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?
That’s what I’ve always understood too.
KF is unhappy:
KF,
I haven’t mentioned “benighted fundies”. What I marvel at is your stubbornness in clinging to an intuition that is clearly wrong and can easily be shown to be so.
Everyone in this conversation, including JoeG, recognizes that the finite is bounded and the infinite is boundless.
Sure it does. The entire endless set of natural numbers is bridged by this simple induction:
KF:
The inference doesn’t depend on the “inherent finitude of cumulative steps”.
Let me repeat this comment from yesterday:
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?
What defeat? Please be specific or admit you are defeated. And what is this alleged spin? You said it. It is here for all to read
kairosfocus:
It’s true that I’m not your admirer, but in this thread I’m focusing on the shortcomings of your mathematical reasoning, not of your theological or evolutionary reasoning.
keiths:
KF:
That’s right, because “ending the endless” is incoherent. What I’m suggesting is the exact opposite: don’t end the endless process, but instead take the results of that endless process as a whole.
You’re not grasping the problems that arise when you regard the naturals as a mere “potential infinity”.
For example, if the naturals were merely potentially infinite, it would mean that they are currently finite. That would mean that there is a largest natural number right now (and at any given time, depending on how far the induction has progressed).
Don’t you see the absurdity of viewing the induction as taking place across a span of time? Do you really want to follow JoeG into the “largest known number” pit?
OK great, keiths cannot support his claim. Not only that keiths seems totally unaware that infinite sets contain finite subsets.
Despite my warning, KF follows JoeG over the “largest known number” cliff:
KF,
If the construction of the natural numbers is incomplete, and only a finite number of them have been generated by the induction so far, then please answer some questions for us:
1) What is the largest natural number as of today, 8:30 AM Pacific Time?
2) How fast is it increasing? What determines this? Is the rate constant, or does it vary? Is the speed the same everywhere in the universe?
3) Suppose we “cheat” and add a trillion to the current largest natural number. Would it be wrong to say that the new number is a natural number, since the induction hasn’t “reached it” yet?
4) Do you have any doubt whatsoever that the new number will be a natural number once the induction reaches it? If so, why not make the leap and call it a natural number now? Why should our minds be limited by the speed of this plodding hypothetical induction process?
KF comes very close to getting it:
Exactly! All natural numbers are finite, but there are infinitely many of them.
But then he spoils it:
No, because all “actual endlessness” means is that there is no end. For every k, there is a k+1.
Every k is finite, and for every k, there is a k+1. In other words, the natural numbers form an endless sequence of finite integers.
KF is still floundering in the “largest natural number” trap.
Interestingly, he stopped “glancing back at TSZ” — or pretended to stop — just after I raised the issue.
keiths,
It’s quasi-semi-latching weasel all over again.
He’s in his safe space.
One month after the discussion began, KF is still hopelessly confused:
No, KF.
For every k, there is a k+1. That means the tape is endless, not finite.
And there is no magic point on the tape where a finite k, when incremented, becomes an infinite k+1. Therefore k and k+1 are finite for every k in the set.
Your statement…
…is wrong, and the reason should be obvious: what makes the tape endless is not the values associated with the cells, but rather how many of those values there are. An endless tape filled with 3s is still endless even though 3 is finite, and an endless tape filled with incrementing values is still endless even though each value is finite.
Given that for every k there is a k+1 it does not follow that the tape is endless.
keiths,
There isn’t any such thing as an endless tape.
daveS, Aleta and ellazimm have been doing a fine job of explaining things, but KF is still irrationally clinging to his faulty intuition:
KF,
Right. The value isn’t endless. It’s the tape that is endless.
That’s what it means for the tape to be endless. For every k, there is a k+1. There is no k at which the tape ends.
There’s your mistake. The fact that the tape doesn’t end — that for every k, there is a k+1 — does not mean that any of the k’s are infinite.
Where’s the contradiction? The only thing being contradicted is your faulty intuition.
Your position is absurd because it implies that for some finite k, k+1 is infinite.
The mainstream mathematical view — the one we’ve being trying to explain to you — suffers from no such absurdities. It simply disagrees with your faulty intuitions.
For one who likes to lecture people about the “principles of right reason”, you are oddly impervious to them.
keiths defines the tape to be endless. Where’s his evidence that it is endless?
What’s the largest known number, mung?
KF is still wallowing in the LKN pit with JoeG:
I know KF isn’t the brightest, but can he really not see the problem(s) here?
KF,
Repeating some earlier questions:
If the construction of the natural numbers is incomplete, and only a finite number of them have been generated so far, then please answer some questions for us:
1) What is the largest natural number as of today, 8:30 AM Pacific Time?
2) How fast is it increasing? What determines this? Is the rate constant, or does it vary? Is the speed the same everywhere in the universe?
3) Suppose we “cheat” and add a trillion to the current largest natural number. Would it be wrong to say that the new number is a natural number, since the induction hasn’t “reached it” yet?
