Hilbert's Hotel: A Comprehensive Response
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- Опубліковано 27 лип 2024
- Stop using Hilbert's Hotel. Here's why.
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OUTLINE
0:00 Intro
1:35 The Kalam
2:29 The Hilbert’s Hotel Argument
19:53 Response #1: Disambiguation
25:10 Response #2: Transfinite Confusion
51:14 Response #3: Galileo’s Non-Paradox
59:09 Response #4: No Intuition of Absurdity
1:08:26 Response #5: Hasty Generalization
1:39:54 Response #6: Btw, Christianity is false
2:25:37 Response #7: Infinite God Problem
2:35:06 Response #8: Fails to Support the Kalam
3:02:42 Response #9: Args FOR Actual Infinites
3:09:06 Summary
3:13:28 Conclusion
LINKS
My website: josephschmid.com
My PhilPeople profile: philpeople.org/profiles/josep...
As someone with a math degree who watches a lot of apologetics, this has been cathartic. So many arguments which boil down to mathematical or linguistic confusion. Thank you
Happy to serve!❤️
the mistakes here are so basic you really don't need to know much about math to see the issues. it's crazy
This. It's a problem with (sometimes intentionally) defining terms.
As someone with a math degree have you thought about infinite in actual existence aka this reality and not just paper assumptions?
@@marksandsmith6778 if there are infinite marks and we shoot one down is it absurd that we’ve killed a mark yet the amount of marks is still the same, think about that in this real world and understand the absurdity
Seeing a 'hilbert's hotel rebutted' video was a bit bizzare because I first heard it as just an explanation of how infinities work, rather than an argument against them, so it almost sounded like you were going to actually call out a problem with the thought experiment rather than just Craig's misuse of it
I'm only 42 minutes in and this is already the best counter apologetics video I've ever had the pleasure of watching. Thank you Joe 🙏
Craig often complains when non-philosophers make elementary mistakes and uses this as justification for not engaging with them. I wish he would recognize his own ignorance in areas that he has no expertise in and save us from the having to listen to him confidently make mistakes that my freshman calculus students can notice.
I don’t think Craig is ignorant. I think he knows that his AUDIENCE is ignorant, and he confidently sweeps the logical inconsistencies in his examples under the rug. He knows that just speaking of lofty ideas like “infinity” will confuse and impress them, so he uses that to his advantage.
@@MichaelJohnson-composer I suspect that in some areas you're right. He's been corrected too many time to not understand at least parts of what claims to be discrediting. In this though I'd be willing to give the benefit of the doubt given he has very little if any formal education in math of the degree.
@@MichaelJohnson-composer I completely agree. Since it has been pointed to him time after time he knows where his logic is trash so he is intellectually dishonest and he is purposely telling lies and misrepresenting science in order to reach his goals.
@@TBOTSS Hahaha. Thanks for a good dose of Sunday stand up comedy and for written evidence that you simply didn't get it. Lol
@@MichaelJohnson-composer Wow. You are the problem with intelligent conversation. Just assume the motivation of the person and run with that.
It gets us nowhere.
Only three hours. You can’t possibly address the argument of the inimitable William Lane Craig, PhD PhD (that’s right two PhDs. Twice as many as the median scholar and infinite times as many as you) that Hilberts Hotel seems a bit funky in a mere three hours
This portion of Craig's Kalam arguments is why I've never liked his work when it gets into math. I'm just enough of a mathematician to have realized he's playing fast and loose with the set theory.
Thank you for describing the sticker so well. It helps those of us who are blind or visually impaired. That’s probably not what you were thinking about when you were describing it, but it’s appreciated nonetheless.
I can't believe you got James Fodor to voiceover your impression of him, I can tell that your production budget has increased. Well done!
When
Excited for this one. I've always had problems with Hilbert's Hotel for the same reason I have problems with the Axiom of Infinity.
50:53 Edward Nelson, my favorite mathematician! You might be surprised to hear that Nelson was a Christian and a finitist (roughly; he calls himself an "ultraformalist"), yet his philosophy resulted in Internal Set Theory which distinguishes the "standard" (things humans can actually construct) and "non-standard" (things too large or too small to construct).
Here's a relevant quote from his essay "Mathematics and Faith", page 7.
"As I lay meditating about numbers, I felt the momentary overwhelming presence of one who convicted me of arrogance for my belief in the real existence of an infinite world of
numbers, leaving me like an infant in a crib reduced to counting on my fingers. Now I live in a world in which there are no numbers save those that human beings on occasion construct."
Nelson has also developed a theory of arithmetic that avoids mathematical induction. Then one can't talk about "all" the numbers (actual), or "any" of the numbers (potential), but only about actually constructed numbers.
I've found his philosophy of math utterly interesting and could talk about it all day...
Set theory can be developed with a true negation of Cantor's axiom of infinity, laid out in "The Foundations of Mathematics in the Theory of Sets" by John P. Mayberry
Of course, Mayberry calls it "Euclidean set theory".
A relevant quote from page 264:
"Cantor's Axiom, the so called Axiom of Infinity, does represent a bold extrapolation beyond our ordinary experience. But everyone must concede, pro and anti-Cantorians alike, that there is nothing obviously wrong with it. If it contains a contradiction, then that contradiction has eluded the notice of the best mathematicians for the past one hundred and twenty-five years."
Mayberry then goes on to give his real reservations towards Cantor's axiom, and why he develops Euclidean set theory instead.
To my personal philosophy, I've concluded that "actual infinity" and "potential infinity" are equivalent ideas, that one proposition phrased with "actual infinity" can be translated into a proposition with "potential infinity" (and vice versa) without any loss of meaning. And, I'm indifferent to whether infinity exists or not, since I've never experienced an actual infinity (a hotel with infinite rooms) or a potential infinity (a hotel with the resources to handle any finite number of people).
Nice connections! Glad I could spark your interest🙂❤️
Craig's claim that actual infinites can't exist and therefore the universe had a beginning but could potentially extend infinitely into the future was pretty intuitive to me, but I can honestly say this video broke it down extremely well and I'm convinced that claim has quite a few errors/issues! This video is really impressive, thank you.
Actual vs potential infinite is not the same, can extend infinitely into the future equals potential infinite
@@japexican007 Watch 1:41:30 to see how Craig defines "actual infinite." Joe explains how the same definition accurately describes the infinite past and the infinite future.
You can construct set A: "all days ever" with two subsets: Set B: "days that have happened" (a growing list of days) and set C: "days that will happen" (an infinitely long list future days). You're confusing set A (made up of set B and set C, and is an "actual infinite") with set B (which is a "potential infinite" in that it is finite but keeps growing).
