Endless and Infinite

Philosopher Wes Morriston and I have coauthored a paper on the Kalam, and it has been accepted publication in the journal Philosophical Quarterly. Once it is actually available on their page access will probably be limited, unless you have an institutional subscription. However, for now you can download it (for free) via this link:

Endless and Infinite

I will probably have to take this page down within a few weeks, so if you want to read the paper, then download it now. Also you can  message me if you would like access to it.

Thanks,

Alex

38 thoughts on “Endless and Infinite”

  1. Great paper, maybe even a coup de grâce for the KCA.

    Loke’s arguing from HH’s supposed metaphysical impossibility seems like a really odd move since on standard theology God creates all of reality outside himself ex nihilo however he likes. He doesn’t play by the rules of metaphysical possibilia, he sets those very rules up, or as Sean Carroll put it “he can do what he wants”. HH is clearly logically coherent since the math works out. Maybe in the future not even the law of non-contradiction holds in the face of paraconsistent theology, that would be a sight to see.

    I left a comment on your The ‘God can do anything’ objection post a few weeks back, if you have time I’d appreciate it if you took a look at it.

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      1. It’s very essay to sign up to Patreon. If you don’t want to, that’s fine. I prefer to not communicate via my personal email address if that’s ok. Thanks.

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  2. Hey Alex

    Great paper, a very succinct response. I was particularly interested in the responses to Loke’s objection.

    I think your responses are successful. But in addition, I think it’s interesting that the ‘in one go’ scenario also seems to commit Loke to an ad hoc limit on the number of rooms God could instantiate in one go that would surely be odd to say the least.

    It seems that there’s no prevention to God creating one more room for any finite number of rooms he creates in a single act, because the consequence of creating all of the possible rooms in one act is absurd. Let’s say that on the number line that we put our potential rooms into correspondence with all along it, we sum the rooms from 0 and halt the sum at _n_ and say only the rooms that correspond with those numbers can be created by God in a single act. Have we reached the limit of what God can possibly actualise in one go? It seems not. And why have we stopped the sum here? It seems arbitrary, and wrong to suggest that n +1 can’t be part of the sum, because some maximal infinite sum implies creating a HH. Isn’t that true if the sum is halted at only two 2? The consequence lurks there too. Yet God could clearly actualise 3 rooms in one go. Just because it could be true that actualising some infinite sum of rooms is impossible (due to it creating a HH) doesn’t seem to imply that actualising the sum up _n_ +1 is impossible. I get that the later than restriction at first appears ‘spooky’ in the temporal sequence, and that the restriction in one go may not be temporal but still seems weirdly arbitrary- as if the sum of a pile of sand couldn’t be added to by some additional grain for fear that if an infinity of grains were added, it would be an absurdity. That consequence says nothing of the ability to add any one grain of a potential infinity to the sum of actual grains, yet that consequence is surely what Loke would say would limit God’s power to create at a given point the limit was reached. Yet no such limit he or anyone could offer is the limit of logical possibility such that ‘one more’ implies an infinity and an absurdity, as we don’t ever lapse from finitude to infinity by adding one more. It seems something akin to a sorities paradox of sorts arises here regarding God’s limits, yet he clearly msut hae them, or else he would be able to actualise without limit all potential rooms in one go.

    It strikes me as problematic if the Kalam proponent bites the bullet and accepts an upper limit of this sort, akin to the sort of objection Loke raised- that it appears ad hoc to stop at a time because of some future consequence, it seems ad hoc to stop at some sum, just because some maximal sum would be absurd. Maybe he would see things differently, but this strikes me as odd to say the least.

    Anyway, good luck with the paper!

    Liked by 1 person

    1. Hi Fox,

      I think the way to understand the restriction on omnipotence is spelled out in the scope distinction, between ‘for all x, it’s possible that…’ and ‘it’s possible that, for all x…’

      What that does is allow God to make a hotel with any finite number of rooms, but not one with a room for each natural number. So the limit you are looking for is aleph0, but it is a limit that is on the other side of what he can create. Think about the real number line. Maybe I can occupy any point on that line strictly less than 1. So I could occupy the point 0.5, or 0.99, or 0.999999, etc. There is no maxim point I can occupy, but there is a minimum point that I cannot occupy, namely exactly 1. Here, there is no maximum finite number of rooms that God can make, but there is a minimum transfinite number that God cannot make, aleph0. So there is a limit, but it is an external limit (one he can approach but not reach).

