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?
I think an infinite number of angels can dance on the head of a pin.
Countably infinite or more?
KF still doesn’t get it:
There is no point, either in time or in space, at which the manager was “transfinitely far from the last room.”
daveS is valiantly trying to convey the same message:
I can’t decide — does KF truly not get it, or is he just too embarrassed to admit his mistake after doubling down so many times?
johnnyb,
Is your god capable of creating a deterministic universe with a two-way infinite timeline?
From where? “The infinite” is not itself a room number. There is no room from which the present cannot be reached in a finite amount of time, because all the numbers are finite.
It might depend on which version of relativity is correct Einsteinian or Lorentzian.
Einsteinian is the dominant one today where space and time are measured in the same units, hence the word “spacetime”.
In Einsteins general relativity time can be measured in meters and there is no absolute clock nor coordinate reference frame nor absolute simultaneity. I think Einstein is mistaken, but if not, then God could make an infinite timeline both directions, and our perception of time is an illusion.
I’m sympathetic to Lorentizian relativity which is more in line with human intuition where we don’t measure time in meters, but it is a separate dimension than space and the Einstein formalism is just mathematical perspective that is usually correct, but is occasionally seen to not agree with experiment or theory, most notably the conflict of General Relativity with Quantum Mechanics. Btw, Special Relativity is consistent with quantum mechanics. One can google
https://en.wikipedia.org/wiki/Relativistic_quantum_chemistry
If time is Lorentzian, then it’s hard to say. It raises the issue whether time can be created or is as fundamental as God and truth.
Agree, but it does seem paradoxical. That is essentially Hilbert’s paradox.
It seems finitistic reasoning breaks down in questions of infinity — we can extrapolate finite approaches only so far before the conclusions appear outrageous. That’s why I’d expect an infinite God to be paradoxical to limited human minds.
ASIDE:
It was nice to see set theory construct the idea 1+1=2. Von Neuman’s method could show the proof in one page (I think), but it must assume set theory.
It’s much harder using logicism of Russell, it took 379 pages:
https://en.wikipedia.org/wiki/Principia_Mathematica
I think Russell started with other axioms than set theory.
The printed edition of Russell’s book sells for 1000 dollars, the pdf is free.
That’s the nature of a true paradox.
It would be easier to use natural numbers to make the argument, imho.
Aleta, to KF:
KF geschwurbelt wieder:
And without a hint of irony, he writes:
Of course, “poof-magic mathematical hand waving” is exactly what KF does when he writes things like:
This is pure hand-waving poofery. You don’t “pick up ω” by “going through the ellipsis” to a new “zone”.
keiths,
What kind of education could KF have that he can’t handle a basic, HS level mathematical proof, even if his mistakes are presented to him time and again? My money is on home schooling
I finally figured out what the heck kf has been trying to say, and thus am clearer about how confused he is. I just found this site (I googled “transfinite zone” to see if anyone besides kf used that phrase, and UD and here are the only places it showed up), and I’ve read the comments here: it seems like the real mathematicians agree with me.
I written several posts at UD today, starting at #95, about this: it will be interesting to see how kf responds. I’d appreciate any confirmation or correction to my points that any of you here would be willing to give.
Welcome, aleta!
Okay. Mathematician here. I’d say that you have it about right.
I’m really not sure what KF is thinking. Perhaps he believes that his god would not have created transfinite numbers if they were inaccessible.
Hi Aleta,
Welcome to TSZ.
Here’s an argument that even KF should be able to follow. Perhaps you could post a link at UD.
The set of natural numbers can be informally constructed this way:
1. 0 is a natural number.
2. If n is a natural number, then n+1 is a natural number.
Applying those two rules yields N = {0,1,2,3,…}.
3. Every natural number n has exactly one successor, n+1.
4. Every natural number n (except for 0) has exactly one predecessor, n-1.
5. If A is the predecessor of B, then B is the successor of A.
6. if A is the predecessor of B, then A is less than B.
7. If B is the successor of A, then B is greater than A.
Let’s assume KF is right and that there are transfinite numbers among the naturals. Then there must be a smallest transfinite number S in N, below which all the numbers are finite.
The predecessor of S — let’s call it P — must be among those finite numbers, since predecessors are always smaller than their successors. If S is the successor of P, it is equal to P+1. But P is finite, and so P+1 is also finite, which means S is finite, contrary to the starting assumption.
