If you want to start with curves (or continuous objects) in order to derive points you need a systematic way of talking about them. — jgill

Agreed. But you've got to start somewhere and fortunately it's easy to understand what I'm talking about, even though I'm talking informally.

If you plan to assemble contours or curves by gluing together tiny straight lines, then you are doing nothing more than is done when one calculates the length of a contour in complex analysis. — jgill

No, I'm going the other way around. I'm not proposing a whole-from-part (curve from infinitesimal line segments) construction but a part-from-whole (smaller curves from larger curves) construction.

So my comment is what difference does it make how you deal with that? — jgill

In one sense, physics would be unaffected by a new philosophy of mathematics because applied mathematics doesn't deal with actual infinities. However, in another sense a new philosophy of mathematics might yield deep insight into the philosophy of physics (in particular, Quantum Mechanics). Philosophy has a powerful way of shaping our worldview.

We've had posters here who spend years working up what they consider astounding revelatory articles, only to be more or less ignored. They become so enraptured with their ideas they get caught in that spiral in which the more effort you exert the more you think your product is of value, losing their objectivity. — jgill

No matter how good I think some of my ideas are, they are informal and several years old and over the years I've shed all my great expectations. At this point, I'm just here to talk about it.

Depends on what you mean by 'manipulate'. — TonesInDeepFreeze

I mean what a computer does, either to the virtual objects or the bits themselves, when it computes.

Indeed, if no one mentioned a connection between the paradoxes and set theory, I'd have no objection at all! — TonesInDeepFreeze

Let's start with my main claim - Nothingness (i.e. points) cannot be assembled to form a something (i.e. a continuum), no matter how much of it you have. Even infinite nothingness is still nothingess.

You are completely missing the point. And now your lily pad jumping from Thompson's lamp to Zeno.

I gave you a rigorous answer regarding Thompson's lamp, but instead of comprehending that, you skip it as if it doesn't exist and frog hop to another pad.

You're not in good faith. — TonesInDeepFreeze

I honestly thought you were covering both Thompson's Lamp and Zeno's Paradox simultaneously since you included the geometric series in your description. I took it that you were giving an algorithm on how to construct the following table:

Step #,incremental distance, current state of lamp
0, 1, on
1, 1/2, off
2, 1/4, on
3, 1/8, off
etc.

I would like to add an additional column to the table to represent the total distance travelled.

Step #, incremental distance, current state of lamp, dist. travelled
0, 1, on, 1
1, 1/2, off, 1+1/2
2, 1/4, on, 1+1/2+1/4
3, 1/8, off, 1+1/2+1/4+1/8
etc.

The pressing question is whether there is a last row to this table - one might call it row aleph_0. On one hand, we say that there is no last row because we want to say that there is no final state of the lamp. However, on the other hand, we imply (but don't explicitly state) that there is a last row because Achilles completes the journey and the total distance travelled equals exactly 2. There is no finite Step # for which the distance travelled is exactly 2.

mathematics shows how to calculate that Achilles did finish the race. — TonesInDeepFreeze

I argue that any valid demonstration uses a parts-from-whole (points-from-continuum) construction.

With the mathematics, we have that certain functions are continuous. This allows that mathematics is modelled by such things as Achilles crossing the finish line. And we know that Achilles does cross the finish line. So the mathematics works. The paradox though is a model in which Achilles does not finish the race. So we're not so attracted to that model, and indeed thankful that it is not a model of the mathematics. — TonesInDeepFreeze

Aha! Your answer rests upon continua, not points! The mathematics works because you're considering the journey as a whole. The complete journey exists first and only then can we choose to talk about what happens at t=0.5 or t=0.95 or any other instant. But we cannot talk about the journey starting from t=0 proceeding to the adjacent instant.....because there is no adjacent instant. Atalanta cannot even begin her journey.

@TonesInDeepFreeze: You complained that I skipped some of your arguments so I'm going back to them.

All bijections are injections. So you're confused to begin with. — TonesInDeepFreeze

I knew I'd make a mistake with those terms, but at least you knew what I was trying to say.

You're wasting our time. We already know that in set theory, infinite sets differ in this salient way from finite sets. — TonesInDeepFreeze

The way I would phrase it is that we already know that in set theory, infinite sets don't conform at all to the intuitions we've developed from all sets that we've actually worked with direction.

There's no law of the universe that one can't just chug along proving theorems without declaring a philosophical position. — TonesInDeepFreeze

It reminds me of the 'shut up and calculate' phrase that sometimes is said when people try to interpret the meaning of Quantum Mechanics. I agree that we can shut up but I think philosophy is important. I believe that with a proper philosophy of mathematics we can refine our intuitions and apply them in our quest to understand our universe. I also believe that paradoxes are the most important guidepost in our quest to see truth.

so you get a contradiction but outside the original terms you set: an imaginary world corresponding to set theory. — TonesInDeepFreeze

I believe all worlds (physical and abstract) are simulated by computers. Not necessarily a computer made of silicon, just an entity that computes. One might say that 'God is a mathematician'. I'm not trying to squeeze set theory into a specific world, I'm just trying to imagine a computer with enough capacity to hold it. If you say that set theory is a finite set of rules, axioms, properties, etc. from which theorems could be derived then Set Theory could exist on a finite computer and all is good.

However, if you say that set theory describes the behavior and asserts the existence of another computer which directly works with the sets themselves then such a computer would necessarily be infinite in capacity (and infinite in speed as well to complete infinite operations in finite time).

