I'm going to respond in two posts. This post covers topics not directly related to my ideas and my next post will cover topics more directly related to my ideas. I hope this covers the important points we've been discussing but let me know if I skipped over something important.

Topology: I now have a basic understanding of the definition of a continuum as compact, connected, and Hausdorff. Connectedness seems straightforward, and I believe it’s a necessary part of any definition. Compactness is more intriguing because "the continuum" itself isn't compact, which is surprising since it means "the continuum" isn't a (topological) continuum. While I understand that compactness is important for results like the Extreme Value Theorem, I don’t yet fully grasp why it's a crucial property of continua. Given that I want define the sort of object suitable for calculus, I'm inclined to not include this property in my definition of continua.

The Hausdorff property, on the other hand, is a more subtle aspect I hadn't considered, likely because most (if not all) spaces I’ve encountered are already Hausdorff. I think it's important and makes sense, but I am unsure how it fits in with my view. This is something I’ll need to explore further in the future. You're right that topology is relevant to this discussion.

Philosophy vs. mathematics: Based on your criticisms, I need to rethink my position. I believe that basic mathematics operates on two parallel tracks: (1) the objects and operations themselves, and (2) the descriptions of those objects and operations. To use a Matrix analogy, this would be like (1) the lived experience within the matrix, and (2) the underlying code or digital rain that describe/define it. Both tracks are fundamental. In basic mathematics, the objects are what we depict, such as points, continua, etc., while the descriptions are the formal logical statements we write down.

Though I’m still learning the basics of logic, I find intuitionism most appealing—especially since you've made it clear that ZFC leaves no room for flexibility when it comes to the ontology of infinite sets (and my focus is on transitioning from actual infinities to potential infinities). To be frank, I'm not willing to accept infinitely many axioms or statements holding a truth value so it's not just a matter of avoiding the Axiom of Infinity. So you're right, I do have an issue with ZFC.

However, my primary focus isn’t the description itself (or the supporting philosophy), but rather the objects and operations they describe. Specifically, I’m interested in the philosophy that justifies the existence of mathematical objects and our operations on them. The orthodox view is a whole-from-parts (point-based) philosophy, which aligns well with ZFC. I, on the other hand, propose the opposite: a parts-from-whole (continuum-based) philosophy. Perhaps this view might align better with CZF, I don't know.

Infinity plays a central role in both (1) mathematical objects and operations, and (2) logical descriptions. I realize now that I’ve been unclear in distinguishing between the two. For instance, my concerns with infinite sets and Cauchy sequences likely fall under (2), the realm of logical descriptions. To address these issues properly, I recognize that I need to deepen my understanding of logic and intuitionism. However, my primary focus is on the objects themselves, such as the Cartesian coordinate system. I believe this system needs a parts-from-whole, continuum-based reinterpretation, as the current understanding relies heavily on the notion of actual infinity.

In short, I would like to see the replacement of the actual infinity based whole-from-parts + ZFC mathematics with the potential infinity based parts-from-whole + ?CZF? mathematics. However, I'm in no position to talk about ZFC vs. CZF.

For reference, here are the axioms of Z set theory — TonesInDeepFreeze

The idea behind these axioms seem sensible yet I'm wary of schemas which imply infinitely many instances of those axioms.

the answer is that a calculus book is about setting forth the most basic mathematics that is put to use in different fields of study; it's not about very much understanding the foundations of that mathematics or its broader mathematical context. — TonesInDeepFreeze

Understood.

And you skipped recognizing that you strawmanned when you said you'd have to study topology to understand my definition. — TonesInDeepFreeze

I thought I indirectly addressed this when I said I was going to go back to your earlier messages on topology and respond to them. But if you're looking for something explicit, yes I unintentionally strawmanned you.

f you wish to engage me with this, then know that first I need for you to determine what are all the possible configurations and then to say exactly which are a continuum and which are not, as I mentioned. — TonesInDeepFreeze

I do wish to engage you with this. On top of being busy these days, I expect it to take me a few days to read, digest, and respond to everything you've said. Please stay tuned.

As an aside, in our previous discussions I felt hurt and turned off by your tone but for some reason in this thread I'm actually quite appreciative of our interactions. Thanks!

If I had to guess where you are headed, I might say that taking a continuum (a line,say) as axiomatic somehow you are cutting it into a fine mesh using the S-B Tree — jgill

I think the S-B tree is just one particularly pleasant way to cut a continuum.

