The question is analogous to the Grim Reaper Paradox, described by Chalmers here:
A slightly better example of prima facie without ideal positive conceivability may be the Grim Reaper paradox (Benardete 1964; Hawthorne 2000). There are countably many grim reapers, one for every positive integer. Grim reaper 1 is disposed to kill you with a scythe at 1pm, if and only if you are still alive then (otherwise his scythe remains immobile throughout), taking 30 minutes about it. Grim reaper 2 is disposed to kill you with a scythe at 12:30 pm, if and only if you are still alive then, taking 15 minutes about it. Grim reaper 3 is disposed to kill you with a scythe at 12:15 pm, and so on. You are still alive just before 12pm, you can only die through the motion of a grim reaper’s scythe, and once dead you stay dead. On the face of it, this situation seems conceivable — each reaper seems conceivable individually and intrinsically, and it seems reasonable to combine distinct individuals with distinct intrinsic properties into one situation. But a little reflection reveals that the situation as described is contradictory. I cannot survive to any moment past 12pm (a grim reaper would get me first), but I cannot be killed (for grim reaper n to kill me, I must have survived grim reaper n+1, which is impossible). So the description D of the situation is prima facie positively conceivable but not ideally positively conceivable.
As you point out later in the thread the light can never touch any given sphere, since no matter which one you pick there will always be another sphere in front of it to block the light. At the same time the light beam must eventually hit something because the centre sphere is in its way. So your light beam must both eventually hit a sphere and never hit a sphere so your system is contradictory and thus ill defined.
You could make the question answerable by instead asking for the limit of the light beam as number of steps of packing done goes to infinity in which case the light reflects back at 180°, since it does that in every step of the packing. Alternately you could ask what happens to the light beam if it is reflected of a shape which is the limit of the packing you described, in which case it will split in three since the shape produced is a cube (since it will have no empty spaces). (Edit:no it doesn’t the answer to this question is again undefined via the argument in the first paragraph, since the matter it bounced of of had to belong to some sphere)
Thank you I fixed it. I think the same argument shows that that question is also undefined. I think the real takeaway is that physics doesn’t deal well with some infinities.
Suppose there was some small ball inside of your super-packed structure that isn’t filled. Then we can fill this ball, and so the structure isn’t super-packed. It follows that the volume of the empty space inside of your structure has to be 0.
Now, what does your super-packed structure look like, given that it’s a empty cube that’s been filled?
EDIT: Nevermind, just saw that Villiam gave a similar answer.
The answer “cubes with no empty space are filled cubes” was perfectly decent, as was the verbal argument about limits that Viliam provided. But in case those weren’t satisfying, I’ve written a more explicit version of the proof* that the ray travels zero distance between reaching the cube and reaching a sphere. It utilizes the solution to the geometric sum, which is proved using limits:
https://en.wikipedia.org/wiki/Geometric_series
I think you can just argue by symmetry that the ray must retro-reflect.
By the symmetry you can argue that the ray hasn’t reach the sphere.
Lumifer suggested something very similar as you, but this retro-reflection is equally possible as the retro-shadowing. Which do you prefer and why? Can’t be both.
My intuition says the ray should bounce back immediately. The reason is how the spheres are packed in the cube: if you look along the main diagonal, there is the largest one in the center, and then there is a sequence of smaller spheres which are all aligned along the main diagonal. The ray shot into the cube along that diagonal would thus be aimed at the center of the corner-most sphere and so should just reflect at 180 degrees.
You are suggesting a kind of Einstein-Bose condensate, where a collective of particles becomes one.
Not at all.
I’m just saying that you have an infinite sequence of spheres with the property X. You’re saying that because the sequence is infinite I can’t point to the last sphere and therefore can’t say anything about it. I’m saying that because all spheres in this sequence have the property X, it doesn’t matter that the sequence is infinite.
I’m just saying that you have an infinite sequence of spheres with the property X. You’re saying that because the sequence is infinite I can’t point to the last sphere and therefore can’t say anything about it. I’m saying that because all spheres in this sequence have the property X, it doesn’t matter that the sequence is infinite.
