KF tackles the transfinite

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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?

387 thoughts on “KF tackles the transfinite”

  1. Tom,

    I still can’t figure out exactly what you are objecting to.

    From your comments, I gather that in your view I have taken a wrong tack by “joining in” KF’s process of using “freestyle rhetoric” to make sense of mathematical concepts, making illegitimate use of the word “true” along the way.

    Do you mean something else?

    I’ve already addressed your objection to the word “true”:

    Tom:

    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?

    keiths:

    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 really object when someone says that “it’s true that 4+3=7” or “it’s false that 4+3=6”?

    As for “joining in” with KF’s use of “freestyle rhetoric”, I’ve presented an explicit argument against KF’s intuition regarding the natural numbers:

    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.

    That argument doesn’t strike me as “freestyle rhetoric”.

    Could you explain, a bit more explicitly this time, what you are objecting to and why?

  3. Mung:

    I’m more interested in knowing why Tom has ” loads of respect for Keith.”

    That makes sense. As someone who isn’t respected, you’d like to understand what others do to earn it.

  5. daveS asks KF point-blank:

    So that we both understand each other, is it true that every natural number is finite? Yes/No?

    KF dodges the question, of course, and doubles down on his misconceptions:

    I am not satisfied that pointing out that successive addition attains to natural number k then k + 1 adequately addresses the endless succession to arrive at transfinite cardinality.

    You don’t “arrive at transfinite cardinality”. When you go from a finite k to a finite k+1, you have simply arrived at k+1. The transfinite cardinality isn’t “arrived at” — it’s a property of the endless succession taken as a whole.

    Surely, if the past is endlessly remote, at some point it must have been transfinitely remote if such is so.

    No. An infinite past would not mean that any point in the past was infinitely remote. Every point in the past is finitely remote; it’s just that there’s no end to those finitely remote points. Likewise, the fact that there are infinitely many natural numbers does not mean that any of them are infinite. It’s just that there are infinitely many finite natural numbers.

  6. I, as an experiment, have written kf at UD with a very specific set of points and questions to see if he can focus on the simple topic of the nature of the natural numbers.

    However, others have again brought up how this relates to the idea of a infinite past, and whether an infinite past is possible or not.

    I have no interest in wading into a metaphysical morass with kf about the nature of time, but I’m willing, for the sake of discussion, to consider the number line as a model for time, with the integers being a model for seconds of time.

    But I do have some thoughts, so I’ll post them here (out of kf’s sight), and get them out of my mind for the moment.

    At UD, daveS made the interesting comment that “the existence of an infinite past just means that given any natural number n, the universe already existed n seconds ago.”

    That seems right to me (although I’m thinking of time as extending before the start of the universe: that is one of the metaphysical issues I want to avoid.)

    Just as an infinite future means there is no last moment of time, an infinite past means there is no first moment of time. Every moment has both a successor and a predecessor.

    But kf, Spitzer et al add another element to the discussion that they see as inextricably connected: that time has within it a set of causally connected events.

    That is, at every second k, there is an event Ek that is caused by the preceding event Ek-1 and causes Ek+1. Causality is one-directional: time has an arrow.

    Therefore, to them, to say that time has no beginning is to say there is no first event, and thus no first cause, and as we all know, that is anathema to the “God as first cause” argument so predominant in Christian theology.

    But if you separate the issue of some abstract infinite number line model of time from the issue of events happening within time, the theist could claim (note: I am not arguing for this), that time has an infinite past, but at some moment the first event happened.

    To summarize, the question of an infinite past, using the number line as a model for time, is a separate issue from an infinite causally-connected uni-directional set of events with time. It is really the latter that is driving kf et al’s concerns. But since they are assuming that time and events in time are effectively co-incident, they can’t see the difference in the issues.

    My thoughts for the morning.

  7. aleta: Therefore, to them, to say that time has no beginning is to say there is no first event, and thus no first cause, and as we all know, that is anathema to the “God as first cause” argument so predominant in Christian theology.

    I thought you wanted to avoid metaphysics. 🙂

    You’re wrong here, but I don’t want to detract from your real purpose so won’t get into the theology or metaphysics of what is meant by a First Cause or Unmoved Mover.