But how this has a bearing to elementary calculus is a bit foggy. — jgill

...we never did get to calculus in the last thread. I don't know whether we'll get there in this thread either. Let's see.

I am working on putting some of your illustrated explanations into actual mathematics. Might take me some time to assemble into a post, hopefully I will finish and post.

But I've been down this road already with you in another thread. I took a lot of time and effort to turn your gibberish into communicative mathematics. Then, all along the way, you revised your idea, so I revised in response, which is fair. But eventually, your proposal came to an impasse of illogic, yet you wouldn't budge and merely insisted on your notions though they had been shown inconsistent. A dead end with you. But maybe this time it could be different. Hope springs eternal. — TonesInDeepFreeze

I see you've already followed up on this. I haven't read it yet as I really want to spend sufficient time digesting it and responding. As I mentioned in my last post, my time is short tomorrow so it may be a couple of days before I respond but I do play to respond to that and the topology comments you made in earlier postings. At this point I'll just say a big thanks to you!

What misinterpretations of the meanings of foundational mathematics? What writings by mathematicians or philosophers are you referring to? — TonesInDeepFreeze

Let me restate the examples I mentioned:

Naïve infinite set theory is thought to be about actually infinite sets when I think it is really about potentially infinite algorithms for constructing the infinite sets. (I want to stay clear of axiomatic set theory since I haven't read the required material.)

Cauchy sequences are thought to be sequences of actually infinite terms when I think they are really about potentially infinite algorithms for constructing the infinite sequences.

When we draw a cartesian plot it is thought that there exist actually infinite points in the plot when I think there really are only finitely many continua, each having infinite potential for partitioning.

First off, why do textbooks for courses in U.S. Civics not mention John Locke, William Blackstone, the Federalist Papers, John Marshall or Plessy v Ferguson? — TonesInDeepFreeze

...not an American.

I didn't say you have to study topology to understand the definition of 'continua'. — TonesInDeepFreeze

All right, I gave up too soon. I'm not going to be able to respond to everything tonight as I'm running out of time and there's a lot to respond to (which I'm greatly appreciative of). Tomorrow is looking like it will be a busy day for me as well but I do plan to respond to everything. Included in that, I'll spend time trying to understand your topological definition of continua and provide a response.

Next time I'll make it even more outlandish for you so that it is inescapable. — TonesInDeepFreeze

Sadly, sometimes I need people to explain jokes to me....

I don't think you're sincere in wanting to communicate. If you were, you would give people the consideration of clearly articulated concepts. — TonesInDeepFreeze

Mathematicians hold a high bar for clarity. Might it simply be that I'm not a mathematician?

I asked you already: Who do you think it's suitable for? Especially if not for a mathematician, then who? — TonesInDeepFreeze

At this point, a mathematician who can piece together informal ideas. At a later point (once I've read more), a mathematician.

Anyway, I speculate that the reason you won't read the substantive material in my posts is psychological. You divert to the false claim that the definition I gave is too specialized....You are so busy espousing that you don't read that to which you respond. — TonesInDeepFreeze

I admit that sometimes when it gets too heavy I glaze over the details. But have I really not adequately responded to many of your points in this thread?

How many examples do you need to appreciate that that bot flat out lies — TonesInDeepFreeze

It works well sometimes though. I see it moreso as a handy tool to use with caution.

That's not the axiom of infinity! It is nonsense to say that you don't object to set theory by recourse to agreeing not with the axiom of infinity but with something very very different! How stupid do you take people to be? How stupid do you think people are not to see the sophistry you just pulled? You're insulting. — TonesInDeepFreeze

I haven't studied axiomatic set theory but I have taken axiom of infinity to mean that there exists an inductive set. Is that not it? What I want to reinterpret this as is 'there exists an algorithm to construct an inductive set'.

And not evidence that even the most patient and open-minded Bodhisattva of a mathematician wouldn't tell you, "Get back to me when you've worked out some math". — TonesInDeepFreeze

Well, that's basically where things ended with you, fishfry, and jgill in the last thread I was active on. That seems to be where things are heading in this thread. I'm starting to get the point.