This isn’t true in general. Each natural number is finite, but the limit of the natural numbers is infinite. Just because each of the intermediate shapes has property doesn’t mean the limiting shape has property X. Notably, in this case each of the intermediate shapes has a non-zero amount of empty space, but the limiting shape has no empty space.
Oh, dear. So all the spheres are shadowed, meaning the ray cannot reach any of them, so it just passes through the cube without even noticing it was there. Fee-fi-fo-fum, I smell Zeno!
I didn’t specifically mean the paradoxes, I meant Zeno as the first guy who said “Let’s throw some infinities into a physical-world scenario and see how many amphoras of wine do we need until it starts to make sense...”
Oh, I see. I strongly suspect, that the abstract world of pure mathematics doesn’t handle infinities any better than the real world does. It’s a fundamentally broken. I mean the infinity is a fundamentally broken concept.
Well, I don’t even believe in mathematics outside the physical world. It’s just a game of particles in Zeno’s head. Or Cantor’s or any other human’s head. Okay, heads, papers, blackboards, computers etc. Mathematics lives inside of physics only.
This sounds familiar. Since every point between your nose and your computer monitor has the property X, that there exists another point between it and your computer monitor, no matter how many points you move your nose through, there’s always more points it must go through before reaching the computer monitor. This is why, no matter how you try, you can never touch your nose to a computer monitor.
It’s not a Zeno’s paradox. You CAN touch the screen, if there is no obstacle between.
But here it’s a different story. Every sphere is shadowed, has an obstacle between itself and the incoming ray.
The finite amount of obstacles where there is then the outer obstacle—and everything is fine. Ray bounces back of this outer/unshadowed reflective sphere.
Pretty much like the Yablo’s list. If there is the last member of the list, everything is just fine.
If there is the last member of the list, everything is just fine.
There is no last point between your nose and your monitor.
Let me pose a different problem, to demonstrate what I think is wrong with your argument.
You start with a reflective square slab that is half as thick as it is wide. You place another slab, which is half as thick as the first on top, then another half as thick as that. You continue doing this until your stack of slabs is as tall as it is wide. If you direct an ideal ray at this slab, along the symmetry axis of the stack, what does the ray do?
This situation is, once constructed, identical to the first ray described in your problem (the one that hits the face of the cube), but your argument would imply that the ray won’t reflect back, because whichever slab it would reflect from has another slab shadowing it.
this is your transparent cube, packed with distinct reflective spheres of different sizes. There is no room for another sphere inside, no matter how small it would be.
Taken literally, this means there is an infinite amount of spheres, some of them (actually, the vast majority of them) smaller than the size of an atom. What are they made of?
Now you beam a ray in the direction of one of the cubes main diagonals – from outside toward the center.
The machine that simulates our universe will crash with an “out of memory” error, because this would require the simulator to spawn infinitely many small spheres at the corner of the cube when trying to determine which sphere will the photons hit first.
If the ray is supposed to be just an infinitely small line, the question is somewhat similar to asking “how much is zero divided by zero?”—there are multiple different answers that can be defended by approaching the problem from a different point of view. In general, the answer is undefined because the question is… wrong.
With real photons, there is the additional question of how they would interact with the hypothetical objects of extremely small size. Maybe they would just skip the first few “too small” spheres, and only interact with the first one larger than Plack size? I give up here, because I am not sufficiently familiar with physics to answer this.
In mathematics we deal with infinities by using limits—essentially by definiting a sequence of finite scenarios that gradually approach the desired situation, and observe how the answer develops for these finite scenarios.
So, by gradually adding the reflective spheres we find that: 1) the diagonal ray bounces back; 2) from a place that becomes nearer and nearer the corner as we keep adding the spheres.
The answer is that limit of the ray behavior is the ray bouncing back from the corner of the cube.
However, the answer depends on how the sequence of the scenarios was constructed; namely that we keep adding the spherese, but the ray already aims to the diagonal perfectly. Small changes, such as assuming that the ray is also getting gradually more precise as we keep adding the spheres (becoming perfectly precise only at the time all the spheres are added), could justify reflection in any angle. Or bouncing between the spheres.