  8. I wanted to avoid talking metaphysics with kf, because I want to stay focussed on just the question of the nature of the natural numbers, so I posted these thoughts over here. The main point here is that the question of time and the question of events in time are two different issues. It does seem kf is concerned about metaphysical issues and first cause notions when he writes in the OP, “For, it seems the evolutionary materialist faces the unwelcome choice of a cosmos from a true nothing — non-being or else an actually completed infinite past succession of finite causal steps.”

    As I said, I don’t want to discuss these issues – I’m just making the point that they are in the background of kf’s mind, and make it hard for him as he discusses the purely mathematical nature of the natural numbers. I think the idea of “an actually completed infinite past succession of finite causal steps” is the idea that he is really arguing about, and I think that idea is two different inter-mingled ideas, one about pure numbers and one about causally-connected events.

  9. KF edges closer to an epiphany:

    I think the best we can see so far is that ascending count from 0 or a finite neighbourhood of that in an attempt to attain w etc is inherently futile…

    In other words, the natural numbers, which are constructed by “ascending count from 0”, are all finite. ω isn’t a natural number.

    But confusion remains:

    …w is in effect an emergent value once endlessness of succession is in play.

    The set of natural numbers has a transfinite cardinality, but contains no transfinite numbers, “emergent” or otherwise.

    My concern remains, that N — which contains the counting numbers as extended in order endlessly — as a set has transfinite cardinality, and the weight of that should not in effect be left to a three dot ellipsis.

    Would he be happier if we added a fourth dot?

    I am even uncomfortable with the argument If k is finite and k + 1 is its successor, then any pair k, k + 1 will be finite, and 1 is finite so all numbers in succession thereafter are finite. The problem being, the matter in view to discuss successions is exactly dependent on finite succession when the ellipsis points to endlessness. I think we are here close to begging a question or two, uncomfortably close for me.

    When your intuition clashes with a well-known mathematical fact, and you cannot find a flaw in the reasoning behind that well-known fact, then it is time to abandon your intuition.

    How do I put it.

    Something like, a finite increment to a finite is a finite indeed, however we are arguing to the endless and in a context where an endlessness is transfinite. So, a transfinite number of finites where the very numbers themselves are what is in view to attain to cardinalities, sits uneasily for me.

    It isn’t the numbers that “attain to cardinalities”. Cardinality is concerned with the size of a set, not the size of its elements.

    And finally, a welcome bit of humility:

    Perhaps, I am being overly scrupulous or needlessly concerned but it will not shake.

  10. keiths: In other words, the natural numbers, which are constructed by “ascending count from 0”, are all finite. ω isn’t a natural number.

    You’re just now figuring this out? And where did you post your proof?

  11. Mung: You’re just now figuring this out? And where did you post your proof?

    The finite cardinal numbers are by definition the natural numbers. That’s not a proposition to be proved true or false. Keith has tried to construct persuasive quasi-arguments for the reasonableness of the definition. I disagree with that approach, because I think it contributes to the sound and the fury. But I don’t care enough about the matter to get into a fight about it.

  12. Keith has tried to construct persuasive quasi-arguments for the reasonableness of the definition.

    Tom,

    For the life of me, I can’t figure out why you regard my argument as a “quasi-argument”.

    As far as I can tell, you don’t disagree with the premises, the reasoning, or the conclusion. So what makes it a “quasi-argument”?

  13. KF is still off in the weeds. Aleta and daveS are trying to get him back on track, but he’s really, really confused.

    What’s funny is that he’s becoming self-conscious about it. He can feel the eyes of the audience upon him:

    F/N: I see we are in the high hit for the past month club here now.

    Any new participants, please understand this is an exploration, live, messy, incomplete, patently vulnerable.

    Relax, KF. You’ve already blown it in the eyes of the audience. What’s left is for you to figure out why the mathematical community is right and you are wrong.

  14. Highlights from the continuing KF trainwreck.

    It’s like pulling teeth. Aleta and daveS are trying to get KF to state, unambiguously, whether he finally agrees that every natural number is finite.

    Meanwhile, KF is inventing a new, entirely redundant symbol that he calls the EoE, or “ellipsis of endlessness”:

    Let me add to the set of symbols, to symbolise endless continuation with: . . . EoE . . .