For the first half of the semester I had hardly a clue what was going on, while some of my classmates seemed to understand the material. Then halfway through all of it suddenly made sense. After that introduction, when I got into the regular curriculum for the next semester it seemed almost trivial — jgill

That seems to be the way with a lot of things. I'm certainly hoping to finally reach such moments of clarity in logic, and eventually topology.

What misinterpretations of the meanings of foundational mathematics? What writings by mathematicians or philosophers are you referring to? — TonesInDeepFreeze

For example:

• That set theory is about infinite sets.
• That Cauchy sequences are infinite sequences.
• That reals are numbers in the same sense that rationals are numbers.
• That the Cartesian (and related) coordinate systems lie at the heart of basic calculus.

The mathematical definition is given in topology. How could the actual mathematical definition not be at the very heart of comparing the mathematical definition with alternative definitions? It seems to me that you're rationalizing your unwillingness to inform yourself on the subject. — TonesInDeepFreeze

Why is it that the intro to calculus/analysis textbooks I’ve read never mention topology? Is it because these texts don’t need a general definition of continua since they only work on the continuum, whereas topology is needed for a defining continua? Keep in mind, you’ve already given me a reading list that I’m just a few pages into. Adding topology isn’t a problem—I’d even prioritize it if it made sense. But I think it’s fair for me to question whether expanding my reading list is really necessary.

What? You didn't immediately apprehend that was a spoof? — TonesInDeepFreeze

Ha! No, I didn’t. But I was being honest with my response. There was so much technical jargon that I had no idea what you were talking about, so I asked ChatGPT. That only made me more confused, so I stopped. And I told you that. The same goes for topology—I stopped because I’m not informed on the subject, and I told you that too.

but you haven't the slightest inclination to even glance over a mathematical definition given to you by a person who has, at extreme length and in extreme detail (in at least two other threads) engaged your notions. Why is that? Could it be in your personal characteristics? (Some variation of being so overly infatuated with your own mind that there's little intellectual juice left in you to bother learning much about the mathematics that other people have given lives of intellectual labor to?) — TonesInDeepFreeze

I think the main reason we're not fully connecting is that I’m not presenting my points in a way that’s suitable for a mathematician, and I’m not fully understanding some of your points because they’re not framed in a way that’s accessible to a non-mathematician. I’ve been using ChatGPT as a tool to help me grasp the more complex ideas, but you discourage that. To be fair, my difficulty responding to your posts on other threads is what motivated me to start studying logic on my own. However, given life's complexities, I only have a limited amount of intellectual juice to dedicate to this. That said, I’m doing my best. But I understand if you decide that we cannot have a worthwhile discussion.

Note: I am passionate about my idea but I don't think that's the main factor here.

What? You started your post by agreeing that it is an adjective. It is an adjective, a predicate in this case. — TonesInDeepFreeze

I agreed with the meaning you intended but not with the exact words you used. That’s my point. I noticed a flaw but filled in the gap to keep the conversation moving. 'Is a continuum' is indeed a predicate, and 'continuum' is a predicate noun. If you had written 'is continuous,' then 'continuous' would be a predicate adjective. But 'is continuous' is neither a verb nor an adjective.

But in this discussion, we see people refer to both 'the continuum' and 'continua', so we should be careful not to conflate those terms. — TonesInDeepFreeze

Yes I get that. I've been consistent with this.

So you have no objection to the axiom of infinity itself, only with philosophizing that there exist "actual" infinite sets? And what do you mean by "actual"? If one views mathematical sets to be mathematically actual but one does not opine as to whether there are physically actual sets, is that okay with you? If one holds that abstractly there are infinite sets but one does not opine that physically there are infinite sets, is that okay with you? — TonesInDeepFreeze

My view is that the Axiom of Infinity represents an inductive algorithm for constructing the inductive set, which is said to have a cardinality of aleph-0. I don’t believe the inductive set itself exists; instead, the inductive algorithm is all we need, and it carries the same cardinality. When someone invokes the Axiom of Infinity, they’re really presenting the inductive algorithm, which is what I believe in. No one has ever exhibited an actual inductive set abstractly in the way that a finite set can be exhibited abstractly. And to be clear, I’m not referring to its physical existence.

Meanwhile, it seems that the point of my parody went past you — TonesInDeepFreeze

I must admit that it did.

It is to be vigilantly mistrusted. — TonesInDeepFreeze

But should it be mistrusted in 2-5 years?

