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

The reason old-fashioned terminology is not so bad. One-to-one onto, etc. Bourbaki may be to blame? Just nit-picking, ignore me.

we never actually manipulate the infinite sets directly. — keystone

Depends on what you mean by 'manipulate'.

Set theory, most strictly, is a certain set of formulas. In mathematical logic, when we say 'ZFC' we are referring exactly to the set of formulas.

Of course, those formulas are "read off", or "rendered" in, say, English as if they are English sentences. But they are not English sentences. Also, most people do have in mind that the formulas pretty much "say" what the English renderings are. However, again strictly speaking, the formal meanings are given by the method of models.

In that context, its hard to say what 'manipulate' means. For myself, I do recognize that I manipulate symbol strings. But I don't at all think that somehow, like a puppeteer, I'm manipulating mathematical objects. Rather, I am writing formulas as lines in proofs. I do intuitively think of those formulas as saying something about whatever objects are in the domain of discourse, but I am not manipulating those objects themselves. Rather, I am writing formulas about them.

But what about having not just a pre-philosophical intuition about what set theory "says" but a truly articulated philosophical position about it? Realist, nominalist, structuralist, fictionalist, consequentialist, instrumentalist...? That's for each mathematician to decide for herself or himself. Or not decide. There's no law of the universe that one can't just chug along proving theorems without declaring a philosophical position. Of course, then one can't provide a philosophical justification for set theory. But then, of course, one can say, "I don't need to provide a philosophical justification. The question of a philosophical justification is a fine one indeed, and I like peeking in on the debates now and then. But no matter how the arguments turn out, I'll still enjoy set theory and I'll correctly point out that it is a recursively axiomatized theory that proves the theorems used for the mathematics of the sciences, and such that, at least in principle, all the proofs can be written in complete formality so that their correctness can be algorithmically verified".

I currently hold an unorthodox view that set theory might not actually be about sets — keystone

Cool. The formulas can be about whatever you want them to be. Hilbert's tables, chairs and beer mugs.

the mainstream view is that it exists in the Platonic Realm — keystone

I don't know whether platonism and/or variations on platonism are the majority view among those who have a view, but I wouldn't bet against it.

While [Thompson's lamp] cannot exist in our world, it should have no problems existing in the Platonic Realm which is infinite. — keystone

In that realm, there is no final state for the lamp. Poof. Done. Still no contradiction.

Your fallacy is in setting up an imaginary world, with states-of-affairs like set theory, but then adding a state that doesn't exist in the set theory and thus rightfully as an analogue not in the imaginary world, so you get a contradiction but outside the original terms you set: an imaginary world corresponding to set theory.

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.

You require that sets can be "built" only in finite "processes". Then, since it's all processes to you, when you see set theory posit objects not built with such processes, you incorrectly saddle set theory as claiming that certain states are achieved only after an infinite process. Your arguments about the realms then are based on conflating your concept with set theory.

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

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

If you reread the posts (and in the other thread) then you'll see exactly where.

In the Platonic Realm, can infinite objects exist but never be constructed? — keystone

Set theory has no theorems about mathematical agents constructing or not constructing things. So you can't plop them into the middle of one of these realms you're making and have it be about set theory. (For an approach to mathematics that does have something like agents constructing things, see intuitionism.)

My view of the Platonic Realm is that it's a world where infinite processes can be completed — keystone

Fine. It's not a realm of set theory. You don't know anything about set theory. But you keep burdening it with what you misunderstand it to be. You imagine a realm that has some similarities with set theory, but also has things not corresponding with set theory. Then you blame set theory.

Basically your realms are like models. But a model of a theory is one in which the vocabulary of the theory is interpreted in the model and every theorem of the theory is true in the model. But you're adding things that are not the interpretation of anything in set theory and even worse, posting states-of-affairs about them don't model the theorems of set theory. Sorry, no go.

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

Let your '...' be "such that P".

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'.

That means, in terms of your "realms" (whatever their ontological or metaphysical character), there is at least one object in that realm that has the "property" denoted by 'P'.