Therefore the assumption is wrong: there are no transfinite numbers among the naturals.
Sunday School.
I thought Keiths proof looks good.
Any mathematicians want to comment? I like the proof, can I use it?
FWIW, I have never seen it laid out like that. Is it because the question is so trivial or just not of concern to anyone? I don’t recall the topic being discussed in math class in those particulars.
That is, I don’t recall any one cared to say, “all naturals are finite”, they focused instead on the cardinality of the naturals vs. reals and Cantor’s diagonalization proof — I guess that’s why Hubey used the phrase “diagonal infinity”.
I suspect KF won’t back down easily. He still hasn’t backed down from his 2nd law arguments for ID.
Myself, I prefer Frankie’s discussion of the largest number.
The issue is that KF tries to prove there’s something intrinsically contradictory in infinite series. It’s absurd
See keiths post here: keiths,
Or another way to look at it: if there was a “transfinite zone” among natural numbers, then it follows that the transition from the last finite number to that zone should be a traversal of an infinity. So the only way to have that transfinite zone in there would be if infinities could actually be traversed, but that’s what he’s trying to prove impossible
Sal,
Sure. I doubt that I’m the first to come up with it.
KF writes:
He’s being misled by the following intuition:
That intuition is faulty. You can’t travel an infinite distance and then stop, finding yourself at a room which is “transfinitely remote”. If you stop at all, then you haven’t traveled an infinite distance.
Isn’t this obvious? If you start counting from 0 and then stop at some point, you haven’t reached infinity. If you start traveling from 0 and then stop at some point, you haven’t traveled infinitely far.
petrushka,
That was classic.
keiths:
Is the largest known number computable?
petrushka,
I know who we can ask.
Hey, wait a minute… Where is that guy?
[latexpage]keiths,
How do you define addition, if not on the numbers you’re trying to define? It is much more difficult to define numbers rigorously than you seem to think it is. To my knowledge, nobody has done it to everybody’s satisfaction.
Everybody that you are dealing with accepts, I think, that the set of natural numbers $\mathcal{N} = \{0, 1, 2, \ldots\}$ is infinite, and furthermore that a set is infinite if and only if it is not finite. So take the ball, and run with it.
Sets $A$ and $B$ are identical in cardinality if and only if there exists a one-to-one correspondence (bijective function, invertible function) $f: A \rightarrow B.$ Set $A$ is finite and of cardinality $n \in \mathcal{N}$ if and only if there exists a one-to-one correspondence $f: \{0, 1, \ldots, n-1\} \rightarrow A.$ With this definition in place, it’s easy to argue that if the cardinality of the set of natural numbers were a natural number, then the set of natural numbers would be finite — a contradiction.
Cardinality is inherently counterintuitive, beginning with the fact that an infinite set $A$ may be identical in cardinality to its proper superset $B,$ i.e., $A \subset B$ does not preclude $|A| = |B|.$ You cannot make the counterintuitiveness go away by saying just the right words when the moon is in just the right phase. Mathematicians (generally) agree on the definition of cardinality they do because it serves their purposes better than anything else that’s come along. People who have not labored with the math have not earned the right to an opinion as to whether the definition makes sense.
A joke among profs who’ve taught discrete math is the student who insists, “I know what the term means; I just can’t define it.” Sometimes people need to be told that if they cannot deal with formal definitions, then they do not know what they’re talking about.
Tom,
Hence my choice of the word “informally”:
Tom:
I don’t define addition. Why reinvent the wheel? If KF agrees with my premises…
…then the conclusion follows: there are no transfinites among the naturals.
Your argument depends on the fact that each natural number is finite, but that’s the very point that KF disputes. He wouldn’t accept it as a premise. I’m starting from premises that he presumably would accept and showing that he cannot be right about transfinites among the naturals.
KF may never be able to reason formally about this; the siren song of his intuitions may be too strong. But I do think he is capable of at least following my argument, which shows that his intuitions are inconsistent.