I'm willing to begin by assuming that such a computer exists, and then explore whether it would explode. I do not have the skill to prove that the computer that holds set theory will explode. However, I think I have the skill to at least discuss whether the computer that holds the infinity paradoxes will explode.

You require that sets can be "built" only in finite "processes". — TonesInDeepFreeze

I'm willing to explore computers that perform supertasks, like one that simulates Zeno's Paradox where he performs infinite steps in finite time. And IF that computer explodes and IF I can form a 1-to-1 correspondence between the processes on that computer and the processes on the computer that holds the infinite sets then it's reasonable to say that that computer also explodes. If you think I haven't proven this yet...well that is true!

In Set Theory we say 'There exists a set...'. What do we mean by this?
— keystone — TonesInDeepFreeze

When I say something exists I mean there is a computer where it is in memory. Today at work I simulated fluid flowing in a mixing tank so during the simulation that virtual mixing tank existed, even though there was no physical counterpart. I just can't envision any computer holding even just the natural numbers without exploding.

So when you write "For a given model M, ExPx is true in M iff there is at least one member of the domain such that that member of the domain is also in the subset of the domain that is the denotation of 'P'", are you assuming the existence of an infinite computer?

'Infinite sets are empty' is a contradiction....
— TonesInDeepFreeze

How is that a contradiction?
— keystone

I'm sorry, but are you serious? — TonesInDeepFreeze

If I write N = {1, 2, 3, ...} it seems that N has infinite elements. But appearances can be decieving. If someone proved that 1=2=3=... then N actually only contains one element.

Here's another answer: Because if it had only finitely many points, we wouldn't be able talk about points greater than any of the positive points you chose or less than any of the negative points you chose. And we wouldn't be able to talk about those that are in between any two adjacent points. — TonesInDeepFreeze

I'm not proposing that the real line is composed of an unchanging, finite set of points. I'm proposing that the real line has infinite potential to 'give birth' to points as they are needed. You never did respond to my post where I drew a circle for you. That post depicts what I mean, but in any case, I think the following paragraph will as well.

If you're a teacher and you tell the students to measure a table, you wouldn't hand them a bunch of nothing (points) to make the measurements. You would hand them a ruler (continuum) which has perhaps tic marks every 1/8 inch. If the table lines up halfway in between the tic marks, then and only then do you add a new tic mark based on the average of the adjacent tic marks. And if matter weren't discrete, you would have the potential to endlessly add tic marks to the ruler. It's just that you never would complete the job. And in fact, if you somehow were able to completely populate the ruler with tic marks the ruler becomes useless. It's just one big black tic mark. In this example, The distinction between numbers is lost and the set of real numbers is no more useful than a set containing only one element.

In my view, the ruler comes before the tic marks. And because we can manipulate the ruler as we go, we can speak about it's potential in greatest generality no matter what degree of tolerance you might need. If we took out the axiom of infinity, then we would acknowledge that we could never produce a useless ruler. We could never cut a string to the point where it vanishes to nothingness.

Here's my airplane analogy. Smart people build a well engineered airplane but before they launch it they go through a non-scientific ritual of blessing the plane to ensure it will fly. I'm acknowledging that they're good engineers and they've built a good plane but is that ritual really necessary? Is an actually infinity of points really necessary?

Your proposal is finitism. It's a cool proposal and an interesting philosophical topic on its own right, but entirely unrelated to the paradox you were trying to solve (i.e. irrelevant). — Kuro

If it doesn't make sense to think that something comes from nothing, maybe we need to revisit the belief that something comes from nothing.

If it doesn't make sense to think that continua come from points (i.e. dartboard paradox), maybe we need to revisit the belief that continua come from points.

What about the "final state"? There is no final state. — TonesInDeepFreeze

In Achilles' journey he arrives at the destination. He takes a final step. Does Set Theory model Achilles' journey or not?

It is the very point that mathematics is not capable of such nonsense. — TonesInDeepFreeze

If mathematics does not allow for such nonsense then it cannot claim to resolve Zeno's paradox.

Basically, you have your concepts of how the statements of set theory should be understood, so you posit a realm to embody those concepts. But your concepts are not analogues of any actual statements in set theory. So it's a cheat. — TonesInDeepFreeze

I don't think you're being reasonable here. You say that many of the paradoxes are resolved by Set Theory (essentially it is you who is making the link between the two) but then you say that criticisms of the paradox can't touch Set Theory. You can't have it both ways.

In Zeno's Paradox Achilles only travels the distance after completing an infinite process. For example, if Set Theory has nothing to say about completed infinite processes then it cannot be used address Zeno's Paradox.

By the way, are you capitalizing 'Platonic Realm' as a proper noun to shade discussion, even if just by hair, about mathematical realism? — TonesInDeepFreeze

Maybe just to make it clear that we're talking about an actual place.

When math starts with points that are then assembled into curves, there is a way of describing those points on the real line, identifying a point with .5 for example. If you are starting out with curves or geometric figures you need to be classify them, order them somehow, for you then wish to create points by intersections I suppose. You need rigorous definitions for curves, then an axiomatic structure. All of which seems far-fetched. But who knows? — jgill

Why do we need to order the curves?

I'm referring to the fact that set theory proves there exists a complete ordered field and a total ordering of its carrier set. And then I highlighted that the carrier set is the set of real number and the total ordering is the standard less-than relation on the set of real numbers. — TonesInDeepFreeze

You've repeated this a lot so it's clearly important. Maybe my view is in disagreement with Set Theory then. In any case, I'm not in a position to challenge Set Theory directly. I'll continue this in a response to another one of your posts.