To say a reality where there are infinite future days but there are no actual infinites is a contradiction. Now, whether or not actual infinites are an ok thing is up to you; but the same logic Craig applies to make an infinite past seem absurd (Hilbert's Hotel) can be applied to the infiniture future to make it also absurd. Joe explains how the Hilbert's Hotel example actually fails to prove an infinite past or infinite future is absurd, leaving the door open for both.
@@josephtnied actual infinite: truly infinite
Potential infinite: has the potential to be infinite
It’s not hard to understand both
God is infinite: no end-no beginning
The future: can keep going forever but hasn’t yet gone “forever”
@@josephtnied or just use a piece of paper, for God’s infinite draw a line across the whole piece of paper, there’s no beginning or end for God
For the future you can draw a line in the center of the paper with one arrow going forward and a point can represent the finite past. The arrow is not touching the edge of the paper, this represents potential infinite
It’s not hard to understand the difference
I also think it is super intuitive - is it possible that a super complicated argument could create some doubt on this? I think so. But to me, the complexity here gives me reason to think that I should default to just what still makes rational sense and is easier to believe due to less complexity.
I never understood if you can add things to infinity how that was infinity to began with? And also if infinity can be composite than infinity should be able to be formed by addition.
Is an infinite past a sufficient argument/premise to adopt? it seems like it makes sense when we talk about determinism, nominalism, etc.
I think it's helpful to point out an example of an inverse operation being undefined that only needs to involve finite numbers: the classic dividing by 0.
Excellent video ! I understood most of the things here but I need to watch Response 2 again.
I wonder do you have any plans for also doing a PhD on Mathematics like Dr Alexander Pruss ?
Hi Joe. I want to know your opinion of Apologetics Squared. What do you think of him?
A rare pleasure to see one’s ideas communicated so accurately
Prof. Malpass,
I hope this comment reaches you, and finds you well.
Can you please help me understand your reasoning with regards to the following? (asked after having read your latest papers "All the time in the world" & "Endless and Infinite")
*Excerpt from the papers:* "Consider the cardinality of the numbers that George will count. Because George isn’t going to stop counting, there is no number such that he won’t eventually count it. Therefore, the cardinality of the numbers he will count is just the cardinality of the natural numbers, Nᵒ."
*1- My question is : Taking Craig's claim that George won't reach infinity at any point in time no matter how many numbers he counts up. (an appeal to potential infinity or future perfect tense, or the NIP). How can we ascertain that the continuity of counting finite numbers **_will eventually_** result in an infinite set?*
*My thoughts :*
--- Simply stating that "eventually" the set will have the cardinality "infinite" is still not convincing to me, because "eventually" means a point in time, and we know that looking at the set of counted numbers at any point in time will tell us it's still finite in cardinality, also you object to this word itself, i.e. "eventually", because it misleads us into thinking in future perfect tense, but you still use it for illustrative purposes, I guess. If I was asked, I would use the word "ideally" rather than "eventually".
--- Most importantly, infinity as a cardinality is a before-the-fact "concept" definition, based on the bijective identity function f(n)=n, for N, the set of natural numbers. This definition of a concept, is an IDEAL concept denoting endlessness. It does not help us see how such finite counting can bring an actual infinite after the fact of counting, due to discontinuity between that ideal concept "actual infinity" and the last number counted. *The jump from the finite number to the infinite ideal concept is still unbridged, and your attempt to bridge it is by using the same logical concept, Nᵒ, seems to me circular, am I wrong here?* I see this similar but not equivalent to the jump from the inanimate to the animate without explaining how life or consciousness occurred. *Merely stating that counting on and on will do the job seems inaccurate to me, as cardinality of any number of numbers counted will always be finite. There is no eventual point where there are no numbers that he won’t eventually count! the dance of infinity continues: George will count all the numbers but also will have other numbers to count, in stable inertia of superposition so to speak, and at any point of measurement, the dance "collapses" to the finite realm.*
--- In your other paper, you stated "It is easily proved by mathematical induction that for every positive integer n, Counter will actualize 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." I would agree that ideally, George will count each number as a potential event that will be, and the cardinality of those counted numbers is infinity, in an ideal or generalized sense. But to say that this idea abstract concept is eventually reachable is why different objections, like potential infinity or simple to future perfect inference, are raised although they stem from the same problem of counting up to reach an actual ideal definitional infinity. I am torn between your misunderstandings of each other and I want to find truth. Until now, I lean towards the idea that successive addition is not a way to reach infinity from a finite, infinity can only be reached before the fact, , not via successive addition), or infinity could be used and assumed as potential ideal concept, not an actual reality, either metaphysical or physical.
2- On the counting DOWN from infinity to our present day. In a metaphysical possible world, we are obliged to believe anybody eternally living in his house and who comes out shouting "TWO, ONE , ZERO,, YES! I JUST FINISHED COUNTING FROM INFINITY Y'ALL!!" because we don't know how we can confirm his claim if we didn't or couldn't get near his house to eavesdrop, or hear some of his relatives' thoughts about him. We have no epistemological way with which we can confirm his claim, or that of another guy who just comes out of his house 1 day later and says the same thing. *So, my question is, is this going to make infinity a malleable concept of no determined (undifferentiated) reality or size? if no why? if yes, then how can we get a sufficient explanation (not partial, or immediate explanation) from either guy's claim ?* I define the sufficient explanation that it is the cumulative group of explanations, after which any additional information is redundant or irrelevant.
I might be wrong but that's why I wrote to you. Thank you in advance for your time.
@@AShaif Firstly, I’m not a professor. I don’t have an academic position at a university.
Secondly, I don’t think I’ve ever said that “eventually” the endless counter ‘gets to’ infinity, or whatever. I think he never gets there.
My point is just that, right now at the start of his count (and indeed throughout it) the number of future counting events - each of which will happen - is not potentially infinite, or finite, but actually infinite. It’s no more an idea than the past counting events.
So you might be struggling to solve a problem that’s not solvable - how will they ever “eventually” get to infinity. They won’t. But just count how many numbers they have left to go through. That’s actually infinitely many.
@@alexmalpass
Thank you for the feedback. You are right about the word "eventually". That was found in the phrase "there is no number such that he won’t eventually count it."