      And I’m not sure that’s terribly arbitrary. After all, if you think HHs are metaphysically impossible, and you think God can’t do anything metaphysically impossible, then it just follows as a logical consequence.

      I agree though that in the temporal case it seems like he would actually have to stop somewhere; there would have to be a time where he doesn’t make a hotel room, and that begins to look arbitrary (why that point and not another?, etc).

      But that is what the very final argument is supposed to address. After all, if time has no beginning, it doesn’t matter if he stops adding rooms at some point in the past. He would have already made a HH anyway (because each moment already has an infinite past). In fact, given the finitist restrictions on how many hotel rooms God can make at any one time, it follows that if there is a HH now, there must have always been one, regardless of what God did or didn’t do in the past. Basically, given his finitist restriction, he can’t actually *make* a HH. The best he could do is just add rooms to a HH that always already existed. And then he may as well not bother, because his efforts make no difference to the cardinality of the set of rooms.

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      1. Hi Alex- thanks for the reply, I hadn’t properly appreciated the distinction in scope. The clarification is helpful, but there’s still one thing I’m unclear on, and apologies for the ensuing post!

        So I get the idea that even if there is an ever increasing spectrum of possibly realisable rooms, given that this simply lacks a point that correlates with an infinite sum of rooms, (because that is not a possibly actualised set of rooms) it doesn’t follow that God has to arbitrarily stop at some point on that spectrum re his power to actualise possible rooms because such a point is not possibly part of that spectrum which he’d have to stop short of. Nor does it follow that the spectrum indicates a maximum point God can inhabit, so to speak, such that the spectrum outruns God’s powers.

        I accept that if you agree with Loke re a HH, then God can’t actualise a HH, so that’s something that’s not on the spectrum of logical possibility. My confusion in part arose about that spectrum of possibility- It seemed to me that if it had a limit it would be arbitrary, and if it lacked a limit, it seemed like it suggested an infinity to me.

        My thinking (incorrectly!) was if we went along that spectrum of possibility, and ticked off the sets of rooms that are potentially actualised, we wouldn’t ever stop (which is true) we could make a one-to-one correspondence between the number of rooms God could possibly actualise and a number on the number line. I had incorrectly assumed that this suggests that there would be a set of rooms with infinite cardinality at some point. That’s a wrong conclusion to draw, because no number on the line is ‘infinity’ such that an infinite set could be in a one-to-one correspondence with it.

        I guess because the spectrum never lapses from a finite set to an infinite set, it doesn’t worry Loke- it’s not as if at some point we lapse into a set of possible rooms that’s infinite, so there’s no worry there about some lurking limit that sets up the worry I suggested above. Loke can happily say that there’s no maximum limit of the possible finite sets of rooms God could create and that God could not create a set of rooms that was infinite in cardinality. That’s a limit not arrived at arbitrarily by halting the spectrum at some point, or a limit in God’s power, but a limit to logical possibility.

        Ok, so what I’m still a little unclear about is this- isn’t there nonetheless potentially an infinite set of ‘finite sets’ of rooms? That is an infinite set of finite hotels? Couldn’t we put a hotel with one room in a correspondence with the number 1, a htel with two rooms in a correspondece with 2, and so on? Now it’s true no individual hotel has the cardinality that’s infinite, as there’s no infinite number to correspond with it- we only ever have finite sets of rooms in any given correspond, at any point on the number line. But it still seems to me that if there’s no limit on the finitude of each set of room that increases all along the number line, then there’s an infinite set of finite sets of rooms when we take into account the entire number line.