'Infinite sets are empty' is a contradiction. And set theory does not have that contradiction. So if it's your intuition, then set theory is not for you. — TonesInDeepFreeze

How is that a contradiction?

Can you provide the simplest possible example in calculus where we need to assume that there are infinitely many points?
— keystone

Day one of high school Algebra 1:

"Students, we start with the set of real numbers and the real number line." — TonesInDeepFreeze

I know that we are taught that the real line is composed of infinite points. What I'm asking for is you to provide an example where we need to assume that the real line is composed of infinite points. Saying we do things a certain way is not evidence that we need to do it that way.

But it turns out his "fair case" is just to wave his magic wand (actually he uses an old drum stick) and say that the natural numbers are the foundation for all the other branches of mathematics. But he says not a single word showing how that would be done except for a chart with 'the natural numbers' as the base of the pyramid of mathematical subjects. — TonesInDeepFreeze

I agree that this is the biggest weakness of his view. He complains about the foundations of math but his foundation is set of tick marks on a white board.

This is Russell's argument. — apokrisis

You seem to be well read in math, philosophy, and science. Out of curiosity, what are you trained in?

But the reverse argument also applies. The representation can be sharper than what it represents. The right facial recognition algorithm could separate a dim CCTV image of Keystone in a hoody from all the other faces stored in a police data bank. Signal processing can extract structural information that stands behind any amount of confusing surface detail. — apokrisis

I disagree. While an algorithm can indeed tune an image to make it better suited for the intended audience (e.g. colorize by infrared for those who are unable to see in the infrared spectrum), it cannot add these colors if the infrared is not actually there in the first place. No computer processing can make the image higher resolution than reality. Or more mathematically, if I tell you that I'm thinking of a number that begins with 3.14159 there is no way that a computer can increase the precision of this number to tell me the next digit.

But how wide are your lines – even mathematically? — apokrisis

For a fully accurate depiction, the line width should be exactly 0. I can't accomplish that with a drawing. Perhaps I should have cut a paper instead, but that wouldn't work perfectly either as the paper would fold and bend and cast shadows and leave small gaps. Even better, I could have made a stereogram whereby each area pops out at a different height. But alas, I am lazy so you have to forgive that my lines have some width when in reality they should not.

How sure are you they are single lines and not a small bundle of lines sharing a neighbourhood with infinitesimal spacing? And when does this vagueness start to matter? — apokrisis

About as sure as I am that there is no teapot orbiting between the Earth and Mars revolving about the sun in an elliptical orbit. Give me a reason to believe otherwise. I don't believe in infinitesimals either so you'll need to also give me a reason to believe in them too. The figure (Apokrisis' face) is exact and so is the measurement (the photo of your face), it's just that they're not equivalent. Do you consider this mismatch to be vagueness?

Doesn't it matter if your rigorous mathematical edifice must also fit a physical world were nonlinearity is in fact the generic condition? — apokrisis

How does my view break down with nonlinearity?

But that doesn't engage with the foundational issue of whether reality itself is vague or crisp at base. And hence what kind of ontology we are correct to import into our "picturing" of math's epistemology. — apokrisis

Perhaps we need to figure out whether the objects of mathematics are vague or crisp at base. This might help us guide our investigations into reality. This is why I think it's so important to have a sound philosophy for mathematics.

How would you catalogue all continuous curves? That would be a starting "point". In order to have derivatives and integrals you would need some kind of function derivable from a catalogued example. Sorry, but the whole approach sounds absurd. — jgill

Can you explain what you mean by 'catalogue all continuous curves'?

There's an Australian mathematician, Norman Wildberger, on YouTube who doesn't accept infinities.
Here's a link to one of his videos.
Difficulties with real numbers as infinite decimals
https://www.youtube.com/watch?v=tXhtYsljEvY
You might try contacting him. — Art48

Thanks. I have admired him for many years. I loved his videos on real numbers. I've reached out to him a couple of times but didn't get much of a response aside from a dismissive response when I posted a comment on one of his videos. I think he's a canary in the coalmine but his proposed solutions don't target what I believe is the heart of the issue - that points cannot be used to construct a continuum. He's also way too much of a ultra-finitist for me. I believe potential infinite is essential in mathematics.

You've described your notion of potential infinity a few times (in another thread especially). And I've replied about it each time. Now, you're coming back to restate it, but still not addressing the substance of my previous replies. As in another thread, this just brings us around full circle. — TonesInDeepFreeze

I apologize if I'm not addressing the substance of your previous replies. It's not intentional, I thought I was.

The thought experiment is suggestive of an analogy with set theory, but suggestiveness is not an argument about set theory itself. — TonesInDeepFreeze

As I mentioned in my other post, it's not just about analogies - it's about the Platonic Realm. It's the only place that can harbor infinite sets. It's also the only place that can harbor the thought experiments so they're inseparably linked.

why would we think they can be completed in reality?
— keystone

We don't! Set theory doesn't say there's a "completion in reality". Set theory doesn't have that vocabulary. — TonesInDeepFreeze

In the Platonic Realm, can infinite objects exist but never be constructed? My view of the Platonic Realm is that it's a world where infinite processes can be completed, where someone can add 1 + 0.5 + 0.25 + ... term by term and in a finite amount of time complete the calculation to yield exactly 2.