Infinity doesn’t exist. Some people say there is a difference between a “potential infinity” and “actual infinity”, but that seems like a wrong way to use language; it would be more precise to say that we can imagine processes that continue indefinitely—ignoring the technical details of running out of resources, heat death of the universe, etc. -- but nothing can be infinite here and now. Or, to put it shortly, “infinite” only exists as an adjective describing processes, but never as a noun.
There are also mathematicians that pretend otherwise and talk about various infinities, effectively extending their maps beyond the territory. But magic always comes with a price, and in this case the price is that there are multiplepossible maps (which all fit the territory, and only disagree about the things happening outside of it) and then the mathematicians start arguing about which one should they use, because there is no way to choose the “true” one (by abandoning the territory, all we can talk about is self-consistence, and you can have multiple different self-consistent maps), so instead they try to choose the most “elegant” one, which is kinda subjective, because even when someone finds weird things happening on a map, the owner of the map will proudly proclaim that this is not a bug, but a feature. -- This leads to a lot of confusion even among people who study mathematics, probably because the few people who understand what this is all about find it too obvious to mention.
Uhm, back to the point: infinity doesn’t exist. If we want to debate something meaningful, we can only debate properties of some indefinite process. For example, a process of “adding more small spheres inside the cube, while a ray shines at it diagonally”. But it is a process without an end. There will never be a moment when there are literally infinitely many spheres inside. We can only ask whether the process has some properties that converge towards some values.
Now I can answer your questions:
Really? What is the diameter of the sphere, which reflected the ray back? And how come, that a smaller sphere didn’t shadow this ray?
In each step of the process of “adding more small spheres inside the cube” it was a different sphere that reflected the ray. In each subsequent step, a smaller one. In each step the first sphere in the way of the ray wasn’t shadowed by yet smaller sphere, because the smaller sphere wasn’t there yet.
This is all there is. The cube will never be actually completed. Debates about what would happen if we send there a ray after it is completed… don’t necessarily have to correspond to anything real. By making the assumption that the infinity already happened we have left the realm of reality. Now it’s all about “my map is prettier than your map”.
But tell this to the 99% of the top mathematicians! For them all those infinities—infinite multitude of them—exist.
...I wonder whether this is true. I mean, as long as someone knows that you can have multiple mutually incompatible systems of axioms, then they should on some level realize that the choice of a specific system is arbitrary.
I mean, I haven’t realized this for a long time simply because I read have some popular books about infinities probably already at elementary school age, and they didn’t start with a big red warning “everything you will read in this book is merely a consequence of an arbitrarily selected set of axioms; other sets of axioms could lead to completely different outcomes; and the choice of the axioms is arbitrary because nothing of this is real, mwa-ha-ha-ha!” So, I naively assumed it was just business as usual. If I didn’t suspect books about prime numbers or complex numbers, why should I suspect the books about infinities?
And when I later heard something about the axioms, the idea of the “one true system of aleph-zero, aleph-one, etc.” was already firmly stuck in my head. So even while I was reading about how different axioms are possible, I still on some level assumed that the ones I was already familiar with are—in some sense—the correct ones (without asking myself what exactly the word “correct” could even possibly mean in this context).
Only a few years later I have finally connected the dots in my head. It was while thinking about natural numbers, the fifth Peano’s axiom, how without it we could have different models of natural numbers, and what exactly that means. And a few years later, my friend complained about the axiom of choice, and suddenly something clicked in my head, and I was like “wait, isn’t that the same thing as with the fifth Peano’s axiom? like you can have different models that all satisfy the same set of axioms, but they still differ at some underspecified places, and you need an extra axiom to distinguish between them?” Because at that moment I already had an understanding of what “having different models” means; it just never occured to me to apply it to infinities. (Despite seeing some hints which in hindsight obviously tried telling me to do so.)
In my defense, though, I don’t remember studying about infinities at university. Probably not even at high school, or at least not so deeply. So all my knowledge of infinities came from the popular books I have read at childhood; it shouldn’t be surprising it was seriously incomplete. I assume that people who actually studied this stuff at university, who were explicitly told about the existence of various axiomatic systems and given specific examples thereof, and then had exercises and exams… I believe they were in much better position to make the connection. So it would surprise me a lot if the best of them just couldn’t connect the dots even better than I did.