Who do you think they are comprehensible to, other than yourself? — TonesInDeepFreeze

A mathematician with plenty of patience and an open mind...so far it's only been ChatGPT...

I'll look at this later, if my time, patience and supply of snacks is adequate. — TonesInDeepFreeze

The next bag of Sweet Chili heat Doritos is on me. :P

I will be interested in what younger and more agile brains make of this. :chin: — jgill

Me too. I often think about giving this up and just spending my free time like most people do. I'm truly open to being pushed in either direction.

I've never been able to see where it is you are going. Maybe it's just me, old and weary of days. — jgill

It's not just you. My ideas are half-baked and I'm not great at communicating them in a way digestible to mathematicians. I appreciate you following along even though you don't see where it is I am going with this.

• fundamental objects: indivisible wholes that serve as the basis for constructing composite objects (which are composed of other fundamental objects) through partitioning.
• by its very nature: the characteristic of concern is fundamental and inseparable from the object or concept itself
• multiple vertices representing the same node: multiple vertices that mistakenly represent the same object (are consolidated into a single vertex).
• broken down into finer and finer elements: Used as a basis for constructing a composite continua composed of more and more objects.
• change the continuity: When a continuum is used as the basis for constructing a composite object, there is a disagreement in meeting the continuity requirements between the original continuum and the composite object constructed based on it.
• acts: procedure
• algorithm describes: Outlines the process of
• The cardinality of an algorithm is determined by comparing it to other algorithms whose cardinalities are already known: I'm trying to capture the essense of Cantor's diagonal argument.
• continua one partition at a time: Continua are fundamental and require little explanation. What matters is how they serve as the basis for constructing composite continua, and how this process connects to established mathematics. This process involves partitioning, which is what I've been describing.
• connected: Hopefully my illustrations in my last post make clear that I mean connected in the exact same sense as it is mean in mathematical graphs.

Ostensive indications and understandings not from explicit definition but rather from gleaning in context are fine and useful, mainly at the stage of basic intuitions. — TonesInDeepFreeze

Consider the following as an ostensive definition. Is this not basic?

As of a recent amendment to site guidelines, using ChatGPT as an academic source is grounds for a warning. You're not being warned for it now, I just thought I'd bring it to your attention, so that you can avoid doing it again. — fdrake

:up:

So, I don't know why ChatGPT "struggled to make sense of it" for you. — TonesInDeepFreeze

The following is for your amusement only. It's not worth our time to discuss the validity of this response.

(1) 'the continuum' is noun that names the set of real numbers. (2) 'is a continuum' is an adjective that we talking about defining. — TonesInDeepFreeze

Agreed.

Then that is mathematics. Or do you mean the study of physical phenomena that calculus is used for? — TonesInDeepFreeze

My focus is on the philosophy underlying the rigorous mathematical foundations that support the informal mathematics used in physics. I have no issue with the informal mathematics itself (e.g., the calculus of Newton and Leibniz), but I believe there is a disconnect between these informal methods and the rigorous foundations (e.g., Analysis of the continuum) meant to support them. This disconnect isn’t due to flaws in the rigorous foundations but rather stems from a misinterpretation of their meaning and philosophical implications. Specifically, I argue that the concept of actual infinity, which plays a central role in these interpretations, should not be part of the framework. With that said, I want to stress that the work of Cantor, Cauchy, etc. play a central role in my philosophy.

I can't make heads or tails of whatever it is you're trying to say....your notions are not of that basic kind but rather are intricate enough that they need more than ostensive definitions. — TonesInDeepFreeze

I believe an ostensive definition with figures is exactly what you need to make sense of what I'm saying. I'll provide that in a separate post. I understand that this approach isn't for everyone, so no worries if you choose not to respond to it.

Understanding the real numbers and the continuum in context of topology is definitely not a distraction. And why would topology be a distraction but your half-baked verbiage not be a distraction? — TonesInDeepFreeze

I want to learn topology, but in this thread I think it makes sense to focus on the topics that are absolutely essential for understanding continua (such as real numbbers). I don’t believe topology, at least in the way you’re suggesting, is necessary for this understanding. Of course, my current ideas are half-baked, but I believe the direction I’m taking is key to grasping the true nature of continua.

A topological space C is a continuum if and only if C is compact, connected and Hausdorff. — TonesInDeepFreeze

I'm not going to challenge this definition because (1) I'm not well-versed in topology and (2) I still recognize its value. However, the philosophy behind applying this definition to describe non-trivial continua assumes the existence of infinite sets, which I believe is unnecessary.