Your analogy betwen mathematics and theology is not apt.

One can disprove 'there exists an infinite set' by stating axioms that disprove 'there exists and infinite set'. The obvious choice for such an axiom is 'there does not exist an infinite set'.

Anyway, I never asked you to disprove anything at all. — TonesInDeepFreeze

I see your point. In reality there is one truth (e.g. God either exists or not). I have this romantic/naive notion that in Mathematics there is similarly one truth and one ultimate set of axioms that captures it. But as it is today that is not the case with mathematics. So we may both be right when I say infinite sets don't exist and you say that they do because maybe we're starting with different axioms. Point granted.

We don't intend or claim that a domain of discourse for set theory is a world such as a physical world of physical particles and physical objects. At the beginning of this discussion, if asked, I would concede that immediately. — TonesInDeepFreeze

In Set Theory we say 'There exists a set...'. What do we mean by this? I take it to mean that the set must literally exist somewhere and my understanding is that the mainstream view is that it exists in the Platonic Realm. This is the world I want to explore. While Hilbert's Hotel cannot exist in our world, it should have no problems existing in the Platonic Realm which is infinite. Same goes for Thompson's Lamp. And what I'm trying to get at is that in the Platonic Realm, where infinite operations are completed (i.e. each guest flicks the light switch) something breaks because Thomson's Lamp can't decide whether it should be on or off after all switches are flicked. Or in other words, if an inhabitant of the Platonic realm is computing a geometric series and Grandi's series simultaneously term by term, then they both must either sum to a number or not. We are not allowed to say that one does and the other does not. That would be a contradiction that would cause the Platonic Realm to explode. I want to argue that the Platonic Realm necessarily explodes leaving no place for infinite sets to exist.

the very fact that anytime you have probability with infinitely many "contestants", whether it's dense space or whatever, you will necessarily either give the "contestants" a probability of 0 or be faced with adding up over 100% (since reiteratively summing any non-zero quantity indefinitely will approach over 100% at some point).

Your "solution" isn't a solution in that it doesn't talk about what the problem talks about. The "problem" is referring to continuity in dense contexts: it's not at all a "problem" in nondense contexts, this is equivalent to solving the Liar paradox by just saying "what if the guy doesn't lie?" — Kuro

What I'm proposing is that there is no "contest" involving infinitely many "contestants". For example, I'm proposing that to do calculus we don't need to assume that a continuum is built from the assembly of infinitely many points. Can you provide the simplest possible example in calculus where we need to assume that there are infinitely many points?

For me, the problem is not so much that there is anything counter-intuitive about this, but rather that it's rude and bad business practice to keep waking guests up in the middle of the night and make them pack and move to another room, especially an infinite number of times. Not only that, but the poster keystone has added lamps that keep turning off and on, which is extremely annoying when people are trying to get a good night's rest for the next day when everybody is going out to see Zeno's 10K Charity Run where Achilles will have to run through an infinite number of distances and suffer the ignominy of getting beat by a turtle. — TonesInDeepFreeze

LoL.

But when I started grad school at another university in 1962 one of the first required courses was an introduction to foundations using Halmos' Naive Set Theory and the Peano Axioms. It was quite illuminating. — jgill

As this conversation progresses, I've warmed up to set theory (although I currently hold an unorthodox view that set theory might not actually be about sets).

But one can't fairly criticize the road of set theory if one is not addressing it as an axiomatization. And even if not criticizing set theory but instead just saying mathematics can be done with unformalized "potential infinity" instead, then it's not a fair comparison since one is an axiomatization and the other is not. — TonesInDeepFreeze

Ultimately I want a mathematics that is formally supported by axioms and my preference is to keep what we've got (ZFC) because I'm guessing that so much is based on ZFC because it works so well. Also, I don't have the skill to craft new axioms, ha!

But here's my beef.

With geometry (specifically drawings) we can do things that cannot be capture with algebra. For example, I can draw a perfect finite picture showing the intersection of y=x^2-2 and y=0 even though I couldn't do the algebra to tell you the infinite decimal coordinates of the intersection.

With 'algebra' (specifically the manipulation of equations through calculus) we can do things that cannot be captured with geometry. For example, I can perform a finite set of operations (derivative) to easily calculate the slope at a point but I could never actually draw the infinite steps as described by the limit definition.

However, we want there to be an equivalence between algebra and geometry and so infinity leaks in. If instead we see algebra and geometry to be complementary (instead of equivalent), then we don't need infinity. In geometry we would limit ourselves to what is finitely possible. In algebra we would do the same. And sometimes we would need to switch back and forth between the two, similar to how in physics we switch back and forth between complementary wave and particle descriptions of light.

But to accomplish the above, we need to rethink how geometry (specifically how we draw things) and set theory (specifically whether set theory is about points or algorithms). But as I've said before, I don't think this changes much math...I think we've been doing it right all along. I just think what we say doesn't correspond to what we do.

We say that there are infinite numbers of points on a graph when we actually only explicitly draw a finite number. We say that sets are about points when we actually only work with 'algorithms'/descriptions.

I am saying much the same thing. But the question is not where the numbers need to be represented or stored. It is how many decimal places do you really need for the task in hand? — apokrisis

If you're going to describe a continuum with numbers you will need infinite decimal places. But of course, it doesn't have to be decimal. Binary is more pure. But that doesn't reduce the number of digits required. You can't do much math with one bit.