On the other hand, sometimes even highly intelligent and educated people fail to connect the dots.
I admit I am somewhat tempted to write an e-mail to Terence Tao and ask him what he thinks about this. There was never a better excuse to bother people much smarter than me. :D
I think you underestimate the zel of the 1000 top mathematicians about this infinity business. Perhaps there is about 10 dissidents among them, who are finitists.
But for all the others, the concept of infinity is a fundamental one. It’s easier to find a Catholic bishop who doesn’t care about God or Trinity, than a top mathematician who doesn’t believe and love infinity.
The infinity mythology is quite a rich one. Surprisingly rich, indeed.
First, we have the countable infinity—like all the natural numbers, or all the rational numbers. Aleph 0 is the cardinal of those. Then, we have 2^aleph0 or aleph1. The number of all real numbers. Between those two, you are free to postulate some greater then aleph0 and smaller than aleph1. You are also free to postulate that there is no cardinal between aleph zero and aleph one. It’s the Cohen’s solution of the so called Continuum problem.
You have more alephs above aleph 1. One aleph for every natural number. And you are free to postulate (or not) cardinal numbers between each two.
Then you have trans aleph big cardinal numbers.
Then you have non-constructible big infinite cardinals.
And then you have a lot of ordinals.
And then you have the Axiom of Choice. You can accept it or not. But it is proven, that if the whole structure of infinity is broken. then is broken with or without the Axiom of Choice.
I suspect Terence Tao has no strong opinion about ZF or ZFC system. He loves prime numbers and all the natural and real numbers and so on. But I don’t think he is in Set Theory very much.
This is my stupid question:
https://protokol2020.wordpress.com/2016/12/14/geometry-problem/
Do not hesitate to patronize me, or whatever does it take, I’d really like to have an answer.
The question is analogous to the Grim Reaper Paradox, described by Chalmers here:
You are right. This is actually the same problem. The problem of the (math) infinity magic itself.
As you point out later in the thread the light can never touch any given sphere, since no matter which one you pick there will always be another sphere in front of it to block the light. At the same time the light beam must eventually hit something because the centre sphere is in its way. So your light beam must both eventually hit a sphere and never hit a sphere so your system is contradictory and thus ill defined.
You could make the question answerable by instead asking for the limit of the light beam as number of steps of packing done goes to infinity in which case the light reflects back at 180°, since it does that in every step of the packing. Alternately you could ask what happens to the light beam if it is reflected of a shape which is the limit of the packing you described, in which case it will split in three since the shape produced is a cube (since it will have no empty spaces). (Edit:no it doesn’t the answer to this question is again undefined via the argument in the first paragraph, since the matter it bounced of of had to belong to some sphere)
It’s not a cube. The corner points for example, are NOT covered by any sphere. Its a cube MINUS infinitely many points.
On the edges, for example, only aleph zero points are covered and aleph one many—aren’t.
The limit technique you employ here, does not apply at all.
Thank you I fixed it. I think the same argument shows that that question is also undefined. I think the real takeaway is that physics doesn’t deal well with some infinities.
The question may be flawed in a way that I don’t see it.
Or the question may be flawed not by my mistake, but by a mistake already built in R^3 or R^4 math.
I think, it’s the later.
Maybe think about the problem this way:
Suppose there was some small ball inside of your super-packed structure that isn’t filled. Then we can fill this ball, and so the structure isn’t super-packed. It follows that the volume of the empty space inside of your structure has to be 0.
Now, what does your super-packed structure look like, given that it’s a empty cube that’s been filled?
EDIT: Nevermind, just saw that Villiam gave a similar answer.
The answer “cubes with no empty space are filled cubes” was perfectly decent, as was the verbal argument about limits that Viliam provided. But in case those weren’t satisfying, I’ve written a more explicit version of the proof* that the ray travels zero distance between reaching the cube and reaching a sphere. It utilizes the solution to the geometric sum, which is proved using limits: https://en.wikipedia.org/wiki/Geometric_series
The proof (derivation?) is here: http://imgur.com/vSfABHk
I think you can just argue by symmetry that the ray must retro-reflect.