        My next question would be, could God actualise all those possible finite sets in one go? It wouldn’t create a HH- but would create an infinity of rooms (so I guess equivalent to a HH in its logical impossibility). Loke’s limit would again kick in I guess, and disbar this possibility as a limit of logical possibility- but at that point isn’t he either committed to halting on the spectrum of possibility at some point as I originally suggested that’s arbitrary, or accepting a limit of omnipotence? Does the worry I had not just arise again because even though there’s no single set of rooms with a cardinality that’s infinite, there’s a set of ‘possibly actualised finite sets of rooms’ and the cardinality of that is surely infinite?

        It still strikes me that given there’s a one-to-one correspondence with the sets of finite rooms and each number on the number line, the spectrum is still infinite in that sense- though no number is ‘infinity’ and no individual set has an infinite cardinality, there is still an infinity of numbers and an infinity of finite sets. And God can actualise more than one hotel room at the same time. Would Loke say that set of possibly actualised finite sets of rooms by God in one go is finite or infinite? I assume not infinite. In which case there is presumably a limit lurking that either halts God’s power to actualise each member of this set in one go (indicating a worry for omnipotence) or the set of possible sets of rooms actualisable in one is finite- we have to stop at some point, at which point we can ask why it’s not possible to actualise the next finite set. I am not clear how Loke would dismiss the worry at this point, at least not in the same way, given the fact that he thinks God lacks a limit in the number of finite rooms he could create, (as long as the number of rooms is not infinite).

        Anyway, I am sure this just remains muddled in my head. It’s tangential to the mainstay of the argument, which as you suggested is more problematic for Loke.

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      2. Hi Fox. Yeah, I think your idea can be stated in a way that is clearer, and when we do that I don’t think it’s substantively different.

        So the idea is that we change focus and think about the finite sets of rooms that god could make and ask about the extent of his powers there. I think we cover this with the ‘infinite ave’ bit of the paper, but let me try to say it here.

        Consider this scope dictinction:

        A) ∀n(god makes n-many finite hotels)

        B) ∀n(god makes n-many finite hotels)

        The second is what is ruled out by the restriction on omnipotence. The former is allowed. So when you set the question up in terms of how many finite sets could he make, that seems answerable in the same way (we just replace talking about making hotel rooms for talking about making hotels).

        Doesn’t that address your question?

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  3. Hi Alex and perhabs Wes
    First of all its a good Paper (at least the part I understood😄)
    But I think you went wrong in replying to craig
    You said that he is switching the tenses. And you are right, but thats my propblem.
    For example an infinite past is impossiple because:
    consider that 1 minute ago a minute (m1) just has passed away if we now consider an infinite past then this cant happen because there would have allways have been pased a minute before the passing of (m1) and onr before that and before that and so on…
    If this is true we cant get to said m1 passing away
    Im sorry if this is formulated poorly but Im from Germany and this subject is pretty hard to get from German thoughts into English😃
    I would be happy to hear your response since I think its quite likely that I did not understood your paper completly
    Greetings Nico

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    1. I find so many things in those posts that I disagree with, it’s hard to know where to start. MJ sent me them when he wrote them and the sheer volume put me off engaging, as I would be there forever. Maybe I will try to respond to the most substantial aspects of it at some point.

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      1. Hey Alex I’m not sure but was ypur reply to me asking if you will reply to the posts to which i put a link here.
        However if so yeah it would be very cool to read a written response and maybe if you have time a really short critique here in the comments
        Sincerly Max

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      2. Hi Max,

        It’s hard to write a succinct response to MJ. Firstly, he covers so much ground, and secondly so much of what he says seems wrong to me. For instance, he makes a big deal out of distinguishing between the events that will be and the events that are yet to be. But they sound like exact synonyms to me! He thinks the future is a potential infinite, but it’s the past that is ever increasing with infinity a limit it approaches but never arrives at. The future, in contrast, is just all those events that will be. If there is no end to time, we could enumerate all the events that will be by imagining someone counting and never stopping. Sure, they will only have counted to a finite number, but they will count an infinity of numbers. I just don’t understand how anyone can object to that. It’s a blind spot in people who have read too much Craig, in my opinion, here. They just can’t see the bleedingly obvious. They don’t exist, but they will do. They aren’t yet actual, but they will be. Etc etc. So I probably won’t go into the weeds with MJs posts, because I would be there forever.