A circle is a certain kind of set of points. I don't know what you mean. — TonesInDeepFreeze

Yes, the conventional way to think of a circle is as an (actually) infinite set of points. What I'm proposing is that we think of the circle as the description (a finite string of characters) of those points. For example, the equation defining the circle. For example, consider this unorthodox drawing of a circle:



In this drawing there are no points. You certainly cannot identify a point and provide its coordinates. However, this drawing captures everything 'circular' about the green curve because it's associated with an equation. This equation is the description I've been referring to in my other posts. It describes, not the points the curve is made of but instead the potentially infinite points that would emerge IF I add more curves to the system. For example, let's add y=0:



In this case, two points emerge with this 'cut'. And because we know the equations for these two curves we can 'measure' the coordinates of these points.

With this approach we start with a continuum and progressively cut it. At any stage the objects are finite in number and the potential for additional cuts is never exhausted.

Conversely, if we start with points to build a continuum we're going to need infinity of them...and even then, I think such an approach is problematic and paradox filled.

I believe that calculus is more closely aligned with this parts-from-whole approach than it is with the conventional whole-from-parts approach.

A sequence is a set. And it has a domain, which is a set, and a range, which is a set. An infinite sequence is an infinite set with an infinite domain.

Of course, one can leave that unconsidered, not in mind, when working in certain parts of calculus. That is not at issue. But when we trace the proofs of the theorems of analysis back to axioms, then, in ordinary treatments, those are the axioms of set theory. — TonesInDeepFreeze

I concede that there may be no better way to formalize calculus than set theory. I don't want to challenge the efficacy/validity of set theory. I only want to challenge the philosophy/interpretation of it. We don't actually work with the infinite sequence itself, we work with descriptions of the infinite sequence.

Mathematics, in many branches, is brimming with sets. Analysis, topology, abstract algebra, probability, game theory... Can't even talk about them, can't get past page 10 in a textbook, without sets.

But of course, one can use the theorems of mathematics for engineering without tracing the proof of those theorems back to axioms, in particular the set theoretic axioms. That's not at issue. — TonesInDeepFreeze

Okay, I'll concede this point and agree that set theory is used to formalize many branches of mathematics...and I don't think it's going away. But as I said before, we never actually manipulate the infinite sets directly. We 'describe' infinite sets and work with the 'descriptions'.

And yet Universal Turing Computation is a mathematical object – conceived in Platonia. This is the kind of "paradox" we are meant to be figuring out here, not simply saying one is the other as if the differences were moot. — apokrisis

Maybe we both agree that our universe is like a simulation by a finite computer and as such our universe has some limitations in measurement, e.g. as it relates to Planck length. What I want to highlight is that our universe's computation is performed by one computer (I actually think it's more like a cluster of computers but let's leave that aside). Conversely with mathematics we're not constrained to a single computer. The math that I do on my pocket calculator might offer a different precision than the math that can be done on a supercomputer. We need our understanding of how computers participate in mathematics to be general and apply to all computers. As such, we cannot say that mathematics has limits of resolution, but instead that the computer does. And to keep things connected with reality, let's stick with finite computers having arbitrarily fine (but in all cases finite) precision.

Must an ordered field necessarily be a field of numbers?
— keystone

No. But all complete ordered fields are isomorphic with one another. So all complete ordered fields are isomorphic with the system of reals. — TonesInDeepFreeze

Ok, if we've only proved that the reals are an ordered field, then is it possible that we haven't proved that sqrt(2) is a number?

the axioms of set theory are not in concordance with the intuitive notions of 'finite sets'
— keystone

All the axioms are in that concordance, except one. — TonesInDeepFreeze

Not an axiom, but at the heart of set theory is the definition of equinumerosity of infinite sets which is not in concordance with MY notion based on finite sets. Let me try to explain using finite sets A and B and hopefully my use of set theory terminology is correct.

When I think of A being equinumerous to B, I think that there exists a bijection AND no injection between A and B.
When I think of A being more numerous than B, I think that there exists an injection from B to A AND none from A to B.
When I think of A being less numerous than B, I think that there exists an injections from A to B AND none from B to A.

If I apply this intuition to infinite sets A and B, then A is neither equi/more/less numerous to B. The only way that this is possible is if A and B are both the null set. In other words, my intuition based on finite sets leads me to believe that infinite sets are all empty.

I want to highlight again that I'm not saying that set theory is wrong, I'm just proposing that set theory might not be about actually infinite sets, but instead the potentially infinite algorithms that describe the infinite sets.

Hilbert's idea was that we can work in infinitistic mathematics if we have a finitistic proof of the consistency of infinitistic mathematics. Famously, we found out that there is no finitistic proof of the kinds of systems we'd like to use, not only not of set theory but even of PRA, the system itself that we may take as exemplifying finitistic reasoning at its "safest". Yet, if I understand correctly, Hilbert's condition was a sufficient condition not a necessary one. — TonesInDeepFreeze

Interesting!

That's interesting. But, if that is to be a statement in the system, we'd need to see "described" couched mathematically. I have a hunch that your notion is pretty much the same as 'there exist potentially infinite sets', and as I've said, I don't know a system that says it. — TonesInDeepFreeze

Yes, I use the word 'described' without explaining exactly what I mean but essentially I'm referring to the descriptions already used in set theory today to describe the contents of an infinite set. I'm not proposing that we reframe set theory to be based on potentially infinite sets as that would not be satisfactory. For example we can't describe a circle using an endlessly increasing list of points. That list will never suffice (proof Cantor). Instead, we describe a circle using a potentially infinite 'algorithm' for generating the actually infinite list. For example, the equation x^2+y^2=1 describes a circle in completion.