*At least, it looks like a proof to a physicist. It may not meet the standards of a proof to a mathematician.
By the symmetry you can argue that the ray hasn’t reach the sphere.
Lumifer suggested something very similar as you, but this retro-reflection is equally possible as the retro-shadowing. Which do you prefer and why? Can’t be both.
Can you explain?
My intuition says the ray should bounce back immediately. The reason is how the spheres are packed in the cube: if you look along the main diagonal, there is the largest one in the center, and then there is a sequence of smaller spheres which are all aligned along the main diagonal. The ray shot into the cube along that diagonal would thus be aimed at the center of the corner-most sphere and so should just reflect at 180 degrees.
You are 100% right about that.
That one doesn’t exist. It’s always at least one still closer to the corner.
I didn’t say any particular one. The reasoning applies to all of them so it doesn’t matter.
You are suggesting a kind of Einstein-Bose condensate, where a collective of particles becomes one.
An interesting idea, but … I doubt it’s correct way of thinking about this.
Not at all.
I’m just saying that you have an infinite sequence of spheres with the property X. You’re saying that because the sequence is infinite I can’t point to the last sphere and therefore can’t say anything about it. I’m saying that because all spheres in this sequence have the property X, it doesn’t matter that the sequence is infinite.
This isn’t true in general. Each natural number is finite, but the limit of the natural numbers is infinite. Just because each of the intermediate shapes has property doesn’t mean the limiting shape has property X. Notably, in this case each of the intermediate shapes has a non-zero amount of empty space, but the limiting shape has no empty space.
That no ray can come to them from that direction, since it’s eclipsed by the smaller ball toward the corner?
If this is the property X, then all those spheres have it, yes!
Oh, dear. So all the spheres are shadowed, meaning the ray cannot reach any of them, so it just passes through the cube without even noticing it was there. Fee-fi-fo-fum, I smell Zeno!
So it’s looks like.
This does not follow at all.
You should rather smell Yablo.
I didn’t specifically mean the paradoxes, I meant Zeno as the first guy who said “Let’s throw some infinities into a physical-world scenario and see how many amphoras of wine do we need until it starts to make sense...”
Oh, I see. I strongly suspect, that the abstract world of pure mathematics doesn’t handle infinities any better than the real world does. It’s a fundamentally broken. I mean the infinity is a fundamentally broken concept.
Well, I don’t even believe in mathematics outside the physical world. It’s just a game of particles in Zeno’s head. Or Cantor’s or any other human’s head. Okay, heads, papers, blackboards, computers etc. Mathematics lives inside of physics only.
I don’t know about that. It’s a model. “All models are wrong, but some are useful”—George Box
On the other hand: “God created the integers and the rest is the work of man” :-/
This sounds familiar. Since every point between your nose and your computer monitor has the property X, that there exists another point between it and your computer monitor, no matter how many points you move your nose through, there’s always more points it must go through before reaching the computer monitor. This is why, no matter how you try, you can never touch your nose to a computer monitor.
It’s not a Zeno’s paradox. You CAN touch the screen, if there is no obstacle between.
But here it’s a different story. Every sphere is shadowed, has an obstacle between itself and the incoming ray.
The finite amount of obstacles where there is then the outer obstacle—and everything is fine. Ray bounces back of this outer/unshadowed reflective sphere.
Pretty much like the Yablo’s list. If there is the last member of the list, everything is just fine.
There is no last point between your nose and your monitor.
Let me pose a different problem, to demonstrate what I think is wrong with your argument.
You start with a reflective square slab that is half as thick as it is wide. You place another slab, which is half as thick as the first on top, then another half as thick as that. You continue doing this until your stack of slabs is as tall as it is wide. If you direct an ideal ray at this slab, along the symmetry axis of the stack, what does the ray do?
This situation is, once constructed, identical to the first ray described in your problem (the one that hits the face of the cube), but your argument would imply that the ray won’t reflect back, because whichever slab it would reflect from has another slab shadowing it.
Look! Need not to be the last point. Zeno is resolved by infinite sums, which can give us finite numbers.
Yablo isn’t resolved and this is related to Yablo, not to Zeno paradox.