Thompson's lamp.

It's a non-converging sequence.

Set theory doesn't have a "final state" with that.

But here's what set theory does have:

Let N = the set of natural numbers.

Let f be a function.

Let dom(f) = N

Let for all n in dom(f), f(n) = 1/(2^n)

So f(0) = 1, f(1) = 1/2, f(2) = 1/4 ...

0 is not in ran(f).

Let g be a function.

Let dom(g) = ran(f)

Let ran(g) = {"off", "on"}

Let for all r in dom(g), g(r) = "off" iff En(r = f(n) & n is even)

So g(1) = "off", g(1/2) = "on", g(1/4) = "off" ...

So I've mathematically "translated" Thompson's lamp.

What about the "final state"? There is no final state. The mathematics doesn't have a "final state" here, exactly because the mathematics doesn't have nonsense like "an infinite process with a final state". Of course you get a paradox from having an infinite process with a final state. It is the very point that mathematics is not capable of such nonsense.

But this:

Let (h) = g u {<0 "off">}

So the "final state" of h is "off". No contradiction.

Let j = g u {<0 "on">}

So the "final state" for j is "on". No contradiction.

Choose whichever "final state" you like - the "final state" with h or the "final state" with j. But neither is determined by g.

NOTE: I am not claiming that there aren't mathematical treatments of supertasks that, with advanced definitions and construction, formalize notions of "infinite number of steps in finite time" or "final states for infinite processes". But if the treatments are formalized with ZFC, then it can't be the case that they contradict ZFC. I'm not expert in that matter, but I would put my proverbial money on it.

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? — keystone

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).

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

The reason old-fashioned terminology is not so bad. One-to-one onto, etc. Bourbaki may be to blame? — jgill

That's a classic '50s movie, 'Blame It On Bourbaki'. Cary Grant and, if I recall, Anita Ekberg. Grant's character is trying to get Ekberg's character to take some experimental injections for her terminal illness, but she doesn't want the new experimental drug. Her famous line is, "No, sir, jections!"

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

My focus is systems science. Which means all of the above really. But neuroscience in particular.

No computer processing can make the image higher resolution than reality. — keystone

The point was that the processing removes vagueness - the unlimited number of shades of grey - by imposing a binary, black and white, constraint on the image. It boils faces down to a grid of points representing exact distances between the most informative features.

But if you just want to resist the concept of vagueness, that’s your lookout. I can only say it was about the single most paradigm shifting thing I ever learnt.

For a fully accurate depiction, the line width should be exactly 0. — keystone

LOL. Just as the cuts in the line should be exactly 0 length. Your arguments here seem all over the place.

How do you glue actually 0D points together to make a continuous line? How do you glue 0D width lines together to make a plane?

The issue to be resolved is how divisibility can co-exist with the continuum that it divides. That is where my point about the discrete and the continuous being a dichotomy that emerges from a logical vagueness comes in. It justifies treating both the cutting and the gluing as complementary limits on actuality. The infinite and the infinitesimal are two ends of the one spectrum of possibility.

But I can’t see what your answer is at all.

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.

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?

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.

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.

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. — keystone

I think I've said it about two or three times. It is important because it is the very heart of the matter, which is that set theory axiomatizes the mathematics of the real numbers. But, for some reason, I don't see the passage in the post you linked to. If you look at where I said it, then you'll see how it was a response to a comment or question of yours.

'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? — keystone

I'm sorry, but are you serious?

where we need to assume that the real line is composed of infinite points — keystone

[bold original]

I've answered that before. (In this thread or another one.)

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.

If you're the teacher and you tell the students there are only these certain points, then the student comes to class the next day and says, "I wanted to use mathematics to figure out how to build a table, but my mom she needs an overhang between 1/8 inch and 1/16 inch, but there is no number between 1/8 and 1/16 among the numbers you'll allow me to use; it's not in your list of all the points."

Infinitely many points are needed so that we can speak in greatest generality, no matter what degree of tolerance you might think is the greatest degree needed.