Undefinable real numbers have no place in my view.
— keystone

You reject vagueness then. That is certainly the usual thing to do. — apokrisis

I believe that cuts made to a continuum are perfectly precise since I can draw it with no vagueness. For example, consider this drawing of y=0 and y=x^2-2:



There's no blurriness to my drawing. However, when I start to measure it (usually through calculus), my measurements may be imprecise. For example, if I try to measure the decimal expansion of the coordinates where they intersect I will generate a rational approximation which does not correspond to the exact point (however I can get arbitrarily close as long as my calculator has got the time and capacity).

Hopefully this explanation makes more clear my view of physics where the planck constant applies to a limitation of measurement, not of space/time itself. This is how I think space/time can be continuous but measurement discrete. And I like to think of our reality as something like a simulation by a computer which has finite time and capacity. The planck length and other limits of resolution are direct consequences of limitations of that finite computer.

And how do you know there is a rule unless you have ever seen some exception? — apokrisis

I'm confused with where we've landed with this and I think I we need to step back. You said "I've already raised this point, asking if sqrt(2) is the exception or the rule." I don't see sqrt(2) as a rule. I see it as a description, perhaps of an infinite set. As I've discussed with TonesInDeepFreeze, I see no problem in there being a description of an infinite set, even if the infinite set doesn't actually exist.

That is a better analogy. I prefer my own still – the static on the TV screen which is both every show you could ever see, but all at once ... or else just meaningless noise. — apokrisis

I like my stereogram analogy more because it highlights that the observer plays an active role in actualizing the rabbit all the while doing no actual manipulation of the page (not actual cutting of the continuum). But I agree that it's not a perfect analogy because the creater of the stereogram encoded the rabbit in the page, whereas in mathematics it's a true blank slate.

But that doesn't prove that there does not exist a set whose members are all and only the natural numbers or that there does not exist an infinite set. — TonesInDeepFreeze

A finite being cannot exhibit or work with an infinite set directly. To do so requires an infinite being. Since one cannot disprove the existence of an infinite being (e.g. God) one cannot disprove the existence of infinite sets. But does the burden of proof lie on the atheist? Paradoxes are a good option because they demonstrate that actually infinite universes (in which infinite sets can exist) harbor contradictions, such as in Hilbert's Hotel Universe where infinite processes can and cannot be completed.

How is that substantively different from Thompson's lamp?
I already responded regarding Thompson's lamp.
I don't know a theorem of set theory that is rendered as "infinite processes can be completed".
Set theory doesn't axiomatize thought experiments. — TonesInDeepFreeze

I think saying "there exists a set of all natural numbers" is equivalent to writing a program to print all natural numbers and running it through to completion. However, I think set theory can be reframed to correspond to potentially infinite algorithms instead of actually infinite sets. After all, we never directly work with the infinite sets themselves, but instead the finite strings of characters that describe them.

Thought experiments are beneficial because they make it clear what we're talking about - with set theory (as it is framed today) we're talking about actually infinite objects, not potentially infinite processes. And I'm not convinced by your response to Thompson's lamp because your answer lies outside of the thought experiment where it's unclear to me whether your resolution requires the completion of an infinite process. In Hilbert's hotel universe, infinite processes can be completed so there must be a definite final state of the lamp. If you can't provide the state of the lamp then it's worth questioning whether infinite processes can truly be completed in that universe. And if they can't even be completed in an imaginary universe, why would we think they can be completed in reality?

A sphere has infinitely many points in it.

And is there such a thing as a sphere with an infinite radius? If I'm not mistaken the radius of a sphere is a real number, right? — TonesInDeepFreeze

Let's drop a dimension for simplicity. I can think of x^2+y^2=1 without having to think of any points. I can also draw that function without drawing any points (I can show you if you're curious). The points come from the circle, not the other way around.

No, there is no such thing as a sphere with infinite radius...that's explains why I can't imagine it.

No, limits use infinite sets. The standard axiomatization of analysis is ZFC. Ordinary modern analysis is decidedly infinitisitic. Maybe you're thinking of the banishment of infinitesimals? — TonesInDeepFreeze

I can't speak to the standard axiomatization of analysis, but the informal definitions that us engineers were taught didn't use sets. As written on Wikepedia: a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. I take "approaches" to be a potentially infinite process.

My argument is that the whole potential vs actual infinity thing comes from the fact that our ideas about numbers are based on systems of constraints. — apokrisis

I disagree. I think that potential infinity comes naturally to us, whereas we reluctantly hold on to actual infinity. We do so because we are not used to the notion of the observer (mathematician/computer) playing an active role in mathematics. As with classical physics, we want the observer to play a passive role by making passive observations upon the objects of math that exist in completion for eternity in the Platonic realm.

But you're a fan of physics so you should appreciate that in QM the observer plays an active role converting the potential to the actual (by means of making measurements to collapse the wave function). Consider the possibility that mathematics is contingent upon the interaction between the computer and the continuum (by means of cuts), in the same way that reality is contingent upon the interaction between the observer and the wave function (by means of measurements).

How many numbers are there?
A mathematical platonist would have to say infinite. I would go in a different direction. I would ask 'where?' How many numbers are 'where'? In other words, in what computer/mind are you talking about. You have to be specific about where is because there is no Platonic realm. And right now, as I'm getting ready to go to bed, if you ask me 'how many number are there in my mind?' I would answer zero.