I understood your point of the idea of infinity at the start or throughout the count, which is fine. *But this idea of infinity is a before-the-fact definitional ideal concept.* Infinity can only be rationalized before the fact, not via successive addition (after the fact). Your argument from cardinality is a definitional ideal concept, but cannot be reached or instantiated in reality with counting. *No number is left uncounted; that could be a true state of affairs. But also, on each number, n, an infinity of numbers are yet to be counted, which is also true; So, another state of affairs is true at the same time (any n number) when the first state of affairs is true! You have two state of affairs being true, at the same time, which contradict each other; That's the superposition of infinity so to speak.*
We agree that infinity is the cardinality of natural numbers, but that's a definitional ideal concept ; *i.e. it's not reachable or instantiable through successive counting/addition, ever; an impossible after-the-fact state of affairs.*
And that's why the word "each and every" in your 2021 paper "he will count each and every one of the finite numbers" is implying the _inclusivity_ of every number, (or the _non-exclusivity_ of any number, same as your "eventually" statement above) but those words are misleading, because they indicate the non-exclusivity of any number, an idea that cannot be a state of affairs in an infinite counting scenario. Such infinity concept can only be before-the-fact concept, as an ideal limit, because the instantiation of infinity in reality is something impossible as an after-the-fact state of affairs.
"But just count how many numbers they have left to go through. That’s actually infinitely many."
That is always finite in reality, and infinite in ideality; due to the reasons I laid out above. Am I making a mistake here?
@@alexmalpass
I tried to put it in a syllogism. Can you please review it ?
1- If George starts counting and never stops, then he will count each and every natural number, n.
2- At each and every n counted, the count is denoted by a finite number.
3- If at each and every n, the count is denoted by a finite number, then infinity is never instantiated.
4- The cardinality of natural numbers is infinity.
5- With 3 and 4, the cardinality is never instantiated by counting.
6- Therefore, cardinality is a state of affairs that can't be actually instantiated via counting. It might be instantiated via non-couting or any non-successive composition.
@@TBOTSS as explained in Response #6 of this video, Craig’s response to Alex rests on confusions. The only one who needs to read set theory is you.
I haven't watched yet, but I'm assuming this is something about the infinite regress of the Kalam argument? I'm interested in what you have to say about it.
You are correct
@Quantum Passport Because theists seem to think there has to be a first cause. (Which is what they are trying to prove.)
Joe addresses that in the video. I know most of us don't have three hours, but you can listen in smaller doses. The Kalam argument depends upon the impossibility of traversing an actual infinite in order to conclude that the universe has a demonstrable cause or beginning. On the other hand, Aquinas' arguments assume _arguendo_ the infiniteness of the universe to the conclusion that God must be its ground.
Wow, I thought you were directly refuting the original Grand Hotel analogy. Was a bit confused.
Isn’t there a difference between an object being infinite and a conscious mind being infinite, I dunno much about being infinite but that’s a reason why I don’t correlate the infinite used in objections because they are object centered
One other question I have is whether there's a problem defining a full infinite, in this sort of context.
I'm guessing there's considerations of the meaning of full, that come into play.
Isn't it a bit telling that in the process of a philosophical proof, when trying to establish an actual proof in the realm of mathematics, he leaves the concepts in less precise linguistic form.
I don't know whether he has the mathematical expertise to actually use the math correctly, but if he doesn't, does he have any business presenting his concept as a proof of a mathematical concept?
I know this is forever later, but one thing I wondered, is do you actually have to prove that subtraction of sets vs subtraction of cardinality of set? Doesn't the subtraction assume that addition is valid, and is addition actually valid in this case?
Set A = {1,2,3}
|A| = 3
Set B = {3,4,5}
|B| = 3
Set C = Set A + Set B = {1,2,3,4,5}
|C| = 5
So since his supposition that the addition is valid is not valid, then the subtraction not working doesn't prove anything.
The move that Craig makes that doesn't seem right to me is the generalization from the absurdity of HH to the absurdity of actual infinites. I haven't seen him give an argument for why it's rational to generalize
Craig gives 3 arguments for this which was discussed in this video at response #5 1:08:26
Entire books have been written on the topic of infinity. However, to provide an example of how an actual infinity cannot exist in any possible world, the thought experiment known as “Hilbert’s paradox of the Grand Hotel” will suffice. In contemporary Western academia, there is a popular case of a hotel with infinitely many rooms. My immediate response to such a thought experiment are:
1. “Infinite” is not a countable number.
2. If the hotel had infinitely many rooms, it would necessarily be infinitely-large in size, and therefore, there would be nothing BUT the hotel rooms in existence (and obviously, the entire universe must necessarily be infinite in space). There could not be any guests staying in the hotel, since that would render the room limited to a certain area of space.
3. As an aside, the hotel would also be limited in time, since someone would need to have built the hotel in the first place. And from where would the building materials originate if there was naught but a (so-called) infinite hotel in the spatio-temporal universe?
The reason why some Western philosophers believe that such a phenomenon is actually possible in some kind of world, I suspect, is due to their longing for INFINITY, which is, in fact, their very essence! If only they would study what is (by far) the most important work of literature ever composed, this “A Final Instruction Sheet for Humanity”, in association with a genuine philosopher (that is, a guru/yogi), that longing may be satisfied via the accomplishment of gnosticism (see the paragraph on “jñāna yoga” in Chapter 16)!
Pretty cool that you used Louis Swingrover! He’s one of my closest friends. I had some input on that paper you referenced.
Yesssss! His paper is wonderful. It’s awesome that you’re close friends with him! I actually wanted to share this video with him but struggled to find a contact method. Maybe I just found one in you😉🙂❤️
@@MajestyofReason
I sent it to him 😊
@@mathematicaluniverse7493 thank you!!!
@@MajestyofReason
You bet!
@38:25 hmmm aren't relative completmentation and subtracting cardinalities same, when the set being subtracted is a subset the set from which it is subtracted, which is so in Craig's case?
I think the problem is that Craig isn't subtracting cardinalities in his examples. Craig is using 'infinity' as if it is like any other natural number when he is doing his 'subtracting' - that's why he thinks the results are weird. But as Joe and James (quoted in the video) point out: he's misunderstanding the operations. You can perform the operations if you are defining the set by its cardinality and the results are consistent with what one would expect. ie: the infinite sets are different where as Craig is thinking they are the same.
Somehow I completely missed this video until now. No idea how that happened. Time to give it a watch.
Time to turn on notifications😉❤️❤️
@@MajestyofReason
That as well. I tend not to use notifications for anything, but I quite enjoy your videos and the occasional back-and-forth enough to warrant it being the first time I try to use that function.