When you posted a purported "proof" by ChatGPT, I even exactly showed you that ChatGPT doesn't know what it is doing*. * You didn't acknowledge that. — TonesInDeepFreeze

I didn't? If I didn't, I accept that it gave an invalid proof.

My definitions are very straightforward use of just the basic logical symbols and a few basic symbols of set theory. And if you don't understand the notation...then you can ask. — TonesInDeepFreeze

A few questions wouldn't be enough. I suspect I'd have to read at least one textbook to truly understand what you mean by "A continuum is a decomposition of a hyperspace module with sidewise-partitionable step-wise radii."

The original post in this thread purported to prove that the continuum does not exist. The argument wasn't that it doesn't exist in nature but that it doesn't exist mathematically. — TonesInDeepFreeze

Maybe I should do as jgill recommends and create a new thread. Thoughts?

Then I would be right to ask you to defined the undefined terminology in those definitions, and again until (1) You finally fail to reach primitives or (2) You end up in a circle or (3) You do reach primitives. — TonesInDeepFreeze

Yes, but readers often offer a certain level of leniency to keep the conversation flowing smoothly. For example, you stated that 'is a continuum' is an adjective, which it isn’t. I didn’t want to nitpick that because, as a reader, I can easily fill in the gaps and understand what you meant.

As I understand, you reject using infinite sets. But you say that we encounter continua. So continua are finite? — TonesInDeepFreeze

Yes, that's the first part of my definition - continua are finite objects. I believe all objects (mathematical and physical) are finite. At the same time I believe that the rigorous mathematical foundations of set theory, including all the work of Cantor, are valid, and that my only qualm with set theory is the philosophy underlying it (centered around actual infinities). But perhaps we should focus on continua for this discussion...

So we begin by defining such curves as "inherently continuous". That seems to solve the problem. Why proceed? Why dabble with sets of points that may fill up a curve - or not?
Just chop up such curves and there we are. Bend them a bit to go to two or more dimensions. — jgill

I detect some sarcasm here. Isn’t the conventional approach to start with points, which are inherently discrete? How is that any less of a leap than starting with curves, which are inherently continuous? In fact, I would argue that you've encountered far more actual continua in life than individual actual points.

Why proceed? Because an unpartitioned continuum only holds the potential for usefulness. In its unpartitioned state, it’s effectively a void.

Just as a point can’t be manipulated to form a curve, a curve can’t be manipulated to form a surface. An open-ended surface is its own fundamental object...

I can tell you've taken care to read my words closely. Even though we sometimes clash, I truly appreciate your effort.

• complete set of attributes constituents: All the fundamental objects that make up the continuum
• can be fully described constructed: Can be built or generated from its constituents
• invoking infinite processes: Using a finite series of steps or operations.
• open-ended curve: A curve that does not have endpoints.
• inherently continuous: By its very nature is continuous.
• duplicates removed: In a mathematical graph representation, multiple vertices representing the same node are removed.
• connected to: As defined in the mathematical graph context.
• 0-2 curves: 0, 1, or 2 curves. (What I'm doing is perhaps in the spirit of Euler's polyhedral formula but for mathematical graphs representing continua)
• 0-2 points: 0, 1, or 2 points.
• refined composite object: broken down into finer and finer elements
• increasingly refined: Progressively more subdivided
• fundamental elements: The basic building blocks or primary components that make up a larger object or structure. In this 1D context, they are open-ended curves and points.
• maintaining continuity: Does not change the continuity (as defined).
• partitioning process: The act of partitioning (as described).
• algorithm devised allowing further division: Such an algorithm describes the partitioning of the object such that further partitions are possible.
• algorithm reveals / aligns with: The cardinality of an algorithm is determined by comparing it to other algorithms whose cardinalities are already known (or assumed), in this case, the reference algorithm is that for indefinitely partitioning a continua one partition at a time.

My experience on this forum is that people don't like figures, but let me just include one as maybe it will help.

My Claim: You asked me to define continua, which I assume was prompted by my earlier claim that "THE continua" is not actually used by applied mathematicians when they describe "physical continua." By "stage," I am referring to the basis on which we can discuss this claim.