4) Do you have any doubt whatsoever that the new number will be a natural number once the induction reaches it? If so, why not make the leap and call it a natural number now? Why should our minds be limited by the speed of this plodding hypothetical induction process?
KF has gobbledygook thinking on this matter.
If one is writing about math, try to write like its from pages of a textbook. That’s the standard of communication, imho.
KF has written a comments closed ‘FYI/FTR’ post which includes him drawing on someone else’s diagram. Victory must be in sight! http://www.uncommondescent.com/mathematics/fyi-ftr-on-ehrlichs-unified-overview-of-numbers-great-and-small-ht-ds/
More LKN inanity from KF. He quotes Wikipedia on the axiom of infinity, with his own comments in square brackets:
KF,
1. What is the largest “finite specific value” that has been “operationally” realized?
2. What is the smallest “finite specific value” that has not yet been “operationally” realized?
3. Have your answers to #1 and #2 changed in the last few seconds?
Those who remember KF’s hilarious “quasi-latching” implosion will be amused to see that he is now referring to ω as a “quasi-member” of the set of natural numbers:
daveS is having none of it:
So KF appeals to faith:
Cantor is rolling over in his grave.
JoeG is back with his set subtraction argument:
He can’t even get KF to agree with him. That’s gotta sting.
When I asked Joe/Virgil, “Does E [the evens] have exactly 1/2 as many elements as N [the naturals]?”, he replied
“As infinity is a journey it would all depend on when you looked. However the relative cardinality can be determined by the bijective function.”
Seriously.
I though about asking him if Tuesday would be different than Wednesday, but I didn’t want to be snarky. Of course, if there is a bijective function than E would have the same number of elements as N. But how in the world could the number of elements in E vary according to “when you looked”, even if he meant something different than Tuesday?
Aleta,
And even though they disagree on the cardinality issue, KF makes the same “it depends on when you look” error as Joe, whether he realizes it or not. He argues that the set of natural numbers N is only “potentially” infinite, since at any given time the “finite stepwise process” hasn’t “actualised” the infinite. That means in turn that N is finite, and if it’s finite, then at any given time there is a largest natural number n, and n+1 is not yet part of N.
It’s ludicrous, but quite entertaining.
Aleta,
That Dionisio business was pretty odd, too, but then, so is he:
Of 1585 comments in this thread (so far), 1494 are by Dionisio
Oh, he’s that guy. I didn’t know of him: it took a few posts for me to catch on that he was just trolling me for who knows what reason. I’ve got it now.
keiths,
Haha, he’s being doing that for years – ‘here’s something science can’t explain’, followed by a link to some enzyme or other, with excerpt. Rather than complain at spamming, I recall someone thanking him for the valuable resource he was building up! I wonder if he reads them.
Yes it does, it literally follows deductively. Given the law of non-contradiction, it is absolutely certain (to the extend that any deductive proof can be certain) and beyond all rational doubt that it does not have an end.
In so far as you declare an end k, you can k+1 to violate it. There cannot be an end.
KF writes:
Aleta,
You should give the priggish KF a warning on language. 🙂
KF:
No, KF. If you take the +1 succession as a complete whole, then it is complete. There are no missing members. Each member k is finite, and its successor k+1 is also finite. There is no k at which the set ends.
You are saying, in effect, “Let’s take the set as a complete whole, then treat it as incomplete. We can then add a member to it, and that member will be infinite, on the “copy the list so far principle of succession”.
It’s a blatant contradiction. The set can’t be complete and incomplete at the same time. And if the set is complete, there is no need to add a successor to it.
Also, by acknowledging that the successor is infinite, you are conceding that the +1 succession itself is itself infinite — an infinite succession of finite elements.
Stop fighting the logic, KF. The succession is endless. For every k, there is a k+1. Yet each of those k’s, and k+1’s, is finite.
Honestly, KF, stick to apologetics. You are better at that than you are at math.
Neil:
It was apologetics that got him talking about infinity in the first place.
I think this is one reason kf can’t give up – he’s tied his wagon to the argument that the impossibility of an infinite past is an argument for God, re Spitzer. We haven’t been discussing that at all, but kf occasionally reminds us that the infinite past issue is really the issue of the thread.
I wrote the following summary of Virgil’s mathematical position
“As far as I can tell, Virgil’s view is that every infinite proper subset of the natural numbers has a different cardinality. Much as Cantor named aleph null (A0) as the cardinality of the naturals and A1 as the cardinality of the reals, and then built a sequence of further levels of infinity from there, Virgil seems to have the idea that we can start with the cardinality of the naturals and build down from there.
The evens have fewer members than the naturals, the set of all squares would have even fewer members, the set of all factorials even fewer yet.
Interesting enough, the integers would have more members than the naturals, by the same argument Virgil uses for the odds and evens.
Let I = the integers = {0,1,-1,2,-2,3,-3, …}, N = the naturals = {1,2,3,…), and N- = the non-positive integers {0,-1,-2,-3, …}
Then I – N = N-, so N and N- have fewer members than I.
