I'm currently feeling unwell and will reply shortly. Cheers, and thank you for continuing this dialogue. — keystone

No worries, as they say. Get well soon. — fishfry

You can see that if x is in (0,1), then x is in a least one (actually all but finitely many) of the sets (1/n, 1 - 1/n). — fishfry

I would have to give this some thought. Would it make progress if I stipulate to your metaphysics? I don't know what to say anymore. — fishfry

Yes ok, so if you have an alternate way of getting to the same real numbers, what does it matter? — fishfry

Not a line, a nested collection of lines. The point zero is (-1, 1), (-1/2, 1/2), (-1/3, 1/3), etc. — fishfry

Every interval containing a given real number, necessarily contains other real numbers. That's the definition of (not) being isolated. — fishfry

The inability for dimensionless points to be reconciled with the continuum is what motivated Whitehead's point-free geometry, a precursor to the field of Pointless Topology, as for instance formalised using Locales whose distributive law characterizes the meaning of a "spot". (It might be useful to test this law in relation to the SB tree, for both the truncated and infinite version). — sime

Thanks! Mostly better now. — keystone

Allow me to further clarify my position. I can write (0,1) as the union of arbitrarily many disjoint intervals. However, I cannot write (0,1) as the union of infinitely many disjoint intervals. — keystone

Do you think you understand my position so far (and perhaps don't agree with it) or do you have no clue what I'm proposing? — keystone

I don't have an alternate way of getting to the real numbers. What I lay claim to is the real points, not the real numbers. — keystone

Consider the ruler depicted below. — keystone

It features 96 tick marks
... — keystone

The fact that the length of each line in your sequence is getting shorter is a red herring. Every single line in your sequence is composed of exactly 2ℵ0
2
ℵ
0
points. The point count isn't converging to 1. What you've exhibited is not actually a nested collection of lines but an algorithm for generating such a collection (or at least the essence of an algorithm). This distinction is crucial because the algorithm, if executed, doesn't halt. If you chose to execute the algorithm, the best you can do is wait for a long time and interrupt it when the last line produced is sufficiently small. In other words, the output of the algorithm is an arbitrarily small line, not a point. — keystone

Do you believe individual rational numbers can be isolated? — keystone

I believe they can. I'm going to use the SB tree to illustrate my view, — keystone

not because it's essential but because it's familiar. — keystone

I can cut this tree such that left of the cut is (0,1/2) and right of the cut is (1/2,inf). With this cut, I've isolated 1/2. I cannot do the same for irrational numbers. — keystone

I found this paper that adopts intervals instead of points in its framework, which is quite relevant. — keystone

It's a very complicated game requiring perseverance and dedication. Are you in it for the long haul? — jgill

Not bird flu I hope. Jeez the medical propaganda is everywhere these days. Are we all doomed? Like not eventually, but as soon as next week? — fishfry

Correct, but why does that matter? (0,1) is already the disjoint union of open intervals, namely itself. — fishfry

But points and numbers are entirely synonymous in this context. The "real line" is just the set of mathematical real numbers. — fishfry

I literally have no idea what we've been talking about the past several weeks. Which makes me feel foolish sniping at it. — fishfry

As a sickly child, when I felt ill, I would imagine myself as heroically fighting severe illnesses, attributing my survival to extraordinary strength. Turns out, I'm just wimp. I was probably just dealing with a common cold last week. Fortunately I wasn't in tune with any of the bird flu news...anxiety doesn't usually help... — keystone

Correct, but why does that matter? (0,1) is already the disjoint union of open intervals, namely itself.
— fishfry

The issue revolves around whether the part or the whole is primary. — keystone

If parts precede the whole, then logically, I should be able to union such parts to create the whole, which you acknowledge is not feasible. — keystone

Conversely, if the whole precedes the parts, then I should be capable of bisecting the whole into smaller sections, continuing to do so until I have arbitrarily small parts. This approach is feasible. — keystone

Yes, individual points are entirely synonmymous with numbers. However, continuous bundles of points are synonymous with intervals. And what I'm saying is that it's these continuous bundles of points (described using intervals) that are fundamental, not the individual points (described using numbers). We start with a continuous bundle of points (described using an interval) and when we cut it (ie. bisect this interval), we not only create smaller continuous bundles of points but also isolate an individual point in between (described using a number). Hence, the individual points and their associated numbers emerge from the bisecting process; they do not exist as independent objects before it. Individual numbers and points are emergent. — keystone