Taken literally, this means there is an infinite amount of spheres, some of them (actually, the vast majority of them) smaller than the size of an atom. What are they made of?
The machine that simulates our universe will crash with an “out of memory” error, because this would require the simulator to spawn infinitely many small spheres at the corner of the cube when trying to determine which sphere will the photons hit first.
If the ray is supposed to be just an infinitely small line, the question is somewhat similar to asking “how much is zero divided by zero?”—there are multiple different answers that can be defended by approaching the problem from a different point of view. In general, the answer is undefined because the question is… wrong.
With real photons, there is the additional question of how they would interact with the hypothetical objects of extremely small size. Maybe they would just skip the first few “too small” spheres, and only interact with the first one larger than Plack size? I give up here, because I am not sufficiently familiar with physics to answer this.
We are talking about mathematics here, an abstract (unreal) world. Not about physics. It’s geometry in the sense of Euclides.
It’s not everything “division by zero error”. Still, it’s a nice try, thank you.
In mathematics we deal with infinities by using limits—essentially by definiting a sequence of finite scenarios that gradually approach the desired situation, and observe how the answer develops for these finite scenarios.
So, by gradually adding the reflective spheres we find that:
1) the diagonal ray bounces back;
2) from a place that becomes nearer and nearer the corner as we keep adding the spheres.
The answer is that limit of the ray behavior is the ray bouncing back from the corner of the cube.
However, the answer depends on how the sequence of the scenarios was constructed; namely that we keep adding the spherese, but the ray already aims to the diagonal perfectly. Small changes, such as assuming that the ray is also getting gradually more precise as we keep adding the spheres (becoming perfectly precise only at the time all the spheres are added), could justify reflection in any angle. Or bouncing between the spheres.
They are tightly packed. How could they “bounce between spheres”?
Really? What is the diameter of the sphere, which reflected the ray back?
And how come, that a smaller sphere didn’t shadow this ray?
Okay, I’ll try to be more explicit about this:
Infinity doesn’t exist. Some people say there is a difference between a “potential infinity” and “actual infinity”, but that seems like a wrong way to use language; it would be more precise to say that we can imagine processes that continue indefinitely—ignoring the technical details of running out of resources, heat death of the universe, etc. -- but nothing can be infinite here and now. Or, to put it shortly, “infinite” only exists as an adjective describing processes, but never as a noun.
(Historical trivia: I found this argument in Alfred Korzybski’s Science and Sanity. That’s the “the map is not the territory” guy.)
There are also mathematicians that pretend otherwise and talk about various infinities, effectively extending their maps beyond the territory. But magic always comes with a price, and in this case the price is that there are multiple possible maps (which all fit the territory, and only disagree about the things happening outside of it) and then the mathematicians start arguing about which one should they use, because there is no way to choose the “true” one (by abandoning the territory, all we can talk about is self-consistence, and you can have multiple different self-consistent maps), so instead they try to choose the most “elegant” one, which is kinda subjective, because even when someone finds weird things happening on a map, the owner of the map will proudly proclaim that this is not a bug, but a feature. -- This leads to a lot of confusion even among people who study mathematics, probably because the few people who understand what this is all about find it too obvious to mention.
Uhm, back to the point: infinity doesn’t exist. If we want to debate something meaningful, we can only debate properties of some indefinite process. For example, a process of “adding more small spheres inside the cube, while a ray shines at it diagonally”. But it is a process without an end. There will never be a moment when there are literally infinitely many spheres inside. We can only ask whether the process has some properties that converge towards some values.
Now I can answer your questions:
In each step of the process of “adding more small spheres inside the cube” it was a different sphere that reflected the ray. In each subsequent step, a smaller one. In each step the first sphere in the way of the ray wasn’t shadowed by yet smaller sphere, because the smaller sphere wasn’t there yet.
This is all there is. The cube will never be actually completed. Debates about what would happen if we send there a ray after it is completed… don’t necessarily have to correspond to anything real. By making the assumption that the infinity already happened we have left the realm of reality. Now it’s all about “my map is prettier than your map”.
I agree, that this is the solution. I completely and absolutely agree with you. It’s a false (mad if you wish) construction.