Formal Logical System: I fully accept the recommendations from you and others in this forum to study mathematical logic (and intuitionism). I'm in the early stages of actively pursuing this in my spare time, but I haven’t yet reached the point of formalizing my ideas into a logical system.

Established Mathematics: I agree that the more established math I learn, the better. However, my primary concern is the continua used in basic calculus, so I wonder if delving into topology might be more of a distraction. I also question whether your definition of a continuum is unnecessarily complicated. For reference, even ChatGPT struggled to make sense of it and had a humorous take. I do appreciate that the mathematics presented earlier in this thread seems more relevant to this thread's elementary discussion on continua, but that math is rooted in actual infinities. I just don’t see how applied mathematics has any need for or use of actual infinities. I see an alternative.

In any case, I’ll provide the requested definitions in my next post. I hope you continue that discussion at least a little more but if I’m heading down a path you'd prefer not to explore, no worries.

Define 'continua'. Preferably a mathematical definition. And most preferably not free-floating, hand-waving verbiage. — TonesInDeepFreeze

The following is not a formal, finalized definition, but I hope sets the stage for the discussion:

A continuum is a finite, continuous object with the potential for arbitrarily fine partitioning, characterized by the cardinality $2^{\aleph_0}$.

Finite object: Finite in the sense that its complete set of attributes can be fully described without invoking infinite processes.
Continuous object: In 1D, the proposed fundamental objects are of two types: (1) open-ended curves, which are inherently continuous, and (2) points. A composite 1D object is the union of these fundamental objects and is continuous if, when duplicates are removed, the following conditions are met:

• Points are connected to 0–2 curves (but not to other points).
• Curves are connected to 0–2 points (but not to other curves).
• No objects are disconnected from the composite structure.

Potential for arbitrarily fine partitioning: The continuum can be subdivided into an increasingly refined composite object made up of arbitrarily many fundamental elements, maintaining its continuity.
Characterized by the cardinality $2^{\aleph_0}$: The partitioning process can be described algorithmically, such that no algorithm can be devised allowing for further division. Although this algorithm would not halt if executed, the structure of the algorithm itself reveals that the potential for infinite subdivision aligns with the cardinality $2^{\aleph_0}$.

I have refrained from providing examples or illustrations for the sake of brevity, though they could help clarify my position.

Define 'continua'. Preferably a mathematical definition. And most preferably not free-floating, hand-waving verbiage. — TonesInDeepFreeze

I keep writing a response to this and then deleting it. I'll be back on this ..

@TonesInDeepFreeze: Apologies if this isn't the best forum etiquette, but I'm genuinely curious about your background. Would you mind sharing a bit about your education and career?

The argument that MoK gave involved the real numbers and their ordering, and real intervals, and his own confused notion of infinitesimals. He gave a definition of 'continuum' that sputtered. And he argued that the reals are not a continuum. His arguments were a morass. And given his personal definition of 'a continuum', he was refuted that the reals are not one. — TonesInDeepFreeze

I believe his arguments indirectly captured the spirit of the Zeno's Paradoxes. I believe Zeno and MoK communicated their ideas informally but nevertheless there's something profound about them. Anyway, I don't want to further defend MoK's position as I have enough on my own plate to defend my own position.

Post removed

I think it is good you are getting back into the discussion. Who knows what might come out of this thread? My only reservation - and ignore if you like - is to perhaps not bring up the Stern–Brocot tree. — jgill

I hadn't lost interest but I needed time to reflect and read. Actually, I still need a lot more time to do that but this forum is too hard to resist. Already my return to the forum has set me on a new reading trajectory so it's already a win. As for Stern-Brocot...no promises :razz:

But the point of the argument by MoK was to first simply show that the continuum does not exist. That argument by him was shown to be ill-premised and confused. — TonesInDeepFreeze

@MoK I know your wording taken literally is in agreement with Tones' view, but were you originally trying to prove that "the continuum" does not exist or that continua in general do not exist? I believe it was the latter.

Df. the continuum = <R L> where R is the set of real numbers and L is the standard ordering on the set of real numbers.*

So, of course, there are points involved. — TonesInDeepFreeze

I accept this definition of 'the' continuum. It's a definition after all so there's nothing to question. My issue is with using 'the continuum' to describe 'a continuum'. I believe when applied mathematicians describe continua they may think they're using "the continuum" but in reality they're using something else (very closely related).