One more example: let N = the naturals and let N1 = {2,3,4,..}, so that N – N1 = {1}. By Virgils reasoning, the cardinality of N1 is one less than the cardinality of N, although still infinite.
Therefore, Virgil’s belief, based on his set subtraction method, is that there are an infinite number of levels of infinity, both less than and greater than the level of infinity associated with the natural numbers.
This seems to be what he believes.”
Virgil seems to agree I have his position correct when he writes,
“Aleta, I would start with some accepted standard, yes. But I would allow for a cardinality greater than, equal to or less than that standard. It is all relative, hence the name.
That way you don’t need any special pleading to get around the ramifications of set subtraction.
And that follows from Cantor’s reasoning behind small and big infinity- and again I am not sure if the naturals is what I would use as a standard. I haven’t given it much thought but that is what I would most likely start with and then see if another standard is better.”
So there you are: every infinite proper subset of the integers has a different “relative cardinality” that, although he hasn’t thought much about it, if he did, he would start with the cardinality of the naturals and work up and down from there, as I suggested.
One of the things I find interesting about conversations like this is that I, and people like me, seem to think harder about the unconventional ideas that some people have than the people holding those ideas do.
aleta,
This will be a huge climbdown for him, if he ever manages to acknowledge his error.
He does seem to be perilously close to a realization:
daveS:
KF just needs to realize, and accept, that T contains nothing but “specific defined terms.” There are no “non-specific undefined terms” in the set.
aleta,
Yes. As a glaring example, ‘Virgil’ never seems to have considered the implications of his “set subtraction method” for comparing the cardinalities of finite sets such as these:
A = {hen, fox, sheep, wolf}
B = {1,2,3,4}
“Cantor math” is able to determine that A and B have the same cardinality. “JoeMath” fails at even this trivial task, because his set subtraction method only works when one set is a subset (proper or otherwise) of the other.
Mathematicians think. Joe doesn’t.
LoL! How do you know what he thought of? I am sure that whatever you can think of he has already considered.
OK so Aleta doesn’t understand that infinity is a journey nor the implications of that.
keiths,
LoL! @ keiths- No, you don’t use set subtraction when it doesn’t apply. And no, you don’t use a methodology for infinite sets on finite sets.
Mathematicians think. Obviously keiths cannot think beyond his strawmen
Joe,
I think you need to redo your homework assignment from 2013:
Don’t forget to do your homework, Frankie/Joe. See above.
Set subtraction proves there is a different number of elements between two countably infinite sets. Nothing you can say will ever change that fact
keiths,
0 has been removed. You obviously didn’t think that through
keiths,
Set subtraction says the cardinality changed. That is if you define cardinality as the number of elements.
keiths:
Frankie:
‘0’ wasn’t removed. It was changed to ‘1’. *
No elements were removed from the set and no elements were added to it. Yet according to JoeMath, the set’s cardinality has changed. It’s smaller than before.
That’s why no one besides Joe (and his eerily similar friends Frankie and Virgil) is interested in JoeMath: it fails in situations that are easily handled by Cantor Math.
Why don’t mathematicians adopt JoeMath? The short answer: they’re not idiots.
* And if you’re tempted to invent a new “rule” of JoeMath that forbids elements from changing, don’t bother. JoeMath would still fail. See if you can figure out why.
Is addition something you also don’t understand?
Here is a list of numbers. Add 1 to each number.
Complex stuff!
keiths,
That means it was removed.
0 was removed as it is no longer there.
LoL! Throwing your hands in the air and declaring them equal is not math.
Look, no one uses Cantor’s concept of equal cardinalities for all countably infinite sets for anything. It is a useless concept. That you refused to answer my question about it proves my point.
Getting technical, I’ll have to agree with Frankie on this point. 1 is a different number from 0. So changing 0 to 1 entails removing 0 and inserting 1.
Keiths has made a subtle change by putting quotes around the “0” and “1” in what I quoted above. That seems to be treating 0 and 1 not as mathematical objects, but as properties of some otherwise unspecified objects. But he did not use such quotes previously, so this seems to be an inappropriate move.
Neil,
I addressed that already:
To spell it out: If instead of changing n to n+1, you insist on removing n and inserting n+1, JoeMath will still fail. Why? Because for every element in the set, you remove that element and replace it with a new element. It’s a one-for-one substitution, so the cardinality does not change.
What could be more obvious? If you have the set {0,1,2,3} and replace every element n with n+1, then 0 gets replaced with 1, 1 gets replaced with 2, and so on. Every element gets removed and replaced with exactly one element, so of course the cardinality doesn’t change. The orginal set {0,1,2,3} and the new set {1,2,3,4} have the same cardinality — 4.
It works just as well for {0,1,2,3…} as it does for {0,1,2,3}. If you remove each element, replacing it with exactly one element, then the cardinality doesn’t change.
Yet JoeMath says the cardinality does change. That’s why mathematically literate people laugh at JoeMath and continue using Cantor’s superior ideas regarding cardinality.