    And:

    Endlessness within the counting succession is involved inside the set, via the ellipsis again:

    { 0, 1, 2 . . . EoE . . . }

    And, we are here dealing with the very set used to count.

    And:

    I am coming to a point of comfort by introducing an explicit ellipsis of endlessness and holding that it is INSIDE the definition of the set of naturals.

    daveS tries to get a straight answer:

    Can you give a good reason for not going ahead and flatly stating that all natural numbers are finite, period? Is it possible that we will at some point discover an exception?

    KF provides this non-answer:

    DS, look at the set itself: the counting numbers, which is by definition endless and would contain all numbers ordered from 0, 1, 2 . . . EoE . . . I am looking instead at a definition, N is the least inductive set (set of successors to 0, 1 etc in effect). KF

    Aleta:

    The … already means endless continuation. What do you gain by adding another symbol EoE to also mean endless continuation? It all means the same thing – there is always another finite integer.

    Aleta:

    N = {1, 2, 3, …}

    N does not equal {1, 2, 3, … w, …}

    In the interest of clarity: Do you agree that N does not contain w? Yes or No?

    Another non-answer from KF:

    First the ellipsis is ambiguous.

    Second w is successor to the naturals, per the usual understanding.

    The issue is on the subject, endlessness is within the naturals.

    Aleta writes…

    There is nothing ambiguous about the …. All it means it that for any k, k + 1 is also a natural number. That is all it means.

    …and then repeats the question:

    In the interest of clarity: Do you agree that N does not contain w? Yes or No?

    KF gives a non-answer concerning his non-answer:

    Aleta, I already gave the answer: w is successor to the naturals, which has a meaning in ordinal context that reflects taking in what is before then capping it. Elsewhere, I spoke of the issue that the endlessness is already in the naturals, expressed in the EoE. But that’s the problem/point right there, the endlessness is in the naturals. KF

    Aleta:

    I don’t get it, kf. I don’t see the problem. I don’t see that you have answered my question, either.

    Is w in the set of natural numbers or not? Saying “w is the successor to the naturals” doesn’t answer the question – is it a successor in the naturals, after the ellipsis, or is it a successor to the naturals – beyond but not in the naturals.

    Which is it? Why won’t you/can’t you say?

  15. KF finally makes it clear: he agrees that ω is not a natural number.

    Aleta asks:

    Then what is wrong with saying every natural number is finite?

    No response yet from KF.

  17. The wortgeschwurbel is metastasizing.

    Aleta provides an apt diagnosis:

    You keep using phrases which betray a deep misunderstanding – you can’t “reach” endlessness, and there is no “extreme zone” within the natural numbers. It’s “inherently futile” to think about “reaching the extreme zone” because the idea itself is erroneous.

    And:

    But when you write that you are “fishing for the inherent, apt description of the essential meaning of a phenomenon, i.e. I am feeling for the innate nature of these number …”, I think you are talking about a psychological issue, not a mathematical one. The mathematical issue is clear, as stated (mostly) in the non-bolded part above: the psychological issue – what I referred to in an early post as the mystery of the infinite, is what you are grappling with and fishing for. [Emphasis added]

    Meanwhile, Mung stumbles in and draws an inapt analogy with divine simplicity:

    It would appear that the infinite must be simple and cannot be composed of parts or anything that can be counted.

  18. keiths: Highlights from the continuing KF trainwreck.

    Doesn’t it just burn you up that your lack of self-control got you banned from UD (again) and that you can’t post there and show kf just how wrong he is and how right you are? That must be seriously irritating. Here you are with all the right answers, yet … EoE …

  19. Mung:

    It would appear that the infinite must be simple and cannot be composed of parts or anything that can be counted.

    Here’s a learning opportunity for you, Mung. It’s math, but don’t be intimidated. The concept is quite simple:

    Countable set

    Pucker up, Sunshine.

  20. This is hilarious:

    zxc
    _____
    Aleta & DS:

    Aleta, 196: >>back to 192: if all known transfinite numbers (cardinal and ordinal) are not actually members of N, then does it not follow that all numbers in N are finite? What alternaives are there? Either all natural numbers are finite, or there are transfinite numbers in N. If we have eliminated the second possibility, the first statement is true.