So, I'll remove to the safer ground of my definitions of 'the continuum' and 'continuous function' and leave 'a continuum' alone. — TonesInDeepFreeze

I like this subtle distinction as it draws a clear line between your interests/arguments (related to the continuum) and mine (related to continua in general).

I think it is better to first get good at working in first order logic and then study the meta-theorems about first order logic. That's why I recommend this three-step sequence — TonesInDeepFreeze

I had debated Enderton vs. Mendelson at the start and had landed on Mendelson because the topics in the TOC looked much more interesting. However, it's been quite slow progress with Mendelson so I'm still in a position where it's not too demoralizing to switch. Okay, I'll give you're recommended reading a try. Actually, I do have Enderton's book on Set Theory which I read part of many years ago. I might skip this book. Thanks.

AI might become reliable in the future, or it could get a lot worse. A bot skims Internet articles that are themselves of dubious authority. The bots re-propagate the misinformation and even fabricate new misinformation. Then people re-propagate the misinformation re-propagated by the bots. A vicious sewage circle. — TonesInDeepFreeze

Yeah, cleaning training data is certainly a challenge that gets harder with time. I wonder if they'll end up giving pre-GPT data more weight. I'm optimistic though that they'll figure it out.

You were talking about learning basic mathematical logic. You said you are supplementing the book you are studying with AI. — TonesInDeepFreeze

Fair nuff.

Who is "the other poster"?
What standard mathematical definition of 'continua' are you referring to? — TonesInDeepFreeze

I'm not sure who Mok meant by other poster but I assumed it was you. For example you wrote the following:

An ordinary mathematical notion is that the continuum is the set of real numbers along with the standard ordering of the real numbers; then a continuum is any set and ordering on that set that is isomorphic with the continuum. — TonesInDeepFreeze

I suspect you'll say that's not point-based since points are not explicitly mentioned...

You're referring there to MoK. He argued that the continuum does not exist. I don't recall that he mentioned paradox (maybe he did?). — TonesInDeepFreeze

I should not have used the word 'paradoxical' but rather logically impossible.

You are really going down the wrong road by resorting to AI for explanations. You are bound to take misinformation and confusion from it. — TonesInDeepFreeze

Thanks for the analysis. That's disappointing. What's unfortunate is that my textbook by Mendelson has many examples yet provides answers to only a small subset of them no explanation is provided. I'll have to tread carefully...

Yes, my definition of the continuum is not adequate. Another poster gave a definition continuum close to mine but it is correct. I can search the thread and find the definition for you if you are interested. — MoK

Both you and the other poster's definition of continua were point-based. I acknowledge that that's the standard mathematical treatment of 'mathematical continua'. But if you're trying to prove that continua do not exist because mathematical continua are paradoxical then I would argue that there's simply a problem with using 'mathematical continua' to model continua. If I recall correctly, another poster mentioned point-free geometry. There's also Euclid's geometry for which continua are fundamental.

Or have you concluded that your argument is nonsense?

Of course, people are different, so my personal route might not serve everyone. But I venture to say that if I had relied on chatbots to supplement the books, I would have been quite confused and misinformed. — TonesInDeepFreeze

Not only are people different, but times are changing. If now is not the moment for AI, the near future will be.

Of course, one doesn't have to come within a million miles of a PhD just to learn basic mathematical logic. — TonesInDeepFreeze

I suppose it depends on what we're talking about. If we're talking about developing original ideas on continua I think more than a basic understanding of mathematical logic is required.

See if you can get it to provide a proof that doesn't assume what is supposed to prove. — TonesInDeepFreeze

I don't know what chatbot you used and I'm not skilled enough at logic to evaluate the output of OpenAI's latest o1, but I'd be keen to know what you think:

Here is the argument:

D) By continuum I mean a set of distinct points without an abrupt change or gap between points — MoK

I like your argument but I would say that the conclusion that follows is that D is an inadequate definition of a continuum. A continuum cannot be completely described with points. I've had ramblings in another thread on an alternate view where continua are fundamental and points are derived but those conversations ended similar to your thread where others (rightfully) recommended I read more.

Well that's what I'm doing! I'm reading a book on the fundamentals of mathematical logic. I disagree with those here who say that ai chat bots are not helpful. As I'm working through the text and problem sets, gpt4o and now o1 have been instrumental in me understanding the material. It's like having a personal teaching assistant. It's not perfect but its benefits far outweigh it's drawbacks if you can't afford the time or money to focus on a PhD.