Wow this is pretty epic. I wish more people did comprehensive videos like this.
Hmm , Joe in your video "Why am I agnostic" you said that causal finitism and paradoxes about infinities are evidence for theism, do you still accept that?
Excellent! I'll surely include this one in my counter-apologetics playlist.
Can I have access to that playlist?
@@asad9042 Sure, it is a public playlist. Anyone who sees my UA-cam profile can access it.
@@asad9042 only well behaved boys can see it.
One thing I would want to point out with respect to cardinalities and set difference is that the claim that *in general* it is not true that, for two sets X and Y, |X \ Y| = |X| - |Y| is correct. However, this equation does indeed hold if Y ⊆ X. Now, in Craig's examples, this subset relation does hold. However, this still doesn't rescue him because the equation holds only for finite sets, whereas Craig is of course performing operations on infinite sets. I just wanted to clarify that Craig's reasoning actually is valid for finite sets, contrary to what Swingrover claims.
Your comment is correct with one important caveat: in my video, I was careful not to include Swingrover’s point in Appendix A of his paper that Craig’s reasoning also entails contradictions in the case of finite sets. This is because I shared your criticism: Swingrover’s point there requires doing a relative complement of X in Y where X isn’t a subset of Y, and yet a natural objection that Swingrover doesn’t consider in said appendix is that Craig isn’t doing relative complements where X isn’t a subset of Y; instead, Craig always uses subsets. (Now, Craig’s argument still obviously fails, as you rightly note, since the diff b/n cardinalities of Y and X and the cardinality of a relative complement of X in Y are only the same (even when X is a subset of Y) in the case of *finite* sets, and to assert that it must also hold of infinite sets (and to thereby insist on contradictions in the realm of infinity) is simply to beg the question - it is to assume that there cannot be infinite sets. But my point is that Swingrover’s criticism here misses the mark, and we’re in agreement there!)
Instead, in the video, I only discussed Swingrover’s point - a correct and relevant point - that even in the case of finite sets, the *operations* of relative complement and cardinality subtraction are different. (This is true even if, in cases where finite set X is a subset of finite set Y, the resulting set from the relative complement has a cardinality identical to that of the difference between X and Y’s cardinalities - just as any number n + 0 is n while any number n - 0 is also n, and yet + and - are still different operations.)
So, in short, your comment is correct, and I agree with your criticism of Swingrover’s point in Appendix A. But, importantly, I was careful not to include Swingrover’s mistaken point in my video.
@@MajestyofReason My mistake! After watching the video, I went and read Swingrover's paper and must have blurred the two together in my mind. Thank you for the peer review :D
With respect to the operations being different for finite sets, I think this is true, though I think we can still ask after performing set difference on finite sets, how many elements were removed from the original set after performing the operation, and how many elements remain in the set? If the aforementioned equation held for infinite sets, we could reason that ℵ0 elements were removed in both cases, but ℵ0 elements remain in the one case whereas 3 remain in the other, and this is a contradiction. Of course, this would still commit the fallacy of treating ℵ0 as a determinate quantity. As far as set difference goes, however, is your point that, given that it is an abstract mathematical operation, it isn't clear that we are necessarily "removing" elements from a set in the way that we remove items from a collection in the physical world? In a sense, I suppose we could view set difference as simply a way of defining a new set without actually "removing" elements from an existing set.
@@colincausey5980 no worries my dude! I’m happy you checked out Swingrover’s paper after watching the video🙂
In short, my main point is that this is all just abstract mathematical functions. The concrete world is different from such functions. The concrete world can be *modeled* by the functions, or be *represented* by them, of course. But they’re different. And so we need to be careful how we’re representing the relevant manipulations of objects in the concrete world. We cannot assume, simply because we’re taking away or removing concrete objects, that we’re doing something that is accurately modeled or represented by arithmetic subtraction. It may, instead, be only accurately modeled or represented by a different abstract function. And this is, indeed, the case with manipulations involving infinitely many concrete things - as I argue in the video, the relevant abstract operation being modeled in concrete reality is relative complements, *not* arithmetic subtraction.🙂
Great video Joe. I've never really understood what is supposed to be wrong with the concept of Hilbert's Hotel.
Personally I think the problem is with the management of the hotel advertising that their own hotel is "full" - it's as if the management have forgotten or don't realise how their own hotel works: they will always have room for new guests. ie: why advertise your hotel is "full" if you know you can always accommodate more people!?
This was excellent. Thank you for your Herculean effort.
It's frustrating to me that Craig's problem with cardinalities is that they don't seem to satisfy arithmetic like axioms. As far as I understand subtraction is not defined for infinite cardinalities and Craig seems to be confusing set exclusion between a set A and a proper subset of A, B for subtraction between their respective cardinalities. It's not at all clear to me why set exclusion that follows this form should always be identified with subtraction between cardinalities. And it doesn't seem at all clear to me why cardinalities must satisfy some arithmetic-like-axioms
Edit: Oh you addressed this already. I wrote this right at the beginning of the video when you were explaining Craig's views x3
Absolutely amazing video! Subscribed.
When I first saw the title I laughed, but then I saw who posted it, shut up, and prepared for war
One neat way of thinking about cardinality is "what would happen if you counted them"
So then it makes sense of the claim that the set of positive integers and the set of positive even integers is the same.
The same thing would happen if you counted them. That is. You would start counting and never stop.
That was the first thing that came to mind regarding the attempt to apply Hilbert’s Hotel to the real world via an indestructible counter. By definition, you can’t count down from infinity and trying to suggest it raises concerns about their ability to understand what they’re arguing.
Not being able to tell the difference between two things is different from two things being identical. To equate the scenarios would be philosophically stupid and lazy. Defining a category and sticking them both in it is fine. But just do go around saying they are the same or "have the same size" for gods sake. And as a definition of cardinality, this is lacking because it would put real numbers and integers in the same category when they have different cardinalities.
@@Elrog3 Is this a response to me? I don't think I said anything that you say I said.
I didn't offer a definition of cardinality. I didn't say anything about sameness of thing or size.
Aleph null is a cardinality, not a quantity. Infinity is not a number. Arithmetic doesn't work on infinity.
There's actually a whole minor branch of mathematics, called trans-finite numbers, which are smaller than infinity but larger than any integer. Aleph-null is part of this.
I don't think WLC understands the math.
On top of that, the observable universe is finite. If the entire universe is infinite (and we have no idea if it is, and maybe never will), we are isolated from it by an event horizon, just like how we can't see the infinity inside a black hole. The physical universe is not subject to infinities.