    “Jarring” concerns about an endless number of finite numbers may express a natural (which is sort of a double entendre, I guess) sense of the mystery of infinity, one which Cantor et al plumbed for us. But unless you can actually establish something mathematical (which my post at 192 highlights), your sense of being jarred is waiting for you to come to terms with that mystery.>>

    I long ago learned to respect my sense of cognitive dissonance, of logical incongruity. Remember, as primarily a physicist I had to swallow the transition from the classical to quantum and relativity.

    In this case, it is not that logical chaining from initial values and an assertion C-n for case n that then entails C-n+1 that is a problem. Yes, it chains on, unlimited. But it cannot exhaust, it is stepwise and subject to the ellipsis of endlessness.

    With what is on the table here being the very counting set that serves as first yardstick of endlessness in the first degree, Aleph null.

    The claims being made come too close to ending the endless.

    I elaborate a bit more in . . .

    DS, 197: >>It seems to me if you reject the inductive proof that all natural numbers are finite, then you reject the validity of mathematical induction in general. And since N itself is generally defined inductively/recursively, I don’t see how you make sense of the existence of N as a set itself . . . . Every natural number is finite, but no one is saying that N is in any way finite.>>

    I took a careful look at what the usual proof by induction I have used since 6th form days actually establishes.

    It finds a case 0 or 1, then establishes — on a framework for case-n (C-n) that C-k => C-k+1. This hangs a logical chain on the first fact and proceeds to do so STEPWISE. This establishes unlimited extension indeed but again we cannot end the endless through a chaining of successive finite steps. We can only append a pointing ellipsis of endlessness.

    The issue is thus endlessness as the heart of the transfinite nature of the successive counting numbers.

    Which can be pointed to but not exhausted.

    Now, look at the set {0, 1, 2, . . . EoE . . . }

    This is, endless. It is also the sequence of counting sets per the von Neumann type assignment

    {} –> 0, {0} –> 1, {0,1} –> 2 . . . EoE . . .

    such that any value such as 5 emerges as the order type of its predecessors collected:

    {0,1,2,3,4} –> 5

    This means that the counting, 1:1 match etc properties of the successor and its cardinality are established by what is on the LHS of the assignment. It can be shown that this example is WLOG.

    Continue to endlessness and w emerges as supremum:

    {0, 1, 2, . . . EoE . . . } –> w

    The endlessness is in the LHS, the set defined as the natural numbers.

    Endlessness is the heart of being transfinite and we need to face it. Where, our process of counting here must be endless on the LHS.

    By definition, ordinary mathematical induction points to but cannot exhaust endlessness.

    The claim that all naturals are finite hangs from finitude of the first and an endless succession that can only be pointed to.

    But in looking at k and k+1 for instance, the endless chain begins again any number of times and points to onward endlessness that can be put in 1:1 correspondence with the set starting from 0, 1 etc. The set is transfinite, on the LHS.

    Does finitude then chain to and exhaust the members?

    By mere force of instant logical extension to the whole?

    Thus entailing, all counting sets in the succession are finite, never mind the endless chain of such counting sets that scale ever upwards in succession, a process which must — at least, it seems it must — in the far zone go on to endlessness in the sets?

    [Or, the successive count has not become transfinite, so far as I can see to date. How do we get a transfinite collection of ever-mounting distinct counting sets where each and every one is finite, including at the far zone of endlessness? To my mind so far, I can see that we have a stepwise incremental algorithm of chaining, that is in an endless loop that it cannot exhaust. It spits out a step per loop, and points on to endlessness but is in itself inevitably finite, it can only step forward a finite, countable number of times. It points to potential infinity, it does not actually exhaust it. It also establishes that at each step we have the next successive number as label for the set that collects sets thus far, and only if the counting sets in the far zone scale to the transfinite within their internal membership lists can we have an overall set that is endless and transfinite. In short, there is nesting of the emerging transfinite character of the whole. At least, that is what I am seeing.]

    The problem is, ending the endless.

    Finitude implies ending, chaining in steps is inherently finite but points to the endless succession. And w is the limit ordinal that summarises that endlessness and is successor to the endless chain.

    There is no finite k such that k + 1 = w.