Premises 1-7 above, which are highly intuitive, are incompatible with KF’s intuitions regarding a “transfinite zone”. Something’s gotta give, and the “transfinite zone” is the first thing a sensible person would jettison.
petrushka:
keiths:
He’s finally learning how to program. We may be waiting a while:
https://www.youtube.com/watch?v=_DyGP4Yk4Zo
What argument? You were trying to clarify things by describing the ordering of the natural numbers. I proceeded under the assumption that we all agree on the ordering, and used the ordering to define finite set and cardinality of a finite set in the usual way. Loosely speaking, this is what makes the natural numbers — all of them — into finite cardinal numbers. We haven’t even gotten to transfinite numbers yet. Someone who wants there to be an infinite natural number must define finite number in a way that does not “use up” all of the natural numbers.
KF will not attempt to define the finite numbers. He’ll just set the matter aside for a while, and then return to treat the onlookers to copy-and-paste enlightenment.
We could ask Frankie, but I think if you add one to the largest known number, you get a transfinite number.
Russell and Whitehead tried and failed. 🙂 It did shock me that some aspects of math transcend logic and hence they could not construct math based on the axioms of pure logic alone. Some articles of faith beyond pure logic must be accepted. I just about fell out of my chair when my math professor pointed out the foundations of math have an element of unprovable faith.
……I like the von Neuman construction, it creates a method for defining addition for natural numbers. The beautiful thing about it is it shows so much math can be constructed from the simple axioms of set theory!
You don’t have to stop there, though. You can multiply it by seven or more and get into the really huge (so-called “super high”) numbers.
Really fascinating stuff if you don’t mind numbers.
W
Articles of faith? No.
We use axioms, but those are not articles of faith.
[latexpage]The “Modern Definitions” section of the Wikipedia article on “Natural Number”:
The set-theoretic definition of the natural numbers usually looks something like this:
{},
{{}},
{{}, {{}}},
{{}, {{}}, {{}, {{}}}},
⋮
Here 0 is a name for {}. For each set $n$ in the sequence, the successor of $n$ is $n \cup \{n\}.$ You all know how to write the successor of a string of digits naming a number. You can see above that 0 = {}, 1 = {0}, 2 = {0, 1}, 3 = {0, 1, 2}, …, 10 = {0, 1, …, 9}, … That is, each natural number is a set containing all of its predecessors. The number named 0 is the empty set because it has no predecessors.
I’m not going to undertake an explanation of the Peano axioms. Perhaps the thing to mention here is that they enable us to do proofs by induction. Basically, you establish that a proposition P “about” a natural number n holds for n = 0, i.e., P(0) is true. Then you show that for all natural numbers n, if P(n) is true, then so is P(n+1). Having established those two parts,
P(0) is true
P(n) implies P(n+1) for all natural numbers n,
it is valid to conclude that P(n) is true for all natural numbers n. It seems to me that there’s embedded in KF’s remarks a claim that mathematical induction is not a valid proof technique. But I don’t see at the moment how to make that clear.
keiths:
Tom:
This one:
Your argument assumes that every natural number is finite. That happens to be true, but the point here is to come up with an argument that could persuade someone of that truth. Simply assuming it at the outset won’t convince anyone.
I was exposing the inconsistency of KF’s beliefs by showing that they lead to an absurdity: the idea that in at least one case, adding one to a finite number results in an infinity.
The problem is that KF thinks we can get to transfinite numbers that way. As he says in regard to Hilbert’s Hotel:
He’s wrong about that, as you and I know, and my argument shows why he is wrong without requiring him to assume the point under dispute. My seven premises are pretty straightforward and intuitive, and even a mathematical naïf like KF should find them unobjectionable.
That extracted sentence seems to me to involve some complexities, like well-ordering of the transfinites and how to interpret “below which all the numbers are finite”.
If you’ve got well ordering of the naturals, then you’ve got math induction, since the two are equivalent AFAIK. Then you can prove what KF denies by induction, assuming that you accept that if n is finite, so is its successor. If someone does not accept that, then its hard to see what more can be done. I’d say trying to argue math further with such a person is a Mung’s game.
Have you read Rucker’s Infinity and the Mind? For those who have not seen it, it’s a rich and stimulating popularization of transfinite math and other material (eg Godel). IMHO it would be a better use of one’s time than trying to argue math with someone like how you describe KF to be. But YMMV.
[latexpage]
It does not happen to be true. It is what mathematicians mean. Do you really believe that you are possessed of the preexistent “truth about natural numbers,” and that it is your mission to persuade others to speak only that “truth,” whether or not they understand it? Or have you simply spoken carelessly? What is far worse, in my opinion, than what KF says outright is the latent message that freestyle rhetoric is the way to make sense of mathematical concepts. I think it’s a very bad idea to join in his process, attempting to nail him with the “truth.”