You can't correctly conclude that the universe is not subject to infinites just because you don't see them. That's a non-sequitur.
When I claimed that Craig doesn't understand the math, I was asked , "you think you're smarter than WLC?" I don't think I'm smarter, I think he's either lacking knowledge or dishonest. Tbh its probably both
Very quick question: Have you abandoned your step-by-step argument for causal finitism that you wrote in 2020 or do you just think Craig’s arguments suck?
See my recent discussion with Adherent Apologetics for my take on my Erkenntnis paper 🙂
(That’s in my Kalam playlist)
@@MajestyofReason So I’m guessing you don’t buy Pruss’s arguments for the compatibility of causal finitism and continuous spacetime? What about them don’t you like?
Please do all your videos in the Zizek voice-it’s so good
Joe will you publish papers on Hilbert's Hotel so that Craig may able to response you?
Btw I requested Dr Andrew Loke for a response to this video and said that you responded many of his works in his channel's comments section. He said he will check this video when he has some free time.
Many (if not most) of the points in here have already been published in response to Craig - and yet Craig hasn’t published a response to them.
Plus, this information will be going in the book that’s 65% likely to come out of my series with Stephen.
@@MajestyofReason oh! Thanks for letting me know. It seems Craig have a habit of missing the main points (arguments).
But it will be wonderful if you guys really publish a big academic book on KCA. Maybe not Craig but other theist philosophers will able to make responses to you and I can't find any complete book on Kalam by atheist philosophers.There are only books written by theists.
I think the reasoning behind the 6th point is that things that lie in the future are not real things. Yes, there is an infinite number of theoretical future events but they are not something that has happened in reality. If the universe has beginning, then there will always only a finite number of events that have happened. I think that distinction between real things and things that are just theoretical is crucial and Craig mentions that there are some things that make perfect sense logically and in mathematics but are not instantiable in reality, and since theoretical future events are not something that's instantiated in reality, it doesn't matter that that theoretical set of events would be infinite.
The past is different, because those things really have already happened, they are more real than the future.
Thanks for the comment!
(1) Under presentism, which Craig accepts, past events don't exist at all. They're not real. Only the present exists. Only the present is real. Past events therefore aren't 'more real' than the future. Indeed, there is no such thing as being 'more real'; something is either real or unreal. Only under growing block theory and eternalism is the past real. Yet Craig rejects both of these.
(2) And besides, even if the future isn't real, as I explain in the video, we can still number how many future events there will be, and since the number is aleph-null, we get all the same 'absurdities' that attend Hilbert's Hotel. And so whether the future is real is neither here nor there.
I'm half way through the video and I have to say I'm very impressed. Response #5 went a little bit over my head tho
Now THIS is why I patronise you ❤️
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I love this channel! So much to learn. 💯
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Your Swinburne and Oppy impressions are spot on. I am sure with practice you can nail a Russell impression, too.
Of course infinities does not make sense in causally connected finite geometries ( for example our obeservable universe), but that is not to say it is impossible if there is more than such a causality dependent geometry.
Amazing vídeo :) thank you
That dark background with white text is a true eye killer
About the second absurdity, there is a potential counter-response.
Correct me if I am mistaken.
Let A,B be finite sets, a property that holds is that if B is a subset of A (B⊂A), then |A-B| = |A| - |B|.
Let C be a finite set such that C is a subset of A, and it has the same cardinality as B, but isn't the same as B (there is an element in B that is not in C), (C⊂A ∧ |C| = |B| ∧ C!=B). It holds that |A-B| = |A-C|.
This may be the property that Craig is talking about, and it can be intuitively associated with arithmetic substraction with the natural numbers.
The odd numbers and the natural numbers greater than 4 are subsets of the natural numbers, so it does violate this property.
those only work in finite sets though
@@proloycodes I think that is the point Craig makes, that an "intuitive" principle doesn't work in infinite sets, so there cannot be actual infinity.
What I was trying to point out is that the refutation given to the second absurdity regarding relative complements doesn't work. It is still refutable by questioning how intuitive is infinity or if intuition applies in this context, which is the response provided by Joe to the other absurdities in the rest of the video.
@@jonyloco4134 oh i see, i thought you were trying to refute the response by extending your argument to infinite sets.
Man, I wish you could have this conversation with WLC. Good stuff.
Also Andrew Loke
Dude, you're making awesome content. It will be helpful if you made a video about a practical guideline for critical thinking
Agree!
A critical thinking schema, starting with a basic mental checklist to run through when approaching philosophical questions/ideas. Summat likat.
It's kind of hilarious that an apologist is talking about the absurdities of infinity given that he believes in an infinite being
So do you believe there is no difference between a block of infinite LEGO’s vs an infinite conscious mind?
@@japexican007 you are making a plea to intuition.
Which is the problem that's always pointed out here.
You are the one with the position that these categories are different in ways that making this counter point counts as a category error.
How is this infinite being existing in a way that Is categorically different to avoid the same charges?
To say "an infinite mind" is categorically different than infinite space is a claim. And it's also abstract.
Why doesn't an infinite mind require space? Why is space special for these problems?
You are appealing to an infinite mind the same way that somone would appeal to quantum woo.
You aren't defining what it actually is or how it works.
So the difference here is the Legos are actually tangible things that I can understand and think about.
While an infinite mind isn't and is never defines or put into terms that can be discussed.
Putting out the question "don't you think these things are different?" Does no work for you. You haven't accomplished anything and this is just a lazy way to avoid a special plea
@@ninjaturtletyke3328 a physical mind doesn’t require “space” because a mind has no physical parts
@@ninjaturtletyke3328 yes that’s kind of the point of God, if you were inteligible enough to understand your creator then you would probably lower him to your level of understanding
As for why infinite objects are different than an infinite mind well objects can’t make decisions and God can freely choose to create or not; it’s very simple nothing complicated but atheists use all kinds of theories and objections to reject God; thank you, someone has to fulfill prophecy
@@ninjaturtletyke3328 btw you can spend your whole existence studying and theorizing God and his nature/being/composition but at the end of the day none of that will lead you to trust and accept God so I see no point in your fruitless endeavor
Set Difference sounds like it could be explained pretty easily using a Venn Diagram
Looks like I'm a little late for the giveaway, love that sticker. Suppose that's my punishment for only subscribing and not hitting that notification bell.
Cheers Joe!
I now know actual infinity is real because that's how long this video is haha!