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    1. Ok Alex lets look at your counting example I agree that he will count an infinite number of numbers (a potential)
      And he will allways have counted a finite number.
      But shouldn’t we look at the past as the things allready counted, (so a finite number) and the future as the things that will be counted (a potential infinite)
      If this is the case then the past would be finite.

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      1. If he never stops counting, then it is true (right now) that he will count an actually infinite amount of numbers. We can equate the amount of numbers Jones will count with proper subsets of itself. The amount of numbers Jones will count is equinumerous with the amount of even numbers he will count, etc. So I am not saying that the numbers he will count is potentially infinite. It’s not. It’s actually infinite.

        What’s potentially infinite is the amount of numbers he has already counted. That goes up as time passes, approaching infinity but never getting there.

        So you don’t have it quite right in your summary above.

        Liked by 1 person

      1. Ok Alex I think now I start to see what you are getting to.
        But still wouldnt the Past be equal to the number jones has allready counted, just with moments passed instead of.numbers counted?
        From your previous response:
        ,,…Sure, they will only have counted to a finite number, but they will count an infinity of numbers”
        Maybe I am missing something but at the moment this sounds to me just like an argument for a finitude of the past.
        Sincerly Max

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  4. Alex Thanks for your response.
    I will think about what you said (and your paper) and probably will write my thoughts in the next few days.
    Sincerly Max

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  5. Ok Alex I read your paper.
    But still I have some troubles with it, that you can maybe help me with.
    First one you argued that claiming that an infinite future cant be an acctual infinity, since it has (not yet) been actualizied is question begging, but why is that I dont understand how you came to this conclousion.
    My second problem is that think we will need an argument why we should use functions to describe events in the rel world. I’m happy to accept that we can use them but still I think we need an argument for this.
    And one final and maybe my biggest problem, is in the part with the counting example and I quote

    (((Consider next an ‘endless count’ scenario:
    Counter will begin counting one minute from now; one minute after that and after every other future act of counting, he will add one to his count.
    Again, we can think of this as a series of ‘pure potentialities’, each of which will be actualised. And again, we can ask for the number of potential ‘counting-events’ that will – at some time or other – be actualised. It is easily proved by mathematical induction that for every positive integer n, Counter will actualise an nth potential act of counting. Here, then, we have an infinity of potentialities – which (note well!) is not to be confused with a potential infinity.)))

    In the last sentence you said that this should not be confused with an potential infinite, but why not? For me this seems exactly like an potential infinite.
    If counter has allways counted the nth amount of numbers and he counts the real numbers, then counter will never stop counting and therefore never arrive at infinity. (If at least we dont assume right away that he reached infinity which for me would be question begging).
    But I think there is another Problem, as far as I can tell the connection between past and present is that the present is equal to the number of events that have allready been acctualized.
    If we now take counter as an example and place him in the infinite past he would need to count from -infinity to now (0) which (I think) is impossible. I know that this does describe the number of events that HAVE happend but for me this is exactly the way we should define the past (which makes it disanalogous to the future) ((and I dont see how this would change if we consider the growing block hypothesis or some other))
    Some closing remarks
    I’m a layman in Philosophy, I dont have a education in Philosophy and I am quite honestly not the biggest brain on earth😊.
    This said it is entirely possible that I conpletly missread your paper or missrepresented you in some way or another (which I hope i did not)
    Anyways I would love to hear your thoughts on my comment because it would help me to get my Head around the iasue of infinity.
    Hoping for a response and sincerly Max

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    1. “First one you argued that claiming that an infinite future cant be an acctual infinity, since it has (not yet) been actualizied is question begging, but why is that I dont understand how you came to this conclousion.”