    Instead, I suggest a more modest interpretation: induction shows an unlimited, reliable logical chain that will work for any particular n you please, which by being specified becomes inevitably finite.

    But it cannot exhaust the endless.

    Just as w cannot simply succeed any particular k.

    Just as, the successive counting set looping algorithm can only ever stepwise attain to the finite and in so doing ever extends the scale of the sets, but points to the endless succession.

    The claim that finitude spreads through the whole chain by succession seems to me to suggest a claim to exhaust, or end the endless by an algorithm that can only ever be actually finite.

    For the moment, I think it safer to say, that finitude propagates down the chain, without limit as our algorithm loop counter increments and labels progress since case 0 so far. But that is very different from claiming it can exhaust the set as a whole.

    If it did so, it would indeed confer finitude upon all members, but to do that it has to end the endless, which by its very nature it cannot, it ever invites another clock-tick and step.

    Further to this, we are counting using this particular set [here, I envision the loop counter and a printer spitting out the assignment endlessly on Mr Turing’s paper strip or a modern update thereto], and a finite number has the character, it is ended and surpassed even. If every counting number in succession is finite, how then can we consistently claim the set as a whole is transfinite, endless?

    So, I put my qualms on the table, I asterisk the claim.

    I do not dismiss it as absurd — but I am concerned as to its coherence given the inherent issue that the counting numbers in succession are inherently endless.

    I do not know if we are able to bridge to mutual understanding as to why my concern. But maybe the idealised processor chugging away endlessly at a counting and printing loop algorithm but utterly unable to exhaust the set as a whole, stamping the final member as finite — whoops there goes another clock tick — can help.

    Where, I am confident that such is a reasonable mathematical exercise of induction from the particular to a succession rule to the chaining to unlimited extent. Howbeit, the chaining inherently cannot exhaust the whole. And that whole must be transfinite and to do so it looks a lot like the far zone we cannot reach has to have a fractal-like, nested copy of the whole in it.

    And, w is not somehow of distinct quality and characteristics from what lies on the LHS of the assignment of identity and labelling as ordinal successor that sums up and holds the cardinality of what it now tags:

    {0, 1, 2, . . . EoE . . . } –> w

    DS, 201: >>do you accept that the standard inductive proof given by Tao does indeed show that all natural numbers are finite? Your phrasing above suggests not.>>

    Please see the thought exercise of a counting loop algorithm implementing machine to see how I think unlimited reliable succession is not equal to ending the endless.

    Notice the need to embed a copy of the whole to end the process.

    Potential, but not actualised.

    Pointing to, but not completing, in a context where completing is the requisite.

    In short, I fear we are overclaiming what induction per se delivers.

    Aleta, 203: >>The definition of the naturals as being created by the statement that every natural number k has a successor k + 1 is an axiom, but it is not an arbitrary axiom – it is a universally accepted, I think, part of the foundational definition of natural numbers. But when you write that you are “fishing for the inherent, apt description of the essential meaning of a phenomenon, i.e. I am feeling for the innate nature of these number …”, I think you are talking about a psychological issue, not a mathematical one.>>

    Not psychological, but conceptual, philosophical, logical and mathematical by virtue of that discipline being the logical, abstract study of structure and quantity.

    Coherence is the chief test and guardian in such an exercise, but in this case there must also be congruence with the natural experiential root of whole counting numbers.

    Here, the issue seems to in part turn on unlimited though at any achieved stage finite succession of incrementing counting sets and ending the endless.

    Which by extension extends to causal succession from a claimed unlimited extension of antecedents [algorithmic process bridges readily to machine implementation] and to attempted decrementing from the transfinite just as much as incrementing to attempt attainment of the transfinite.

    Aleta, 209: >> we having been using “natural number” in the mathematical sense, not in some informal sense as including all sorts of different types of other numbers. As Dave said, let’s keep our eye on the ball: N = {1, 2, 3, …} is the topic.>>

    The mathematical sense of natural numbers builds on and must not violate what has been established through human experience of the phenomenon of matched counting sets and unmatched ones leading to counting, the concept of counting numbers that label particular standard sets — which are then made abstract — and so forth.

    The formal builds on and systematises the informal.