Seriously though... kick arse vid
I haven't performed this exercise, but your mentioning of different pronunciations of aleph by English speakers reminds me that what one should really do to resolve that debate is ask someone for which the Hebrew alphabet is part of their native language. Purely as a pedantic side note, it seems odd to me to mention aleph naught without saying the word 'countable'. I can understand why you wouldn't mention this for time, but aleph one, the uncountably infinite set of real numbers, seems relevant because it further breaks infinite regress by talking about the infinity between zero and one (see further the grim reaper 'paradox', but I'm responding as I watch, so maybe you get to it, but I wonder if the grim reaper 'paradox' isn't resolved by knowing calculus, or at least convergent sequences, further suggesting it may only be called a paradox because philosophers slept through their maths courses)(I further wonder if the uncountably infinite number of fractional seconds between now and one second from now might say there can exist actual infinities in reality. I further further wonder about the infinite nature of pi or other irrational numbers in this regard)
I haven't written a philosophy paper before, and I'm not sure I know how or if my ideas are worth writing further, but I'm pretty sure I can prove Wm Craig knows jack-all about maths and the possibility of actual infinities by focusing on pi alone. Possibly with some great word play with circles disproving his circular reasoning about infinity.
But I only got a BA in Maths (in 1992), and he got a PhD in philosophy, so I should probably shut the hell up.
I would also like to point out that most, if not all, laws of physics are symmetric wrt time. I would suggest, then, that anyone who believes in an infinite future must believe in an infinite past, unless they simply reject physics altogether. I could be wrong, though. I'm kinda stupid these days after my brain damage.
Joe , I tried to put my thoughts in a syllogism. Can you please review it and give your thoughts about it?
1- If George starts counting and never stops, then he will count each and every natural number, n.
2- At each and every n counted, the count is denoted by a finite number.
3- If at each and every n, the count is denoted by a finite number, then infinity is never instantiated.
4- The cardinality of natural numbers is infinity.
5- With 3 and 4, the cardinality is never instantiated by counting.
6- Therefore, cardinality is a state of affairs that *can't be actually instantiated via counting.* _It might be instantiated via non-couting or any non-successive composition._
This syllogism's emphasis is that the concept of infinity was given before evidence; i.e. it is a before-the-fact ideal concept, assumption, that is presumed in mathematical inquiries, such as in the cardinality of natural numbers. It is not an after-the-fact state of affairs.
If it is okay for me to jump in, I think you're right that infinity is a mathematical idea. In fact I think mathematicians have to admit it is an assumption, right? Set theory's axioms include the axiom infinity, something that is not demonstrated - but stipulated to be true. It is not proven in set theory. I think you are on to something if you are questioning whether or not it is legitimate to apply the mathematical concept of infinity to the properties of time. Not all mathematical ideas correspond to reality.
@@michaelremington5902 Yes. Infinity is stipulated before we prove it to make ideal states accessible for mathematical inquiries. Taking this assumed ideal state and using it to prove metaphysical successive events are in cardinality infinite with reference to the cardinality of natural numbers is rendering an ideal assumption an actual reality. This appeal to cardinality was resorted to by Alex after Craig told Alex that he committed a modal operational shift from (each thing is counted) to (all things are counted), more like the fallacy of composition. Cardinality method seems even more fallacious.
@@AShaif I am suspicious of the concept of infinite cardinality also. I have a question, do you think the future is endless? or will it come to an end?
@@michaelremington5902 it is endless, an open end, but at every point finite from 0 (now).
@@AShaif I see. Do you think the past presently exists?
I appreciate all the work that went into this video, I agree this video is the most comprehensive analysis of Hilbert's Hotel I have seen as it clarifies the many objections while adding new info I was previously unaware of. I have to confess Craig's argument was very intuitive to me upon first examination, but why should my primitive intuitions of the infinite refute the majority of philosophers and mathematicians, and oddly enough this this does seem to be the entirety of Craig's argument. And upon further examination of Hilbert's hotel the conclusion that an actual infinite is metaphysically possible seems to follow " logically and inescapably" in the famous words of Dr. Craig. Finally, I did have a whole "well couldn't the Christian posit an atemporal afterlife" thought typed out so thanks for anticipating and responding to that thought, but of course that isn't going to address all the other forceful objections to Hilbert's Hotel laid out previously.
Very happy to serve you in your pursuit of truth❤️
This rebuttal is as problematic as Craig's. If there are two legitimate ways of defining fulness, etc, then what makes one definition more legitimate than the other? Craig can just claim that the other definition of being full or being vacant not favored in this rebuttal is the more acceptable one. So, I don't see this resolved one way or the other.
Hilbert is a small village in Wisconsin - way too small for an infinite hotel.
Did you intentionally release this on WLC's birthday?
I work in mysterious ways……
@@MajestyofReasonJoe, are you saying you are a God?
2:22:00 This point you make right here about how can the bodies change if they are timeless? How can anything change if there is no time? Timeless in this sense to me sounds like stasis. This is what I bring up when people say god is timeless. I dont know how he went from the state of not having created the universe to have created the universe, and then later coming to earth to sacrifice himself and do all this stuff. How can he have the ability for potential change in a state of stasis?
Around 31:23 the analogy given is dis analogous
In all the examples of Craig what is subtracted is subset of the set from which it is subtracted
|A-B| = |A| + |AnB| if B is subsect of A then the equation can be indeed rewritten as |A-B| =|A| - |B| at least for finite numbers
28:20 But Craig doesn't believe in transitivity. Father = God; Jesus = God; Father != Jesus. So what's he claiming is absurd?
Fascinating and thought provoking, as always, Joe.
Wonderful to see you here in comments!🙂
@@MajestyofReason A pleasure to be here. Appreciate your time and work.
Haven't finished watching yet, so this may come up. But something has always struck me as odd about Hilbert's Hotel as an argument. Because the story is surely about 'space' being infinite in directions and manipulating infinite people in that infinite space.
It strikes me then - *if this is successful WLC should only really be able to infer that 'space' can't be infinite in any direction? I don't understand (though maybe missing something) how from this he can infer the 'past' can't be infinite.
ie: if space is actually infinite then it seems to me Hilbert's Hotel can exist - I mean - there's always extra room. It seems then WLC is then committed to think space also can't be infinite?
@@christaylor6574 he is committed to that! Many find this an implausible commitment - Wes Morriston, for instance, in a clip played in the video, finds this implausible
AS YOU GET INTO #6, your assertion of a real actual infinite series in the space-time manifold. But the mapping breaks down at the "quantum foam" on one end of the measure and the absolute heat death on the other (or you get into aeons, and multiverses which seem to equivocation on the type of reality set from the original space-time manifold we begin our count in, whether "backward" or "forward").