I defined finite set, and also cardinality of a finite set. Taking these definitions along with a set-theoretic definition of the natural numbers, an immediate consequence (not an assumption) is that the natural numbers are themselves finite sets. This gets at what it ordinarily means, in contemporary mathematics, to say that a natural number is finite.
KF apparently does not realize that he disagrees with the conventional definition of a finite set. It is an intuitively appealing definition, not the “truth.” Most people will see the sense in defining the finite before addressing the not-finite. Most ID partisans will excuse their “genius” from laying a foundation for his rhetorical edifice. No rhetoric of yours is going to change that.
Ok Neil, that’s what my math professors said, even likened it to the Apostles Creed!
It’s not like they are alone:
You posted while I was composing. That’s very close to what I’ve been thinking. I didn’t want to come back at Keith with an apparent tu quoque: “No, what you’re doing is subtly circular.”
I have loads of respect for Keith. That’s indeed part of why it bothers me to see him take the tack he has.
Weird.
There’s an “if” condition there, and it is surely false. A religion isn’t just a system of ideas. Otherwise Sherlock Holmes would be a religion.
I’m inclined to say that religion is a deep emotional commitment to a system of ideas. I don’t commit myself to axioms in mathematics. Rather, I temporarily assume them while doing the math, so as to determine the implications of those axioms.
Tom, Bruce,
I think what both of you are forgetting (or overlooking) is that I am not trying to construct a ground-up proof that each natural number is finite. I simply want to show that KF’s intuitions are inconsistent:
Tom:
It’s intended to be circular, in the following sense: If KF agrees with my premises 1-7, then he already implicitly agrees that every natural number is finite. It’s just that he doesn’t realize that yet. He is unwittingly contradicting himself by claiming that there are transfinite numbers among the naturals. The point is to get him, and similarly confused folks, to recognize that contradiction.
Bruce,
Yes, I read it when it came out, which I’m astonished to realize was 34 years ago!
Fair enough. Better you than me.
In any event, my reasons for the post were to make sure people were aware of Rucker and to make the pun on Mung’s userid (although I would not be surprised if I was not the first).
That’s correct. Some secret gestures are also required, along with a magical potion.
Yup. What cannot be derived from one set of axioms is possibly derived from another. Hopefully I will be around long enough to see what comes of the work at founding math on categories. I don’t cherish the thought of having to acknowledge that math has changed dramatically, and that I’m clueless as to where it stands. But I have the decency not to pretend to know what I don’t know.
keiths:
Tom:
It’s true in the same sense that “4+3=7” is true. That is, it follows from the definitions and conventions we accept. “Every natural number is finite” is in fact a true statement, given the consensus definitions and conventions regarding those terms.
Do you agree that my conclusion follows from my premises? If so, why characterize my argument as “freestyle rhetoric”? If not, could you point out the flaw?
https://www.youtube.com/watch?v=8hEYwk0bypY
Am I Right Or Wrong
You are altogether beautiful, my darling; there is no flaw in you.
http://biblehub.com/songs/4-7.htm
Hi math people. Rather than rile kf, I thought I’d ask the real math people over here a question about infinity. (I am a retired high school calculus teacher, so I’m not entirely a layperson.)
In the quote by William Lane Craig that kf cites, Lane writes
Now leaving aside the idea of countable events (let’s think natural numbers), it seems to me that it might not be accepted terminology to say that a sequence goes to infinity “as a limit”. Of course sequences go to infinity, but is it proper to think of infinity as a limit?
Ex: I could write as x -> infinity, x^2 -> infinity, but would it be equivalent, and proper, to say
lim (as x -> infinity) x^2 = infinity?
My understanding is that sequences need to converge to have a limit – something they get “infinitely close to” (I know that can be formalized), but that if a sequence gets infinitely large it diverges and it doesn’t have a limit. So calling infinity the limit of the sequence is perhaps incompatible with the distinction between convergence and divergence.
So is the phrase “goes toward infinity as a limit” acceptable, or confused, or what?
Thanks in advance.
You would not use that in a strict mathematical proof. But that kind of talk is common during informal discussion.