The infinitude of the set of real numbers is a consequence of the definition of 'real number'. Set theory axiomatizes that courtesy of the axiom of infinity. If we took out the axiom of infinity, then we wouldn't have the set theoretical construction of the complete ordered field that is the system of the reals.

Think of it this way: I show you an airplane.

You say, "Why do you need that big plate at the bottom; couldn't you make an airplane without that kind of thing?"

I say, "I don't know whether you can, but it is needed for this airplane. And it's not that just that it has a specific purpose, but that if you took it away, then all the rest of the parts wouldn't work together; the whole thing is an interdependence. If you want to build an airplane without that kind of plate, then go ahead and do it."

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.

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. — keystone

You're not being reasonable. Do you want to inform and enlighten yourself about the subject, or you just want to raise objections premised in not knowing anything about it? Better to know the thing, then you could critique it fairly.

You say that many of the paradoxes are resolved by Set Theory — keystone

I said that, and it's okay. But in the last posts, I realized that the more pertinent point is the one I'm making now.

essentially it is you who is making the link between the two — keystone

What are you talking about? You are mentioning the paradoxes to impugn set theory. Then I reply: (1) The paradoxes don't show a contradiction in set theory, (2) the set theoretical, mathematical methods don't end in contradiction, so, in that way they "solve" the paradoxes; not that the paradoxes dissolve on their own, but rather when we do actual mathematics instead, we don't incur contradiction, (3) the realms in the paradoxes are not models of set theory anyway.

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

but then you say that criticisms of the paradox can't touch Set Theory. You can't have it both ways. — keystone

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.

Zeno's Paradox — keystone

I answered about Zeno's paradox many posts ago (in this thread or another one).

How about having some attention span and look at my rigorous response to Thompson's lamp?

You asked me about Thompson's lamp. I replied about it, now with rigor.

If you then just jump to another subject (and you're incorrect about it also) then I take it you are not in good faith.

Thompson's lamp.

It's a non-converging sequence.

Set theory doesn't have a "final state" with that.

But here's what set theory does have:

Let N = the set of natural numbers.

Let f be a function.

Let dom(f) = N

Let for all n in dom(f), f(n) = 1/(2^n)

So f(0) = 1, f(1) = 1/2, f(2) = 1/4 ...

0 is not in ran(f).

Let g be a function.

Let dom(g) = ran(f)

Let ran(g) = {"off", "on"}

Let for all r in dom(g), g(r) = "off" iff En(r = f(n) & n is even)

So g(1) = "off", g(1/2) = "on", g(1/4) = "off" ...

So I've mathematically "translated" Thompson's lamp.

What about the "final state"? There is no final state. The mathematics doesn't have a "final state" here, exactly because the mathematics doesn't have nonsense like "an infinite process with a final state". Of course you get a paradox from having an infinite process with a final state. It is the very point that mathematics is not capable of such nonsense.

But this:

Let (h) = g u {<0 "off">}

So the "final state" of h is "off". No contradiction.

Let j = g u {<0 "on">}

So the "final state" for j is "on". No contradiction.

Choose whichever "final state" you like - the "final state" with h or the "final state" with j. But neither is determined by g. — TonesInDeepFreeze

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

No, because mathematics shows how to calculate that Achilles did finish the race.

You just keep showing that you don't care about understanding any of this but instead want to continue roving among already answered points as if they weren't already answered.

Does Set Theory model Achilles' journey or not? — keystone

You have it backwards, and show that you didn't bother to read my previous post about modelling.

The realm of the paradox is not a model of the mathematics. Formally, the mathematics is not a model; it's a theory. Models are of theories, not the other way around.

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.

Can you see that? Can you be reasonable enough to see that that is reasonable.

Why do we need to order the curves? — keystone

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

But I guess what you are really interested in is potential infinity and its kin. So my comment is what difference does it make how you deal with that? 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.

But if you enjoy your project as a personal challenge, have at it. I have certainly been in that boat!

'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?

@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?

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.

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.

