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?
If history is any indication, keiths doesn’t know what finite, transfinite, or infinite means.
Knee jerking over knowing, Mung.
Don’t pick on Mung. He’s gone a whole 12 hours without whining about what a persecuted victim he is.
Beg Barry to unban you, then beg Barry to unban keiths.
Why? His ethics are are as arbitrary, capricious and flexible as yours. We’ll laugh from here, thanks.
What about you, Mung? Do you recognize that KF is wrong, and that there can be infinitely many integers even if every integer is finite?
Yikes, this is out of my specialty (not that I’m that good at math) but:
If that’s the case KF is (gag) right, but I’d never count on him ever being clear and simple. He is pathologically incapable of straightforward communication.
If I’m wrong about this I’m wrong, but I wouldn’t feel comfortable saying all integers are finite if we allow ω as an integer.
I don’t specialize in this, but I think that, for the topic under discussion, KF is correct.
The normal context of Hilbert’s hotel is that of whether different types of infinity are equal. The question under discussion in this context is whether or not the comprehension of the infinite list is finite or not.
KF is arguing, rightly, I think (just based on the quotes here, and knowing a bit about the topic under discussion), that the comprehension of the entirety of the negative rooms is not finite. Therefore, there cannot be an infinite past, because we can never reach the present.
daveS *may* be referring to the idea that time starts at a specific point, but then goes into two different directions. In that case, the past is infinite, but, it hasn’t really happened yet. Thus, what you actually have is two opposite forward directions from a beginning, not an infinite time before the beginning. The “time before” will never be infinite, just a perpetually growing finite past.
I am personally a fan of real infinities, but I have not yet found a way past this objection. Infinitesimals are easier to conceptualize, and perhaps there is a reality for infinitesimals and not infinities. Or, perhaps, infinities exist in some other way than traditional physical objects.
Nonetheless, the philosophical argument against an infinite past is powerful, even for us Calculus teachers who like to play with our infinities.
Sal,
That statement is incorrect. ω is an ordinal, not an integer.
johnnyb,
I’m talking about these claims of KF’s:
And:
The first claim is incorrect because the set of all integers can have infinite cardinality even though each element is finite. The second is incorrect because there are no “transfinitely remote rooms”.
Do you disagree?
The paradox of Achilles and the Turtle can be resolved in two ways:
1) mathematically: The sum of an infinite number of rational numbers may be finite
2) physically: Time cannot be divided indefinitely, there is a smallest amount of time
Both solutions are quite different, but as they result in the same conclusion – Achilles can overtake a turtle – no one cares.
It is different for the Temporal finitism:
1) physically: Time is given as a sequence of events, it doesn’t make sense to speak about a time before the big bang
2) mathematically: Allow for an event of infinite duration…
Mind you, as a mathematician, you don’t have to accept an event of infinite duration, you can chose other axioms. Doesn’t make your math wrong, just different (think intuitionism).
Conclusion: everyone is correct, everyone doesn’t want to understand the position of her or his opponent.
(and KF’s Wortgeschwurbel doesn’t help…)
How KF fits Isaiah 43:13 ““From eternity to eternity I am God.” into this, I don’t know.
Actual mathematician here. Keiths is correct, johnnyb and kairosfocus are wrong.
The integers (by this I mean either N, the natural numbers, or Z, the integers) do not contain any “infinite” or “transfinite” members. This is very basic stuff.
Besides being logically illiterate, being clueless about infinity is a necessary prerequisite for the proud creationist. Nothing to see here
So are orbits really polygons with Planck length sides? Are the physical conatants rational numbers? When we move, are we hopping from one location to another without traversing the intervening space?
False to fact. In an infinite past, we can always reach the present from any time in the past in a finite amount of time, and one can reach the present (or any other time) from infinity (even though infinity is not really a point in time) in an infinite amount of time. And an infinite amount of time is available by definition, because an infinite past was postulated.
When the “we can never reach the present” argument is used, you need to define “from where” you can never reach the present. There’s no such time, therefore the objection is bunk
In light of Jeff commenting (Jeff is a professor of computational number theory and was at the top of his class from an Ivy League school and was Bill Dembski’s professor), I have to now agree with Keiths and retract my earlier citation and error.
I looked at this:
http://stanford.edu/~jbooher/expos/ordinals_promys.pdf
I constructed the proof from definition 1 in that article.
Quasi proof of what Jeff is saying:
Is that about the right way of proving all integers are non-transfinite?????
Is the universe quantized? is a fascinating question.
Another actual mathematician here. And I agree.
I’d say that Wolfram made a mistake in terminology, which I don’t see as a big deal. Johnnyb was very tentative in his comments, recognizing the limitations of his knowledge. KF seems blissfully unaware of the limitations of his knowledge. But then KF seems to have appointed himself expert in everything. Whatever happened to that Christian idea of humility?
Prof. Shallit,
Not to Joe G!
keiths – I don’t disagree that you are using terminology more exactly than he is. However, when you look at what he is actually trying to say, rather than engaging in pedantry, he is quite correct.
I find it depressingly typical that KF can write (at length) what nobody seems able to understand. Maybe this is deliberate (and also typical creationist) so as to be able to say “that’s not what I meant” to all objections. Just ask them to define CSI in operational terms!
Niel, Dieb, Jeff Shallit, anyone,
Is my proof correct? Is the correct phrase “order omega” or “order type omega”?