When I presented that table and you wrote 'I would have to give this some thought' but didn't follow up on it, is it that you don't want to consider an alternate view? — keystone

It's a very complicated game requiring perseverance and dedication. Are you in it for the long haul? — jgill

Why do you want to know? — keystone

I find their concepts a bit challenging to grasp just by skimming. It seems like a thorough reading might be required to truly understand these ideas, something I'm not quite ready to dive into — keystone

That's why the Amish communities weren't hit hard — fishfry

Why do you insist on the one decomposition that we can't do? — fishfry

...it certainly is the pairwise disjoint union of ONE interval, and so what? — fishfry

Conversely, if the whole precedes the parts, then I should be capable of bisecting the whole into smaller sections, continuing to do so until I have arbitrarily small parts. This approach is feasible.
— keystone

You surely can't do that with countably many cuts. — fishfry

And your bisection idea doesn't work, you can't get any irrationals that way. But I believe you've already agreed with that. — fishfry

When it comes to the real numbers, I do think building the parts from the whole is difficult, because you'd need uncountably many cuts. — fishfry

But Dedekind has already built the reals from cuts of rationals, so it can be done. But there are uncountably many cuts. — fishfry

I asked you earlier: Suppose that rather than snipe line by line at this paragraph, I just accept it for sake of discussion. Can we move forward? — fishfry

But of course your whole approach is pointless (that's a pun) so maybe I'm getting it. — fishfry

Thankfully, ↪fishfry is there to help guide you. — jgill

I want to make it clear that a line cannot be constructed from/defined by infinite isolated points (numbers) or micro-lines (intervals). If that's clear then what is a line --> — keystone

Yes! Forget about declaring that the line is infinite individual things and instead call it ONE thing, ONE bundle, described by ONE interval. This is an important distinction because it frees us from actual infinity allowing for a stronger foundation. — keystone

I can't do what? How small do you want the bundles to be? I assure you, I can divide them as small as you wish. Of course, I can never cut a line down to indivisible bundles, but I never claimed I could. Why would we even need that? — keystone

You're right that I can't execute a cut to isolate an irrational point. However, what I can do is develop an algorithm that defines an endless cutting of the line such that the line segment containing the desired irrational point gets arbitrarily small. — keystone

As we've agreed, that algorithm is the irrational. There's no need to declare that the algorithm can be run to completion to output an irrational number. — keystone

The algorithm is sufficient in and of itself. And if I need a number, I can interrupt the algorithm to deliver an arbitrarily narrow interval with rational end-points and I can pick a suitably close rational number within that interval. — keystone

Now, I can't isolate a non-computable this way, but that's not a problem. The non-computable points are not missing from my view. They are included - my line is continuous. The non-computable points just cannot be isolated. But we don't need to isolate them. They fulfill their job being constrained to bundles. Do they not? — keystone

My argument is that we don't need completeness. — keystone

Let's embrace our inability to fully execute a non-halting program. Our inability to isolate everything is a feature of my view, not a flaw. After all, why do you need 2ℵ0
2
ℵ
0
isolated numbers? — keystone

Let's lay out all countably infinite rationals in an ordered line. How many gaps are there - countably many? — keystone

What is the difference between a gap and a Dedekind cut? — keystone

If they are the same, how do we arrive at uncountably many cuts? — keystone

The answer is that Dedekind doesn't ever execute the cut. Dedekind Cuts only make sense if they correspond to non-halting algorithms which by definition cannot be executed completely. — keystone

I had asked whether you understood what I was saying and you said you literally have no idea. It's hard to move forward if nothing I'm saying is coming through. — keystone

Ha. My view has points, they're just not fundamental. Points emerge when a cut is made, but the line doesn't come precut and nobody could ever completely cut a line. — keystone

Thankfully, ↪fishfry is there to help guide you. — jgill

Well then the intermediate value theorem is false. Calculus would collapse. — fishfry

I'm really tempted to respond to all your latest comments, but you're getting impatient, so I'll hold back and move forward. — keystone

In later posts, I aim to demonstrate that calculus not only remains intact with my perspective, but is actually built on firmer foundations. However, before we advance I'm going summarize the essentials so far. If you understand what I'm saying (even if you don't agree) we'll be ready to proceed. — keystone

1) Initial Composition: My line consists of the same points and numbers as the real line. However, initially, the continuous points bundle together to form a line, and the continuous numbers bundle together to form an interval. Thus, we begin with a single object (a line) described by a single interval.