    It then leads to extensions: rationals, reals, complex numbers etc. So, onward links are relevant. Especially given the context set in the OP.

    DS, 211: >>How can mirrors be real if our eyes aren’t real?>>

    Mirrors and eyes are both real.

    And my first look into an abstract, virtual half-infinite in principle world was when I looked in a mirror and learned that images can be physically located behind it by parallax.

    I recall once setting as a 6th form exercise, doing the pins and mirrors expt they did in 4th form, then challenging to ponder the virtual half-universe.

    Next step, set up two mirrors in parallel: endless, receding mutual reflections, in an endless in principle loop of light.

    In praxis, fading off as the reflections are not perfect.

    Applicable to laser cavities and creation of coherent radiation — and the place of half silvered mirrors leading to spiking thence q-switching by various means to get controlled, much stronger pulses.

    Mathematical, idealised extension: endless loops pointing to the infinite.

    I recall, looking into the recession as the two little mirror strips were put in parallel was quite a shock that opened up the vista of the infinite. Even, though it could not actually attain it physically.

    A crucial distinction between physics and mathematics.

    Which brings us back to our ever looping incrementing algorithm, which logically is unlimited but is in principle strictly forbidden from claiming to have ended the endless.

    Aleta, 212: >>Why is this argument not valid?>>

    Please see the algorithm loop based illustration of the distinction between unlimited extendability and ending the endless. I am concerned that we may have gone a bridge too far.

    KF

    PS: White screen of captcha death, I have to go back and do the copy, cut out, nonsense phrase and insert real post on edit trick.

  21. All that verbiage, yet zero progress:

    How do we get a transfinite collection of ever-mounting distinct counting sets where each and every one is finite, including at the far zone of endlessness?

    There is no “far zone of endlessness”, KF.

    The entire sequence is endless. Endlessness is a property of the whole, not a property of some “far zone”.

    The entire sequence is a “zone of endlessness”, including right at the beginning. There is no end at 0, no end at 1, no end anywhere else in the sequence. Yet that endlessness, which extends across the entire sequence, does not make 0 transfinite, and it does not make 1 transfinite. Why then should it magically bestow transfinitude on the natural numbers in some “far zone”?

    There is no magic zone where adding one to a finite number somehow produces a transfinite number. Every natural number is finite, but there are infinitely many of them.

  23. dazz,

    Somewhat frighteningly, he has a M.Sc. in physics from the University of the West Indies.

    Your Imperial Majesty.

  25. DNA_Jock:
    dazz,

    Somewhat frighteningly, he has a M.Sc. in physics from the University of the West Indies.

    Your Imperial Majesty.

    Just call me Palpy :p

    But seriously, if he doesn’t understand something as simple as the mathematical concept of infinity it doesn’t matter what credentials he has

  26. petrushka,

    Eternity is a long time, particularly toward the end.

    It’s best to hit the restroom first.

  27. dazz,

    But seriously, if he doesn’t understand something as simple as the mathematical concept of infinity it doesn’t matter what credentials he has

    There are some shocking disconnects between credentials and abilities among the IDers. My favorite example is that of Rob Sheldon, PhD in physics from the University of Maryland, who wrote:

    The derivative of a function has different properties than the function, which was a mistake that I made in the first paper I wrote after getting my PhD. (I had assumed that if f(0)–>0, then df/dx(0)–>0. It took the referee a year to straighten me out. The only time I can say that peer review worked.)

    That’s just… amazing.

  28. keiths,

    You have to be kidding me, and it took him a year to realize such a basic mistake? Didn’t it occur to him to maybe try f(x) = x or something?

    and he has the audacity to dismiss peer review altogether. What a fucktard

  29. Petrushka wrote, “Eternity is a long time, particularly toward the end.”

    That’s excellent.

    And keiths says, “It’s best to hit the restroom first.”

    I believe that last restroom is at the restaurant at the end of the universe.

    Maybe, instead of moving into the zone of endlessness, time just rocks back and forth: k, k + 1, k, k + 1, …

  30. Aleta,

    You forgot the “EoE”. 🙂

    KF is quite the problem student. I applaud you and daveS for your patient pedagogy.

    KF writes:

    And I find myself further uncomfortable with the proposition that on an ordinary inductive proof it is shown that an endless set that counts up has in it only finite members. That runs very close to an outright statement of ending the endless.