In set theory, there is no completeness axiom. Rather, we prove as a theorem that the system of reals is a complete ordered field. — TonesInDeepFreeze

Must an ordered field necessarily be a field of numbers? Could it instead be a field of equations? For example, I would imagine that adding a 1 to each number in the field does not negate the field.

And yes one might want for the axioms to be intuitively correct ("true") even if the theorems might be surprising. And with set theory, people's mileages vary. I find the axioms of set theory to be exemplary in sticking to only principles that are in concordance with the intuitive notion of 'sets'. — TonesInDeepFreeze

I largely agree with you about axioms. My only issue is that the axioms of set theory are not in concordance with the intuitive notions of 'finite sets'. And since the only sets we ever work with directly are finite, I think we should be cautious accepting axioms that oppose them. But as I mentioned before, a small tweak to the language of the axioms will be consistent with my intuitions. Instead of saying 'there exists an infinite set' I would be comfortable saying 'there exists an algorithm that describes an infinite set'.

I can draw a line with open ends on a piece of paper and label the ends negative and positive infinity. This unbounded object is entirely finite.
— keystone

You can draw a sign that you then interpret in a certain habitual fashion. The issue then is how does this sign relate you to the reality beyond. Does is create a secure bridge? Or is it wildly misleading? — apokrisis

ELI5.

If maths has been left behind in this grand and still unfolding adventure, tough shit. — apokrisis

You believe that math itself has some fundamental limits, perhaps a frequency, a duration, or a length. You may be right but I think you're wrong in applying it to continua. Instead, you should include the computer as a fundamental participant in mathematics and apply these limitations to the computer. Apply them to measurement. It makes sense, after all. Every computer has its limits. It has a finite memory so there is a limit to the size of the numbers that it can store. It also operates at some frequency. This is all common sense and it requires no philosophizing. Mathematical objects don't exist eternally in the Platonic realm. They exist when we (computers) compute!

Yes, this will seem very counterintuitive. The simplest way I can explain it in a non-technical fashion is that selecting any non-zero probability for each number will force us to add up way over 100%, because there are infinitely many other "participants" (numbers), which means the only probability we can assign to each participant is zero. — Kuro

I understand how we got here, it's just that to me this screams that what got us here is a mistake.

There's actually a way out of this being nonstandard analysis — Kuro

I have a hard time accepting real numbers, I have an even harder time accepting hyperreal numbers.
But yes there is a very simple way out of this, and that is that points don't exist. When you throw a dart at a dartboard, you don't hit a point, you hit an area. Any discretization of a dartboard into areas produces a finite number of areas each with a finite probability, all summing to a probability of 100%. What's wrong with this view?

I believe that irrationals are algorithms which describe this mysterious other object - continua.
— keystone

I've already raised this point, asking if sqrt(2) is the exception or the rule. Higher dimensional generators could produce generators of some number that looks to be an irrational point of the line. But then these numbers - growth constants like e and phi – are ratios and so are dimensionless unit 1 values more than they are some weird real number.

The status of any regular irrational seems different. They would lack generators apart from decimal expansion. Something else is going on. — apokrisis

Undefinable real numbers have no place in my view. Focusing solely on definable irrationals, I believe an irrational is irrational, no matter how efficient/beautifully it can be expressed. My point is the rule not the exception.

I don't believe there is a fundamental length since any length can be divided.
— keystone

Can the Planck length be divided? Not without curling up into a black hole.

What you believe and what the Universe would like to tell you seem two different things. Who wins? — apokrisis

It has not been determined whether space is discrete or continuous (LINK). I'm inclined to believe that the planck length is a limitation that is applied to measurement, not the divisibility of space itself.

Your trust in physics has made you believe something obviously counter to your mathematical intuitions - that any positive number can be divided.

I can imagine a mind that lives in a 4D universe that can picture a 4D hypersphere as easily as a sphere.
— keystone

Yea, nah. I'm not buying these feats of your imagination. — apokrisis

Seriously? My mind can only hold so many bits therefore there is a largest number that I can retain in my memory. However I can easily imagine a mind whose memory could retain a larger number. It's the same thing with dimensions. No leap of faith is required here. The real feat of imagination is imagining a mind whose memory could hold an infinitely large number. That I cannot do.

the one where the world is "thinking itself" into definite being in ontic structural fashion. — apokrisis

In my stereogram analogy I do not exist within the page, and neither does the computer exist within the continuum. Or perhaps more digestible, neither does the computer exist within the simulation.

The hidden rabbit or seagull is merely hidden while the brain finds a way to suppress the shapelessness of the coloured pixels from the intelligibility of a depth perception-based contour. — apokrisis

It sounds like you're saying that the rabbit exists even when not observed. I would argue that it only has the potential to exist and it actually exists when observed correctly. In any case, I think you're nitpicking my analogy. If I handed you a blank white piece of paper you could argue that it is a picture of a polar bear playing in the snow but I would argue that it only contains the potential to be such a picture, and it would actually be that picture only once you cut out the bear figure with scissors.

The page contains the potential of infinite images,
— keystone

The problem is that it doesn't. It plays on a dichotomous rivalry of brain subsystems. You have to switch off the one and employ the other. The search is for the single hidden interpretation. Only one of the two points of view can spot it. — apokrisis

Perhaps I'm missing your point. Do you agree that given a continuum there's infinite potential to how you cut it up?