Sounds good!

If you stay away from the SB-tree and Niqui arithmetic I might linger a bit longer. — jgill

I see no need to mention those topics any time soon.

Why not go directly into 2D. — jgill

Without sufficient priming, you will almost certainly reject the 2D images I intend to share. I'm going to just spend this message covering the remaining 1D topics and if you're still following and interested I'll follow up with a message on 2D.

----------------------------------------

In 1D, figures adds no value as they are equal to the k-interval union. However, this is an ideal situation as it allows us to ease into unorthodox top-down figures.

Here's a k-interval union:
<-∞, 1-ε> ∪ 1-ε ∪ <1-ε, 1+ε> ∪ 1+ε ∪ <1+ε, +∞>

Below is that same k-interval union placed above a grey box containing an illustration of k-line geometry (with matching components vertically aligned and color coordinated such that k-lines/k-intervals are green and k-points/fractions are red).




I wrote <1.0> as a k-interval and placed it below <1-ε, 1+ε> because they both describe the same k-line.

I want to highlight one important difference between k-line systems and Cartesian systems.

All k-line system properties (such as order and length) are preserved under continuous deformations. In other words, all of the 3 systems illustrated below correspond to <-∞, 1-ε> ∪ 1-ε ∪ <1-ε, 1+ε> ∪ 1+ε ∪ <1+ε, +∞>. It is for this reason that complete k-line systems can fit on a finite page, since 'k-points' at infinity need not be far off in the distance. From our discussions, I've learned not to call these systems topological as that term comes loaded with meaning that doesn't apply. However, to capture the idea that properties are being preserved under continuous deformations I'm going to say these systems are k-topological. So if in 2D you see a graph that looks weird, I will likely justify the weirdness by saying that it is k-topological.



Also, please note that if ε were allowed to equal 0 then a k-topologically different system would result. As such, ε must never equal 0.

What do you think?

Your top post lured me back from my vacation. — fishfry

Our relationship was mostly one-sided, with me being the main beneficiary of our conversations. I'm glad to hear that you found some benefit in meeting me as well. This is a great way to conclude our conversation. Cheers!

I assumed you might begin with something akin to contours in the plane, but you went another direction, and sticking with one dimension I think was very limited, and rather boring I fear. — jgill

Indeed, I went in another direction and 1D is boring.

If you were to return to the beginning and speculate continua that precede points, or something similar, the thread might continue. Just my opinion. — jgill

This is exactly what I want to return to. Here's what I'm thinking:

If we keep going, I'll summarize the 1D story so far and then spend a couple of messages discussing the visual representation (using lines instead of intervals). After that, I'll move on to 2D, exploring higher-dimensional analogues of intervals/lines, namely equations/faces. Then, I'll spend a few messages examining the meaning of roots, derivatives, and integrals from this top-down perspective.

However, I'm not sure if your advice was about how to appeal to you specifically or more about attracting a new audience. If you're not interested, that's totally fine, and I'll move on. If you want to continue but prefer to take it one message at a time, then we want the same thing. I'm not asking for a long-term commitment like the one fishfry gave me. But if we proceed, I want to ensure it's a conversation, not just me 'lecturing' to the bleachers. What do you think?

Or, do your thing and persist until the thread dries up and vanishes. Good luck. — jgill

Well it's been a couple of months but we're finally at the state you predicted. I know you've been on the bleachers largely but now and again you've injected a comment which suggests that you've been following. Do you care play a more active role in the discussion or would you rather leave it at that and let this thread 'dry up and vanish'?

I've done all I can. We're at a point where our interests have diverged. I need to wrap up my end so perhaps if you have any final questions, or perhaps if you come back now and then with more ideas formed over time, we can chat. So a pause, if nothing else. — fishfry

You stuck with me for an incredibly long time, and while I wasn't specifically looking for a sounding board, it turns out that's exactly what I needed. However, we've now reached a point where this discussion requires more than just a sounding board; it needs someone who can truly digest and engage with what I'm saying. Given this, it makes sense for you to disconnect now. This isn't due to a lack of topics to discuss, but rather because you're not interested. As such, I don't foresee us picking up this conversation again in the future, but who knows. Thank you so much for staying with me up to this point. I've gained a lot from our discussions, and I wish you all the best!

[Deleting duplicate]