My focus is systems science. Which means all of the above really. But neuroscience in particular. — apokrisis

Very cool.

But if you just want to resist the concept of vagueness, that’s your lookout. I can only say it was about the single most paradigm shifting thing I ever learnt. — apokrisis

I'll keep it in my back pocket in case it will be useful to me in the future :D

How do you glue actually 0D points together to make a continuous line? How do you glue 0D width lines together to make a plane? — apokrisis

In my view 0D points are not objects so you don't glue them together. You don't make a curve from points. You don't make a surface from curves. You don't build the whole from the parts. In my view you go the other way around. You start with the whole and cut it up. Consider how we draw a line. We don't use pointillism until a line emerges. We draw a line and then put some tic marks on it to identify points. I don't see what part of my explanation you're not getting.

The issue to be resolved is how divisibility can co-exist with the continuum that it divides. — apokrisis

What exactly do you mean by this?

You don't build the whole from the parts. In my view you go the other way around. You start with the whole and cut it up. — keystone

My systems view says you have to go still further into true holism. You are simply replacing one kind of constructive action - gluing - with an opposite one, that is cutting.

The systems view instead opposes construction to the other thing of constraints. And even here, both construction and constraints arise out of a common unity in vagueness. So the ontology is fundamentally complex. And hence not widely understood by folk.

Anyway, what this means is that my view you do indeed start with the whole that you mean to divide into its constituent parts. So rather than constructing the line from a set of points, I would talk about constraining the line to arrive at the limit where it would become indistinguishable from a point. The interval would be made so small that the length of the line was the same size as the width of the line - both being now effectively zero.

You don’t come at it top down by chopping up a whole line - an infinite number of constructing actions. You come at it top down by increasing the degree of the limitation. Length gets shrunk down to the point that it is no longer possible to be certain about describing it as indeed “a length”.

A line in turn would be arrived at as the constraint on the quality of “plane-ness”. Squish the 2D plane from either side and the limit of its compression becomes how a 1D line arises.

This sets up the basic dichotomy of global constraint vs local construction. And whereas a simple metaphysics, when faced with a dichotomy, demands that you now back one side or the other, a systems metaphysics says the two instead make for a unity of opposites. Each entails the other. Reality is what emerges in a triadic fashion because there is now a world in which top-down constraints enable the existence of local acts of construction - be they cuttings or glueings - and, reciprocally, these local acts of construction generally tend to reconstruct the system of constraints that were shaping them in the first place.

It is a synergistic metaphysics. Constraints define freedoms (as that which ain’t constrained). And freedoms construct constraints, on the whole, as that ensures the continued existence of a world that indeed has precisely those kinds of degrees of freedoms.

An electron is somehow the right kind of fundamental stuff for the laws of the universe to produce. They will keep being produced for as long as they prove a productive way to maintain a universe that has exactly those kinds of laws.

But good luck applying this kind of advanced systems logic to the simplicities of number theory. Peirce did try, but only got as far as a reasonable sketch. He never published the logic of vagueness he was working towards. We have only the notes.

'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. — keystone

You answered the question. You're not serious.

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 — keystone

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. — keystone

You asked me about finitely many points, not about potentially infinitely many points. Be clear.

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. — keystone

I bet it was yet another way for you to say that you like the idea of potential infinity. No, I haven't responded to you at all on that. I mean, the dozens and dozens of my posts now on display in at least two threads don't exist.

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. — keystone

You didn't have to waste your time typing that. I know your notion of potential infinity.

And you egregiously obfuscate the terminology. 1/8 increments is not a continuum. You could at least give the consideration of not appropriating terminology in a blatantly incorrect way.

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. — keystone

Another of your arguments by analogy. Mathematics doesn't cover rulers with ink. The existence of the set of real numbers doesn't stop you from considering only a finite number of numbers for a given problem. The rest of the infinitely many numbers are not there waiting to spill themselves like ink all over your favorite finite set.