    Ironically, it is KF who is effectively trying to “end the endless”.

    His faulty intuition seems to run something like this:

    1. If I take 7 natural numbers starting from 1, the sequence ends at 7.
    2. If I take 73 natural numbers starting from 1, the sequence ends at 73.
    3. If I take infinitely many natural numbers starting from 1, the sequence ends at infinity, or in the “transfinite zone”, or in the “zone of endlessness”.

    The error couldn’t be more obvious. In #3, the sequence doesn’t end. If it did, then the set of natural numbers wouldn’t be infinite. To assume that there is an ending point where the final number is infinite is to contradict oneself. KF is effectively assuming an end to endlessness.

    He papers over this a bit by referring not to an endpoint, but rather to a “transfinite zone” or a “zone of endlessness”, but the error is the same. He is assuming that there is a point within the sequence where infinity is attained and completed — an end to the endlessness. It’s just that he starts counting up again from that “point” and calls the new sequence the “zone of endlessness”.

    There is no point within the sequence where infinity is attained. There is no last number, no “transfinite zone”, no “zone of endlessness”. What makes the natural numbers endless is that there is no end. For every finite n, there is a finite n+1. Endlessly.

  31. KF:

    Aleta, the issue is perilous closeness to a contradiction between everybody finite — including the “far end” of the succession of counting sets, and the set is transfinite as the EoE shows. This is a way to try to put it.

    Aleta:

    There is no “far end”. I give up! (But I’ve said that before, and then have come back, so we’ll see.)

    Aleta,

    Perhaps you could point out that when KF speaks of a “far end” to an endless sequence, he is speaking nonsense.

    It’s amazing to me, and fascinating, that he can’t see it.

  32. Meanwhile, Mung is confused as usual:

    George Gamow writes:

    The sequence of numbers (including the infinite ones!) now runs:

    1. 2. 3. 4. 5. …… ℵ1 ℵ2 ℵ3 ……

    and we say “there are ℵ1 points on a line” or “there are ℵ2 different curves” …

    Is this a finite sequence?

  33. Thanks, keith. I’m sticking with the discussion with kf for several reasons. One, I’m interested in the math. Second, I’m a math teacher, and I like trying to explain things to people. One of the things I learned from many years of teaching is that trying to find out what the student misunderstands is often a key to getting them able to learn what they are not getting.

    In addition, I’ve had many “conversations” at UD on many subjects which, I’m sure you know, get caught in a morass of dedicated unwillingness to try to even understand a different point of view. And, more particularly, kf is renowned for some of the discussion habits that we are seeing in his discussion. But, and this is what is keeping me around, is that is this case it’s math, and much more clearcut (one would think) then theology or evolution or metaphysics, etc.

    So the opportunity is here, as an exercise in psychology, pegagogy, and techniques of constructive dialog, to see if kf can move away from his habitual style of discussion. Can he separate issues, and can he possibly say, “Oh, I see what you are saying”, and change his mind from his original position? I think we are seeing many of the same problems, but there has been some differences in tone and a willingness to couch his position as concerns as opposed to pretty inviolable assertions.

    So I keep sticking around despite my claim that I’m going to quit.

    And you’ll notice I steadfastly refuse to start talking about the reals, infinitesimals, etc. That is a not a place that I would be willing to sink my time.

    And it’s been good to have the Skeptical Zone as a place to have some offsite feedback and discussion.

    Thanks,
    “Aleta”

  34. Well, this isn’t my speciality, but I thought everyone here has argued well against KF.

    That said, if I can add my two uniformed cents.

    It seems to me that finite is formally defined to mean bounded. That is, if there exists a natural number n, it is bounded from above by (n+1). That’s all it really means to say than any given n is finite.

    The fact that the set of naturals is infinite then makes it possible to make every n finite. That is for every n, there exists a bound from above, namely n+1.

    To say we have a natural number n equal to infinity is to say there is no n+1 that bounds it from above. That makes little sense as far as I can tell.

    At issue is the meaning of finite is not clearly defined. If we say finite means bound from above by some other number, then it is easy to show all naturals n are bounded from above by some other number, namely, n+1.