This is why the brain is not a computer. — apokrisis

I would prefer to say 'this is why the brain operates different from a modern computer'. I agree with Max Tegmark when he said 'we should reject carbon-chauvinism'. However, this is already a broad discussion, I don't think we should extend it to debate this.

The imaginary hotelier can do that also. — TonesInDeepFreeze

Please allow me to indulge in Hilbert's Hotel a little more just so that we can stick with actually infinite objects instead of potentially infinite processes. As such, I want to avoid resorting to algorithms or properties to ensure that we are clear on our differences. In the lobby there is a lamp that is currently off. Each room has a switch to invert the state of the lamp. When each guest is moved to the next room, the guest flips the switch. After infinite guests move rooms, is the lamp on or off? There's no way around it...Hilbert's imaginary universe (where infinite processes can be completed) doesn't make sense.

What was the 19th century analysis resolution to Zeno's paradox?
— keystone

Infinite summation: convergence of an infinite sequence to a limit. — TonesInDeepFreeze

I believe that, contrary to the mainstream interpretation, 19th century analysis is the mathematics of continua (not the mathematics of points). I do believe that the mathematics of continua (i.e. 19th century analysis) does indeed resolve Zeno's paradox. So in this partial sense I agree with you.

However, I don't believe that a point-based interpretation of 19th century analysis allows for Zeno's paradox to be resolved. If it does, then answer this: If I travel along a line from x=0 to x=1, what is the next point that I travel to? As you know, there is no 'next point' so Zeno would claim that it's impossible to take a first step. As long as we hold on to points Zeno's paradox stands.

Are you not disquieted that a subset of rooms is equinumerous to the full set of rooms?
— keystone

I don't conceive of an infinite set of physical rooms.

As to sets, I already mentioned that I am not bothered that the squares (a proper subset of the naturals) is 1-1 with the naturals. — TonesInDeepFreeze

It sounds like you don't think Hilbert's Hotel is a good analogy then. Is that true? I like discussing in terms of the hotel because it's clear that we're talking about an actually infinite object not a potentially infinite process. For example, I have no problem with programming a computer algorithm to print n:n^2 for all n. Such a program could be written in a few lines of code. My issue is with the notion that such a program could be executed in completion. The complete output of the program would be an actually infinite object which I don't believe in.

I don't think calculus needs actual infinity to work.
— keystone

It does in its common form. — TonesInDeepFreeze

Please provide the simplest example you can think of where it is needed? I'd like to see whether I can challenge it.

I suspect (with no evidence to provide) that ZFC doesn't need actual infinity to work either.
— keystone

It wouldn't be ZFC then. — TonesInDeepFreeze

Just as I can write a program that outputs all natural numbers, I think it is fair game to talk about an infinite set. Might it be possible that ZFC is useful mathematics that talks about things that don't actually exist? Might the useful thing not be the objects but the 'talking'. Might we be able to reinterpret the axiom of infinity as 'we can talk about infinite sets'?

I know so little of cosmology that I don't know how to dispel my bafflement that the universe could be finite or my bafflement that the universe could be infinite. — TonesInDeepFreeze

I think it's quite easy to imagine a closed finite universe, for example a sphere of finite radius. It's a lot harder to fit in one's mind a sphere of infinite radius.

If there is any mathematical reasoning that can be considered safe, then it's manipulation of finite strings of objects or symbols (whether concrete sticks on the ground, or abstract tokens)...if I really had to, I could fall back to extreme formalism by taking the theorem to be utterly uninterpreted, but a formula nevertheless to be used in mathematical reckoning. — TonesInDeepFreeze

I completely agree. I can't imagine a scenario where the work that mathematicians have been doing for decades using set theory could suddenly be worthless. Maybe some ideas would need to be re-evaulated (e.g. Axiom of Choice and the Banach–Tarski paradox) but by and large I am completely convinced that the vast majority of modern math would retain it's value even IF actual infinities were banished.

Adding to my response about the particular paradoxes. Even if we granted that they indicate flaws in the concept of infinitude, then that is a concept of infinitude extended beyond set theory into imaginary states of affairs for which set theory should not take blame. Those paradoxes don't impugn set theory itself. — TonesInDeepFreeze

I would say that those paradoxes don't necessarily impugn set theory itself. Just as actual infinities (rather infinitesimals) were banished from (mainstream) calculus and replaced with potential infinities (through limits) leaving the applied math essentially uneffected, I wonder if something similar could be done to the actual infinities of Set Theory.

In QM we have come to accept a certain level of uncertainty. Why can't we do the same in math?
— keystone

I wouldn't argue that we can't. I suppose people already have made logic systems with values such as 'uncertain' that can be be applied to a different mathematics. And I can imagine that certain scientific enquires might be better served by such systems.

But that doesn't erase the rewards meanwhile of classical mathematics. — TonesInDeepFreeze

One would be crazy to say that classical math is wrong. I'm in no position to say this with certainty, but I believe that our finite descriptions of any given real number is valid and useful. Similarly, our finite descriptions of any given infinite set is valid and useful. I just don't think the real number or the infinite set itself actually exists, nor does it need to. We never work with the decimal expansion of pi, nor do we work with the infinite set itself (e.g. by explicitly listing all its elements). We always work with the finite descriptions. We work with algorithms. And if we want to work with an actual number, we halt the potentially infinite algorithm to generate a rational number and use that in our computation. I believe this is what classical mathematics does. At least that's what engineers do.