      The problem is that its just an equivocation on two senses of the word ‘actual’ here. Craig’s problem is about the Cantorian property, where you can show that proper parts are equivalent to the whole (the natural numbers and the even numbers, etc). That’s what is behind the Hilbert’s Hotel, and infinite library weirdness. So take a bunch of not-yet-actualised-potential future events. How many of them are there? Well, if the future has no end to it then we could assign each one a unique natural number. And that shows that this collection of not-yet-actualised-potential future events has the Cantorian property. Ever since the 18th century that’s what an ‘actual infinite’ means. If you don’t want to call it that, then ok. But it has the property (whatever we call it) which is behind all of Craig’s examples here (the hotel, library, etc).

      “My second problem is that think we will need an argument why we should use functions to describe events in the rel world. I’m happy to accept that we can use them but still I think we need an argument for this.”

      All that’s going on here is that we are being precise about the concept of a potential infinity as Craig understands it. He doesn’t define it as such. He just describes it in different places. So we just put those descriptions together and came up with a way of understanding it. He says the potential infinite is i) growing, ii) always finite, iii) approaching infinity as a limit without ever reaching it. It just seems like the A-function captures these properties. If so, then it is Craig’s idea. It doesn’t feel like an uncharitable way of looking at it.

      “In the last sentence you said that this should not be confused with an potential infinite, but why not? For me this seems exactly like an potential infinite.”

      But this is just what I said in response to your first point here. Whether the events are ‘pure potentialities’ or whatever isn’t the point. What matters is if you can count them, and if so how many of them there are. And the answer is yes you can count them, because you could assign each one a unique natural number. So there are actually infinitely many of them (because they have Cantor’s property) even though each one is a ‘potentiality’. So you can have an actually infinite amount of things each of which is a pure potentiality. All that matters for our argument is that you can enumerate them, not what their ontological status is.

      “If counter has allways counted the nth amount of numbers and he counts the real numbers, then counter will never stop counting and therefore never arrive at infinity. (If at least we dont assume right away that he reached infinity which for me would be question begging).”

      What you need to do is distinguish between “he will count every number” and “he will have counted every number”. The first is true (if he never stops counting), but the second is false. When you say that he will never arrive at infinity, thats fine, because thats the second of the above points. All that matters is that he will count each number, not that he will have counted each number. In most contexts saying ‘it will be that p’ implies ‘it will have been that p’. But this isn’t true in every context. Sometimes you can have the former without the latter. Here is a different example which shows that: imagine time ends at t, and p is true at t for the first time. Prior to t it will be true to say ‘it will be that p’, but because time ends at t there is no point where we can look back and say ‘it was that p’, meaning that now it is false that ‘it will be that it was that p’. So you can obviously have ‘it will be that p’ without ‘it will have been that p’, which is all that we need in our case. Just because Counter won’t ever have counted every number doesn’t on its own mean that it won’t count every number.

      “But I think there is another Problem, as far as I can tell the connection between past and present is that the present is equal to the number of events that have allready been acctualized.”

      I don’t really know what this means. In the present, I can say various things in the past tense, like “I have counted to x” or whatever. But I can also say things in the future tense, like “I will count to x”. So I don’t really know what you mean by the present being equal to the number of events that have been. Why not say it is equal to the number of events that will be? Or just not say either?

      “If we now take counter as an example and place him in the infinite past he would need to count from -infinity to now (0) which (I think) is impossible.”

      Well you will need an argument for why that’s impossible. I’m working on another paper on exactly that, so (spoiler alert) I don’t think there is a good argument there. If counter had been counting for an infinite amount of time, what stops him counting every number? He has enough time to go through all of them!

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      1. Hi Alex first of all Thanks for your response.
        ((And yes I can agree with a lot of your answers esapacialy the part where you explain the issue with my counter analogy part. The only think I might want to say is kind og an answer you said
        I don’t really know what this means. In the present, I can say various things in the past tense, like “I have counted to x” or whatever. But I can also say things in the future tense, like “I will count to x”. So I don’t really know what you mean by the present being equal to the number of events that have been. Why not say it is equal to the number of events that will be? Or just not say either?))