    The fact this can be done for every n implies the set of naturals is infinite.

    At least that’s how I resolve the paradox. The problem is one of an ambiguous definition of finite.

    PS
    One of my 3 undergrads was in math, but I wouldn’t ever call myself a mathematician. I’m not that good.

  35. Sal,

    It seems to me that finite is formally defined to mean bounded. That is, if there exists a natural number n, it is bounded from above by (n+1). That’s all it really means to say than any given n is finite.

    To establish that some particular n is finite, it isn’t enough to show that n is bounded by n+1, unless you also show that n+1 is finite.

    You need to do that from below, not from above.

  36. Sal,

    The problem is one of an ambiguous definition of finite.

    No, the standard definitions are unambiguous.

    And interestingly, the exact definition doesn’t matter, as long as your interlocutor accepts this premise:

    If n is finite, then n+1 is also finite.

    I think KF would be hard pressed to dispute that.

    The same inductive step from n to n+1 that is used to construct the naturals also establishes their finitude, if you accept that premise.

    KF’s position is bizarre and contradictory: he accepts the construction but rejects the induction, when both can be expressed using the same logic.

    The only consistent way for him to maintain his position is to deny the premise above — which would be absurd.

  37. There cannot be an infinite number, because each increase in the number increases the amount of energy required for it to increase again. That is, if we had an infinite number, then there would be an infinite amount of energy, and we would be able to travel at the speed of light, violating relativity, which Einstein proved mathematically to be true.

  38. Fortunately, the astute and articulate Mapou has shown us that relativity is false:

    You can jump up and down and foam at the mouth all you want but relativists are all a bunch of crackpots. Einstein had no clue as to the cause of gravity and the current crop of relativists are even more clueless. Spacetime, as so many thinkers have noted, is a block universe in which nothing happens. You call that physics? As Popper would say, relativity is an exercise in mythical metaphysics.

    Heck, you ignorant jackasses do not even know what causes something as fundamental as inertial motion. The deep, in-your-face ignorance of relativists and their insufferable pomposity are now legendary. All this nonsense about binary stars is a further sign of your abject ignorance and propensity for crackpottery.

    Take a hint from hamsters. When they get tired of running around all day in the wheel, they go into a corner and fall asleep. Just give a rest, crackpot.

    It must chafe his ass to see yet another confirmation of GR in the news.

  40. Tom English:
    There cannot be an infinite number, because each increase in the number increases the amount of energy required for it to increase again. That is, if we had an infinite number, then there would be an infinite amount of energy, and we would be able to travel at the speed of light, violating relativity, which Einstein proved mathematically to be true.

    I’m not sure I get this. An infinite number of what would necessarily imply infinite energy? Are you talking about space-time (vacuum energy)?

  43. https://en.wikipedia.org/wiki/Finite_set#Necessary_and_sufficient_conditions_for_finiteness

    S is a finite set. That is, S can be placed into a one-to-one correspondence with the set of those natural numbers less than some specific natural number.

    Let S be the set of natural numbers less than or equal to a a natural number n. The set of those natural numbers can be placed into a one-to-one correspondence with the set of those natural numbers less than the natural number n+1. Hence n is finite.

    We show this for 1, and then 1+1, by induction we show this for all natural numbers, hence all natural numbers are finite, and this is possible because the set of natural numbers is infinite. Paradox resolved.

    Btw, I care less and less about what KF thinks, engaging him is an exercise in practising how to dealing with obfuscation.

  44. If n is finite, then n+1 is also finite.

    Agreed, we start with n = 1.

    We show n+1 is finite because it is bounded from above by ( ( n+1) + 1).

    Rinse and repeat.

  46. Sal says, “Btw, I care less and less about what KF thinks, engaging him is an exercise in practising how to dealing with obfuscation.”

    Well I’ll be darned – I agree with Sal (and there’s been many a time in the past when I disagreed with him.)

  49. stcordova: At least that’s how I resolve the paradox. The problem is one of an ambiguous definition of finite.

    Meanwhile over at UD they are asking “what paradox.”

  50. keiths: It must chafe his ass to see yet another confirmation of GR in the news.

    As for me, I am waiting for Richardthughes to post a link to a youtube video where Seth Lloyd explains what gravity is.

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