Planck measure is used to be the "meaningful" universal, that would be common anywhere in the universe, and this admits an infinity of directions and the Zeno effect is discussed and it all points to our having a knowledge of infinity that may actual but because of the light speed barrier we don't appear to be able to prove it as "reality", unless it's evidence of platonic reality, omniscient insight, just plain imagination that a thing can just keep going. The absurdity would be in the proposition that that the thermodynamic arrow would not cease, which raises the question of an infinite temperature in the past, even in the infinite density and curvature of a singularity, there is a point of relevant meaninglessness. Unless you are God, this points to the physics we draw from the event of Jesus resurrection, if it suspended and reversed thermodynamics once we may suppose it could do so infinitly, and this seems to be your best evidence in this universe that anything like the aeons Penrose describes may be possible. Or actual and potential infinitesimal. THE decimal expansion of PI may be actually infinite but impossible to know in the age of the universe at its end, or polynomial time.
I was reading your comment and I haven’t watched the video so not sure what your point is.
Just the last thing you about Pi may have an infinite decimal expansion. It just does. We have proven that Pi indeed has an infinitely long decimal expansion. In fact, it is possible to determine any digit you want of Pi, it just might take a while lol.
Instantly subscribed
@Majesty of Reason Do you think that it is absurd to believe that it could be possible that the past is finite and the future is endless? I think there are reasons both for and against this view - but this view doesn't seem to me to be absurd on its own without bringing in other beliefs, such as Craig's theism. I don't find the symmetry arguments against his view alone compelling. Time could be asymmetrical after all. There are reasons supporting asymmetry.
Excellent question. So, I don't see any clear absurdities in affirming the conjunction of (i) the past is finite and (ii) the future is endless/infinite.
What I *do* think is that some of the central *arguments* that Craig et al offer for (i) would, if successful, equally disprove (ii). [But not absolutely *every* such argument.] :)
@@MajestyofReason Thank you for your response! I really enjoy your content. Do you think belief in finitism is compatible with (i) and (ii) ?
@@michaelremington5902 It depends on (i) the motivations one has for believing finitism, (ii) how one cashes out the thesis of finitism, and (iii) the ontology of time.
For (iii), only under growing block and presentism will an endless future not exist (simpliciter), and these are highly contentious views (and not majority views among philosophers and especially physicists). Under any other view, all of the infinitely many future events really exist, in violation of any finitist thesis.
For (ii), whether an endless future (even under presentism or growing block) is compatible with finitism depends on the thesis of finitism. If one just means there cannot be any collection with aleph-null-many members [whether or not such members exist] -- which is how Craig uses 'actual infinite' -- then finitism is incompatible with an endless future, as explained in the video. But if one just means that there cannot be (say) simultaneously existent actually infinite collections, then that *is* compatible with an endless future. (Though it's *also* compatible with an infinite past under presentism)
For (i), the motivations for finitism might equally rule out an endless future, even if the mere fact that the future is endless doesn't contradict one's finitist thesis -- e.g., as I explain in the video, all the alleged absurdities attending Hilbert's Hotel equally afflict an endless future, and so if one's motivation for finitism includes things like the absurdities illustrated by Hilbert's Hotel, then one cannot consistently think the future is endless.
@@MajestyofReason This is awesome - thank you. This is my first ever UA-cam conversation - even though I've used UA-cam since its invention. If you don't continue I completely understand - because I know you're a busy person.
On presentism, do you think an endless future entails that there exists an infinite collection of future days? This would be saying that there exists an infinite collection of non-existent things - which seems counterintuitive. Perhaps "endless future" could just mean that the universe has temporal inertia; once it starts, it never ends. (Yes, that idea is inspired by your ideas of existential inertia - heehee)
50:35 That's a sightseeing boat on the Seine, before the fire at Notre Dame de Paris.
Do you think Dr Craig will respond to this?
He should
Craig response won't be good.
@@senkuishigami2485 He will, he even has a blog named Dr. Craig responds, pretty much like Michael Jones from IP or Andrew Loke. Theist philosophers will never shut up, atheists on the other hand never have the last word
@@logans.butler285 Philosophers usually don't shut up and they never have the last word .
I requested Dr Andrew Loke to review/response this video. He said he will try to watch this video whenever he get some free time.
Awesome video Joe!
Is it not impossible for an "infinite hotel" to be "full"?
This seems to be having a final member of an infinite set; an unsatisfiable pair
Edit: Joe covers this at 20:30
A one mile road has a infinate number of mesurments between it yet you can reach the end of the road
Regarding the issue of an actual infinite set of future events, Craig says that the set of future events is a potential infinite. Wouldn’t that be patently false since god has already decreed all future events according to his understanding of gods foreknowledge? Even if we grant a presentist theory of time, god still has the knowledge of all future events (and even has middle knowledge) on Craig’s view. Unless Craig wants to affirm open theism I don’t see how Craig’s answer gets him out of the problem of an actually infinite future.
I don't think it's necessarily about God's decrees. The problem with WLC's response is that he's getting confused with what a potential infinite is supposed to be.
A potential infinite is supposed to be a collection that is *always finite and always increasing eg: 1, then 2, then 3 etc.
This can't apply to the infinite future events since those are *always infinite and don't 'increase' in number. In fact, anytime a future event occurs it is removed from the collection of infinite events and placed into those of past events.
Ie: the set of infinite events can't be a potential infinite because it doesn't meet any of the requirements to be considered a 'potential infinite.'
Hopefully that helps.
"Philosophy is such an easy class, it's just writing your opinions" Screw my classmates who said this, I've got to watch a 3 hour video on set theory and try to grasp multiple concepts of infinities just to follow along in lecture. But seriously, great video as usual Joe
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Excellent video! My favorite argument is...let's start counting the days today and not stop until God ceases to exist. Since we cannot count to an actual infinity of days, God must cease to exist at some day. Let's have a nice funeral for God.
Aleph is pronounced with the "Ah" sound not "Ay" because that's a Hebrew letter.
I would love to see Craig attempt to dispute this
Waa just watching Alex discuss this in his Ben Shapiro video.
11:35 what??? Lol Why does it sometimes, sound like Slavoj Zizek is speaking 😂
I literally just got what they voice was lol
This is potential infinity and not infinity !
The great mathematician David Hilbert claimed that “the infinite is nowhere to be found in reality.”
Numbers are only potentially infinite, they can be drawn forever .