        In response to me claiming that I think the present is equal to the moments allready acctualized. I may have formulated that poorly, what i meant is:
        When I think about what I will write next there is this one little moment where it is present, one millisecond later it is past and one millisecond earlier it was future.
        So i kinda used the word,,present,, in an unfitting way. What this acctualy meant was I think we should look at the past as the things allready acctualized.
        Sincerly Max

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      2. Hey Alex just another thought shouldn’t there be a difference between counting every Number and counting all Numbers. I think this a fallacy of composition (not in the way craig said it is in the debate)

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  6. Hi Alex

    Yes, I suspected that the answer might be the same!

    When we consider the set (say S1) of possible rooms for a single hotel, Loke thinks God can actualise any of them- but such a set does not contain an infinite set, as they are impossible, so there’s no problem with any member not being realisable.

    My thought was that when we consider the set of possible finite sets of rooms (say S2) this looks to be infinite. So whilst S1 doesn’t contain a set with a Cantorian property, S2 does seem to have a Cantorian property.

    God’s ‘in one go’ power doesn’t then seem to fall short on S1- no possible hotel couldn’t have its rooms actualised in one go. But I assume that God could actualise some subset of S2 in one go. Loke’s limit suggests that he can’t actualise the entirety of S2 in one go (for it results in a HH). So what is the cardinality of the subset of S2 he can actualise in one go? Isn’t it finite?

    But I assume what you’re saying is that with regards to S2, the scope distinction just falls again, and it is such that God can’t actualise the entire set in one go as that’s impossible, but can actualise any finite subset (in one go) and that is without limit such that no finite subset is ever possibly not actualised- the only thing that is not possibly actualised is the entirety of the set in one go. That seems to resolve the worry, so thanks.

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  7. Hey Alex I managed to write my thoughts to multiple other persons.
    Now Someone made a video (I dont know if he specificlly made it because I wrote him my thoughts or if he had similiar ones), however I would love to hear your thoughts on it.
    Sincerly Max

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  8. Hi everyone! MJ (Damore, not Danmore, haha) here.

    I’m the author of those posts at philosophyofthedead.blogpost.com, and the dude in the Democracy of the Dead YouTube video that was linked. Don’t worry, though, about responding to the old stuff in the blog. I’ve veered away from much of where I stood on the issue. The newest post (and my back and forth with Wes Morriston in the comments section of the YouTube video) represent my most updated views on the issues Malpass and Morriston argue for.

    But a quick note to Alex. Agreed that the blog posts were a bit prolix. No biggie. But don’t respond to those. They represented where I was at the time (I don’t even make the will be/yet to be distinction anymore). If prolixity was an issue, I would have been glad to shave down any point you were interested in. What was going on in the posts is that I was basically writing an individual blog for every blog on the Kalam you had, with additional blogs providing my thoughts on Morriston. It wasn’t meant to be this overwhelming quantity of prolixity. But please don’t go into any weeds. I’d be glad to focus on whatever you’d like.

    Of course, it’s possible I have this ‘blind-spot’ as you claim I have (no hard feelings, haha), but then it’s possible we might all have our particular blind-spots. Our suspicions cancel each other out, haha – I could say you have a blind-spot because you’re too critical of Craig and it doesn’t allow you to see something that’s ‘bleedingly obvious’ to me. I try my best to cultivate the intellectual virtues and I definitely try not to uncritically swallow something just because a particular philosopher said it. There have been many points where I’ve disagreed with Craig.

    Anyways, thanks to Max for the links!

    If anyone has any questions or issues they’d like to raise, I’d be glad to try and answer or explain.

    Cheers!

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    1. I watched a brief bit of your videos on the debate because Landon brought them to my attention when we spoke. Don’t know if you’ve seen the debate review video:

      it seemed to us that you had the composition fallacy thing a bit wrong. Was the idea you were arguing for that there is no valid part to whole reasoning when it comes to infinity? If so that’s wrong (Landon gives a nice example in the video). I felt you also struggled to get your head round the simple to perfect inference, why that’s invalid, and why it isn’t just when infinity is present. I give an explanation of a counterexample that has nothing to do with infinity for that in the video too (about the angel playing the saxophone at the end of time). So that should dispel the idea that infinity is to blame there too.

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