But tell this to the 99% of the top mathematicians! For them all those infinities—infinite multitude of them—exist.
What I don’t agree with you, is:
It cannot continue “indefinitely”—it will stop rather soon.
But yes … I gave this solution as the possibility number 6.
6. This is an illegal or ill defined question
You have answered correctly at the bottom line, Viliam. By my judgement. But that’s just my judgement, of course.
I disagree with your reasoning above the bottom line, but that’s not that important as your (and mine) final answer.
Okay, I am happy we came to an agreement, but...
...I wonder whether this is true. I mean, as long as someone knows that you can have multiple mutually incompatible systems of axioms, then they should on some level realize that the choice of a specific system is arbitrary.
I mean, I haven’t realized this for a long time simply because I read have some popular books about infinities probably already at elementary school age, and they didn’t start with a big red warning “everything you will read in this book is merely a consequence of an arbitrarily selected set of axioms; other sets of axioms could lead to completely different outcomes; and the choice of the axioms is arbitrary because nothing of this is real, mwa-ha-ha-ha!” So, I naively assumed it was just business as usual. If I didn’t suspect books about prime numbers or complex numbers, why should I suspect the books about infinities?
And when I later heard something about the axioms, the idea of the “one true system of aleph-zero, aleph-one, etc.” was already firmly stuck in my head. So even while I was reading about how different axioms are possible, I still on some level assumed that the ones I was already familiar with are—in some sense—the correct ones (without asking myself what exactly the word “correct” could even possibly mean in this context).
Only a few years later I have finally connected the dots in my head. It was while thinking about natural numbers, the fifth Peano’s axiom, how without it we could have different models of natural numbers, and what exactly that means. And a few years later, my friend complained about the axiom of choice, and suddenly something clicked in my head, and I was like “wait, isn’t that the same thing as with the fifth Peano’s axiom? like you can have different models that all satisfy the same set of axioms, but they still differ at some underspecified places, and you need an extra axiom to distinguish between them?” Because at that moment I already had an understanding of what “having different models” means; it just never occured to me to apply it to infinities. (Despite seeing some hints which in hindsight obviously tried telling me to do so.)
In my defense, though, I don’t remember studying about infinities at university. Probably not even at high school, or at least not so deeply. So all my knowledge of infinities came from the popular books I have read at childhood; it shouldn’t be surprising it was seriously incomplete. I assume that people who actually studied this stuff at university, who were explicitly told about the existence of various axiomatic systems and given specific examples thereof, and then had exercises and exams… I believe they were in much better position to make the connection. So it would surprise me a lot if the best of them just couldn’t connect the dots even better than I did.
On the other hand, sometimes even highly intelligent and educated people fail to connect the dots.
I admit I am somewhat tempted to write an e-mail to Terence Tao and ask him what he thinks about this. There was never a better excuse to bother people much smarter than me. :D
I think you underestimate the zel of the 1000 top mathematicians about this infinity business. Perhaps there is about 10 dissidents among them, who are finitists.
But for all the others, the concept of infinity is a fundamental one. It’s easier to find a Catholic bishop who doesn’t care about God or Trinity, than a top mathematician who doesn’t believe and love infinity.
The infinity mythology is quite a rich one. Surprisingly rich, indeed.
First, we have the countable infinity—like all the natural numbers, or all the rational numbers. Aleph 0 is the cardinal of those. Then, we have 2^aleph0 or aleph1. The number of all real numbers. Between those two, you are free to postulate some greater then aleph0 and smaller than aleph1. You are also free to postulate that there is no cardinal between aleph zero and aleph one. It’s the Cohen’s solution of the so called Continuum problem.
You have more alephs above aleph 1. One aleph for every natural number. And you are free to postulate (or not) cardinal numbers between each two.
Then you have trans aleph big cardinal numbers.
Then you have non-constructible big infinite cardinals.
And then you have a lot of ordinals.
And then you have the Axiom of Choice. You can accept it or not. But it is proven, that if the whole structure of infinity is broken. then is broken with or without the Axiom of Choice.
I suspect Terence Tao has no strong opinion about ZF or ZFC system. He loves prime numbers and all the natural and real numbers and so on. But I don’t think he is in Set Theory very much.