinite.
Another issue is that he seems to think that if we take the results of the process as a whole, that means that we have “ended the endless”. For example, he writes:
He thinks that the process of adding 1 to the “current” number, again and again, cannot “exhaust” the cardinality of the naturals, but that’s not correct. If the process were unfolding over time, with a fixed, non-zero time per iteration, then sure, at any point in time it would not have exhausted the naturals, but that’s not what we are talking about.
We are taking the entirety of the results of the endless process, “executed” in zero time.
They’re referring to the following kind of ambiguity. The set S = {3,5,7,…,19} could plausibly be interpreted as {3,5,7,11,13,17,19} (the primes in that range) or as {3,5,7,9,11,13,15,17,19} (the odd integers in that range). Even cases that appear straightforward, like {1,2,3,…,9,10}, are actually ambiguous, though the unstated heuristic is “apply the most straightforward rule you can think of”.
keiths,
He’s so ronrey. bless.
I guess all this is about whether it’s possible to “add one more natural number to the set” an infinite amount of times, to “reach” infinity (in the sense of producing the complete set of natural numbers, not some nonexistent infinite natural number)?
Keith, those are good points. I think one of the things that has set the stage for the discussion is that it was originally in the context of causally connected moments in time, and so was thought of as a step-by-step process. If we think of the whole set existing at once, and the ellipsis representing that, as you say, then the whole subject of moving step-by-step is eliminated.
However, since the definition of natural number is defined recursively (I think you posted the axioms earlier in this thread), I don’t think the perspective of creating the numbers one-by-one can be dismissed.
So I think the tension between these two perspectives is a major part of the problems that kf has.
aleta,
Those perspectives aren’t actually incompatible. It’s okay to think of the process as stepwise, provided you keep in mind that it isn’t spread across time or incomplete.
No, and it shouldn’t. The “construction” of the number 2 is logically prior to the construction of the number 3, which is logically prior to the construction of the number 4.
What should be dismissed is the idea that the construction is taking place within and across time and that it is somehow incomplete right now — a “potential” infinity vs. an actual infinity.
I think you inadvertently contributed to KF’s confusion by agreeing with him that the naturals cannot be “exhausted” by a step-by-step process. For example, you wrote:
Ending the endless is incoherent, but spanning or traversing the endless is not. The natural numbers truly can be exhausted by a stepwise process. It just takes an infinite number of steps.