This is the problem: In a context such as this, it's fine to deploy analogies to illustrate intuitions and things like that. But the argumentative force is limited, at best. I shouldn't indulge you more.

Here's my airplane analogy. — keystone

Right. Since you can't be bothered to see the point of my analogy*, you skip it and just jump to your own analogy. Barely read the posts, not taking moments to reflect on what's been said, to ingest, so the posts are just jumping off points for you to say yet again how you think mathematics should be. You've got this down to an art, if not a science.

By the way, my analogy was not offered as an argument but merely lagniappe for you to (hopefully) grasp an idea that is not your own for a change.

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. — keystone

What on earth are you talking about? Sounds like a pitch for an opening scene of a movie or something. Who do you have in mind to direct? Spielberg? Soderbergh maybe?

The mathematicians don't just assume the theory will work. Rather, they prove that it does, by deriving the existence of the real number system, then proving the theorems of mathematics used by the sciences.

Again, we go around in circles, because you keep re-insisting on points that had long ago been answered. You are a sinkhole.

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? — keystone

For heaven's sake! I've answered that and answered it and answered it already.

The answer is that the ordinary axiomatization of the mathematics for the sciences has an axiom that implies that there exist infinite sets. If we remove that axiom from the rest of the axioms, then we don't get analysis. Period. Final answer, Regis. Got it?

But that does not preclude that one can devise a different system that yields the theorems for mathematics for the sciences, with different axioms, and if needed a different logic, in which we don't have the theorem that there exist infinite sets. Got it?

So if you really are interested in a mathematics without infinite sets, then go look at the other systems already!

Whoops, mistake.

Some of your quote links are not going to the posts in which the quotes occur.

You have a framework. You don't have a hint of an idea how to make it rigorous, but that doesn't disallow that nevertheless it might suggest an intuitive motivation toward a rigorous treatment. On the other hand, other people don't share your framework and have different intuitions, and have made rigorous mathematics. It is poor thinking on this subject then to keep trying to put a different framework within your own. I've been saying this over many many posts. Do you see?

You complained that I skipped some of your arguments — keystone

Where 'some' includes the most important ones.

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. — keystone

"We". There are mathematicians and philosophers for whom set theory is intuitive. You don't get to declare for all "we".

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

[...] philosophy of mathematics we can refine our intuitions and apply them in our quest to understand our universe. — keystone

Of course. And I said that mathmematicians and philosophers may choose among many philosophies. You left that out. Probably you didn't even take in that I said it.

paradoxes are the most important guidepost in our quest to see truth. — keystone

I don't know that they are the most important subject, but they have been at the very heart of philosophy of mathematics. It's instructive then how mathematical thoeries are presented to avoid contradiction.

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. — keystone

I have been pointing out that your main arguments are to try to make set theory fit models that are not models of set theory. And your response to that? Another writeup in which you view set theory per a model that is not a model of set theory!

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! — keystone

I'm glad it's not my job to reconstruct verbiage like that so that it makes sense. In the meantime, why don't you learn something about computability theory?

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

When I say something exists I mean there is a computer where it is in memory. [...] 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? — keystone

Wow. You just cannot help yourself from continuing to ask me to make set theory fit your Prorcustean beds.

And you took not a millisecond to think about my own answer.

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. — keystone

Points are not "nothingness".

I honestly thought you were covering both Thompson's Lamp and Zeno's Paradox simultaneously since you included the geometric series in your description — keystone

I exactly stated it as regarding Thompson's lamp.

There is no geometric series in my writeup.

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. — keystone

I said nothing whatsoever about "distance" or "travel". And I said nothing whatsoever about sums.

You're adding things into what I wrote that are not there.

I recommend that you read what I wrote without imposing your preconceptions about it.

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

Meanwhile I showed you math.

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! — keystone

Aha! You're making stuff up again!

There is no notion of continuity mentioned whatsoever in my writeup.

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. — keystone

There is no "journey" mentioned in my writeup. The writeup has nothing to do with "journeys".

I suggest that you clear your mind for just one moment and read my writeup free of incorrect preconceptions about it.