The "undrawn" numbers don't exist yet so they are not ACTUAL infinity, just potential.
As a general take, it would seem Craig has a fundamental misunderstanding of how infinite arethmetic and set theory work. Seeing as he no dedicated math education on the level of his philosophy PhD, I'm not surprised. This stuff is not intuitive.
I feel like I'm missing something here, but to me it doesn't actually seem like Hilbert's hotel ever accommodates new guests without removing old ones. For every guest you add, there is another guest who is displaced from his room and thus either doesn't currently have a room, or is in a room together with another yet to be displaced guest. The only difference is that those states are very short-lived for the individual guests while they are being shuffled around, but it doesn't change how for every guest you have added in the past, there will now be one guest who is currently in the process of being displaced to a new room.
You could even do the same thing in a normal hotel with finite rooms. Just make the guest from the last room loop back to the first one, and for the individual guests the effect will be the exact same as if they were in Hilbert's hotel and there are infinitely many new guests coming in.
The hotel manager could send a message over a PA system, telling all guests to move. It's a bit easier to imagine all guests moving at the same time.
@@amavect for that you have to assume that information travels instantly, which iirc leads to a bunch of problems just on its own
22:43 isn’t this part of the problem though? There ls always a room for someone to go to, but there wasn’t enough rooms to begin with to accommodate new guests, because every room is full. Thus the absurdity, how does pointing out that every guest will always have somewhere to go solve the absurdity, when that was part of the absurdity?
Great video so far by the way. Haven’t finished it but am planning to.
the most indepth indeed
2:01:45 relocating the problem
1:40:00 Craig self refutes Christianity,
2:20:40 Craig refuses to correct himself 20 years later
All the guests + more guests are also there: yes but the point is it’s still infinite that’s what makes it absurd 🤦♂️
These are different kinds of infinite sets. Where as Craig is mistakingly thinking they are same.
infinite set A - all guests in numbered rooms (1, 2 , 3 ...etc)
infinite set B - all guests in numbered rooms 4 and above (4, 5, 6 ... etc)
A and B - yes, the *cardinality of both are infinite. However the sets contain *different members. Set B is missing 1, 2 and 3.
This highlights where Craig's misunderstanding is - he think sets A and B are the same - they clearly aren't.
2:21:00 "it's 2 decades since this point has been made and craig still hasn't corrected himself"
Just like kent hovind keeps saying that evolution says we evolved from rocks. Craig is a professional rhetorician whose job is to argue for christianity. He's not a philosopher. A philosopher would care to correct himself. He, instead, has no love of knowledge.
Misunderstanding infinity also happens when people say: infinite possibilities; therefore everyting is possible. So infinitely many universes means that God exists in a universe.
That's wrong. With 10 digits you can make infinitely many numbers. But if you're only allowed to use 9 digits, you can also still make infinitely many numbers. Just no numbers with the digit 3 in it, for example. So in that infinite set of numbers without the digit 3, you won't find any number you can think of, because numbers containing 3 aren't possible in that set. Therefore: infinite does not equal any/every.
This is my favorite youtube video. So fun.
Jeez. Do they issue PhDs for watching this video?
Joe, you have some good work here, but it keeps diving into semantic equivocation. For example, the universe as the set of all things, and the universe with no beginning before the set of all things, the universe is a space-time manifold, without space there is no thing ness to discuss, no differentiation. This is my problem with the orthodox definition of the trinity, the three-ness implies a space-time manifold, which may be God, but it's not clear that is what trinitarians attempt to say. Was there thing-ness before the planck boundary? Or NON thing ness? The math breaks down and so do words, like a beginning with no time which is non-sense, there's no way to relate it to our sensory experience. You could say the square root of negative one is what was before the beginning, the language is imaginary but it may be useful if thought of in the right way, as a direction on the unit circle in a Cartesian system for example.
Regarding Craig's presentism and events in the past, I'm wondering if we aren't doing a tense switch here. "On Craig's presentism, though, past events are *also* not part of the actual world... if past events don't exist, then they aren't part of anything." Well, the thing they aren't part of is the set of actual events. That said, it seems to me they are still in the set of events that have ever been actual. That is, they aren't in the set of things that *are* real, but they *are* in the set of things that *were* real. That is, imagine the function R|E where the reality of E is only true at the time P(E) - when the event was present. We can still refer to events that go through that function, and can even quantify them or posit that they are infinite.
You're correct -- whether past events are 'part of the actual world' depends on how we're analyzing or understanding 'the actual world'. I was using an understanding which is commonplace in philosophy (and so expected Craig to go with that usage), but there are other ways to understand the term. Importantly, note that this is all compatible with my point in that portion of the video -- namely, that we can count the number of future events (whether they're actual or not, or part of the actual world or not), and that their number is aleph-null :)
1:27:30 Why do Americans use expressions such as "t negative 1" when it's " t minus 1" ? Negativity is a property of a number, so is oddness. To talk of one property and not another is "ahhhbzurd" (in Craigspeak). Nobody would say "t odd 1", would they?
This strange way of expressing the binary or unary operator of subtraction seems common in America - is it taught that way, and if so what is the justification?
Haha!
So, the binary operator of subtraction is always taught - at least for me and everyone I’ve met - as “minus”. So, I say “ten minus three is seven”. But whenever the “-“ is affixed to a number (as a unary operator), I’ve always been taught to say “negative __”. So, for instance, I will say that “negative five minus negative four is negative one”. It sounds like nails on a chalkboard to me to say “minus five minus minus four is minus one”. That also just seems confusing.
So, whenever “-“ is a unary operator, it’s “negative”, and whenever it’s binary, it’s “minus”.
If you 'disambiguate' your own use of cardinality in your first rebuttal, I think you'll find that you're making a similar mistake to the one you're accusing Dr. Craig of.
This is awesome! I will watch it later.
Btw I just finished watching Alex's video on Ben Shapiro I think Alex like Ed Feser's arguments .so I think u can do a discussion with him on those or if you want you can send your book to him since you won a war against Edward Feser .
I’ll be sending my book to him at the end of August - that’s when I’m turning it in to Springer🙂
@@MajestyofReason Awesome
@@MajestyofReason Damn Alex is lucky. When can we buy this book ?
@@deathnote4171 check the community post about this
But but but…it’s absurd!!! By my lights, not by math or anything.
Metaphysics or philosophy ?
I'm going to give these 5 stickers away to the first 5 people who buy them and continue to pay for them...🤔😂🤣
Who is this A. Leff Knowell fellow?