What pedantry? Try articulating your argument against actual infinities and you’ll realize it’s nonsense.
It isn’t the kind of proof that a mathematician would give. Too much handwaving.
Well that wasn’t very helpful, perhaps you can point to a more rigorous treatment? I am trying learn.
Meta-mathematics isn’t my specialty, but it’s not like I have seen the von Neumann construction before in formal classwork nor Cantor’s diagonalization. I just never studied Omega numbers.
The farthest I got was something about the power set of natural numbers not being countable.
I’m not trying to be polemic here, I would like to know a better way to refute Kairos Focus. Terse would be good.
Niel,
How about this proof which I got from here, page 30
Diagonal Infinity
I looks like a proof via Mathematical Induction:
johnnyb,
No, it isn’t a disagreement over terminology, and I’m not being pedantic.
KF misunderstands the transfinite, and his argument is invalid because it rests on that misunderstanding.
More later.
keiths –
Right, that’s where I think you are being pedantic. The subject under discussion is not individual rooms, but the totality of the list of rooms.
Let me rephrase it then. How long does it take to pass by *every* room? This is not a finite amount of time, even though you can finitely define the time for any specific room.
keiths and dazz –
I think the problem in thinking is here, from dazz:
But this is precisely the problem with an *infinite* past. For *any* finite past, then there is a “from where” to ask about. For an infinite past (i.e., where the total ordered set of past events is an infinite number of finitely-sized units), the “from where” actually does become transfinite. At any *given* point in the past, it is finite. But not from the totality of all of the units. That is transfinite.
An infinite amount of time. Problem? Now tell me that’s impossible because an infinite time can’t be available… but then you’d be assuming your conclusion. Isn’t that impossibility what you’re out to prove?
???
Since KF is Christ, no doubt, humility doesn’t apply to him.
🙂 🙂
Word salad.
https://de.wikipedia.org/wiki/Geschwurbel#Etymologie_und_Verbreitung
Translate page.
Yikes! Thanks! I tried google translate box but only got “word” not “salad”
Curious.
hotshoe_,
Word salad or (UK English) twaddle.
Ninja’d
Not to be critical of mathematics, and mathematicians, but isn’t KF’s attempt to use this to disprove evolution as meaningful as arguing about how many angels can dance on the head of a pin?
Interesting. A book by H. M. Hubey.
Is that the same Hubey who used to post on sci.math; the same Hubey that said laughably absurd things about infinity?
Hmm, from Montclair State University. Then it must be the same Hubey.
Diagonal Infinity has a poetic sound.
Like time cube.
I find myself unable to wade through all of kf’s wortgeschwurbel: can anyone tell me – is he distinguishing properly between a transfinite number and a merely infinite number? And doesn’t Spitzer’s argument against an infinite past not apply equally to an infinite future?
Finally, is it just me, or do geschwurbel’s Germanic roots, listed in de.wikipedia.org as “beben, zittern, schwanken, wackeln”, sound a little like a good night
outin in Manjack Heights?I’ll get me coat.
This all started with the Click Whore of Babble-on stupidly quoting a stupid quote of Robert Spitzer, and calling for stupid comments — which she got. According to Spitzer:
Marshaled by whom, you passive-voice putz? No biologist that I’ve ever read. William Dembski got the notion of bounded “probabilistic resources” from Dawkins (The Blind Watchmaker).
Oh, so you actually wanted to talk about cosmic fine-tuning, not evolution.
Obviously false. It is possible that each and every one of the universes in an infinite ensemble of universes has a temporal beginning. It is not necessary that there be some sort of quasi-temporal ordering of the universes.
DNA_Jock:
🙂
Well, I said shake, baby, shake
I said shake, baby, shake
I said shake it, baby, shake it
And then shake, baby, shake
Come on over, whole lotta shakin’ goin’ on
Sal,
That “proof” doesn’t work because it doesn’t cover zero and the negative integers and it makes a bunch of hidden assumptions.
If I were informally trying to persuade someone, I would get them to agree to the following:
If you accept 1-6 above, it follows that every integer is finite.
It just means it can only be yesterday but never today
johnnyb,
There’s your mistake, and KF falls into the same trap.
Negativity infinity is not a starting point, just as positive infinity is not a stopping point. You don’t stop at all. If you did stop, then you’d be stopping at a finite point.
And it’s symmetrical. You don’t start anywhere. If there were a starting point, it would be finite.
There is no “from where” in the negative case, just as there is no “to where” in the positive.
Unrelated, but this is another issue where William Lame Craig contradicts himself: he labels himself an A-theorist of time, considers time as a continuous, and actual infinities impossible. Well, if time is continuous then an actual infinity of state of affairs have been actualized in any finite time span
keiths:
What you are missing is that it makes sense that there is no “to where” in the positive, because it has not been reached. But the interesting thing about calling something the “past” is that it has already been reached.
No, there’s no difference. The only reason you think there must be a “from where” in the past is that you assume that there’s a beginning
johnnyb,
If it hasn’t been reached, then you are at a finite n. You aren’t talking about infinity.
Earlier, you wrote:
…but you’re failing to apply that to the positive case. We aren’t talking about the rooms up to some finite n. We’re talking about the totality.
To reiterate, if you insist on a “from where” — a starting point — then you aren’t including negative infinity. If you insist on a “to where” — a stopping point — then you aren’t including positive infinity.
Negative infinity is not an origin, and positive infinity is not a destination.