2) Isolation Through Cuts: A point/number can only be isolated from the line through a cut. Until the cut is executed, it is meaningless to refer to the point/number as an independent entity.

3) Rational Cuts: A rational cut corresponds to isolating a rational by bisecting the line.

4) Irrational Computable Cuts: An irrational computable cut corresponds to a non-halting algorithm that isolates an irrational computable within an arbitrarily small interval. This cut cannot be executed completely.

5) Irrational Non-Computable Cuts: These cuts don't exist. Irrational non-computables cannot be isolated.

6) Completeness: All the points are there from the start (bundled in the line) so in a sense the line is complete. However, it is impossible to fully cut the line such that all points/numbers are isolated so in a sense the isolated points/numbers are incomplete.[/qouote]

Without disputing point by point (for example, you're wrong about noncomputables, because if you don't have the noncomputables, you can't have completeness), it doesn't matter. If you have the real numbers, they're the same real numbers.

And your claim of completeness but no noncomputables is inconsistent. One of those has to go.
— keystone

PLEASE try to understand the following example (including the figures!). This is essential for me to make any progress explaining why calculus doesn't collapse with my view. — keystone

Notice that in these 1D examples the figures contain the same information as the unions. It contains no additional information, but when we move to 2D, the figures become much more significant. — keystone

π is the familiar irrational number and ε1 and ε2 are arbitrarily small positive numbers. — keystone

Their independent values are not important as they are never used in isolation. What's important is that π-ε1 and π+ε2 are rational numbers and π lies within the arbitrarily narrow interval (π-ε1,π+ε2). — keystone

Do you follow what I'm saying? — keystone

So I will stipulate that you have a construction of the real numbers. — fishfry

Well if you have the intermediate value theorem and the least upper bound property -- ie, completeness -- then what you have, whatever it is, is isomorphic to the standard real numbers. — fishfry

No, I don't need to. If calculus works, then you have the standard real numbers. — fishfry

No such thing as an arbitrarily small positive real number. — fishfry

def small_number_generator():
    n = 1
    while True:
        print(n)
        n /= 2

But then, so what? I keep asking you that. — fishfry

I don't have an alternate construction of the complete set of isolated real numbers. — keystone

I don't have the intermediate value theorem or the least upper bound property. — keystone

I acknowledge that for the bottom-up view, calculus requires the complete set of isolated real numbers, the intermediate value theorem, and the least upper bound property to "work"...I use quotes because it also requires some mental gymnastics. However, that's just not the case for the top-down view. It works perfectly in absence of all of the above...including the mental gymnastics. — keystone

No such thing as an arbitrarily small positive real number.
— fishfry

Consider the following Python function:

def small_number_generator():
n = 1
while True:
print(n)
n /= 2

If executed, this function will print 1, 1/2, 1/4, 1/8, 1/16, and so on to no end. By saying that there is no smallest positive number you are essentially acknowledging that this program does not halt. — keystone

I agree with that. What I'm saying is that for any positive number you provide, x, I can run the code in a finite amount of time to print out a number smaller than x. In other words, it has the potential to print out a number as small as you want but it cannot actually print out the smallest positive number, any more than it can halt. Do you understand this distinction? — keystone

Moreover, without ever executing the program I can describe the function's potential. Assuming it will run for at least a little while, at any time the last number it will have actually printed, ε, necessarily cuts the line (0,1) as depicted below: — keystone

Again, until we execute the function ε doesn't hold an actual value. In this illustration, ε is simply a placeholder. The fact that I drew it approximately 1/3 between 0 and 1 is inconsequential. All that can be said is that if executed, ε will correspond to a point somewhere between 0 and 1. That's how you should interpret the drawing. — keystone

In this light, I ask that you revisit the example from my last post and see if you understand step (3) where I plan an irrational computable cut at π. I specifically wrote plan there instead of execute because I wanted to focus on the potential of the cut, as I have done for the program illustrated above. — keystone

I keep trying to advance forward but your responses continue to either directly or indirectly show that you're not following. If you don't understand what I'm illustrating when I plan an irrational computable cut at π then you won't understand my 2D illustrations that demonstrate that the IVT and the LUB property are not required for the top-down view. — keystone

I acknowledge that for the bottom-up view, calculus requires the complete set of isolated real numbers, the intermediate value theorem, and the least upper bound property to "work"...I use quotes because it also requires some mental gymnastics. However, that's just not the case for the top-down view. It works perfectly in absence of all of the above...including the mental gymnastics — keystone

You claimed completeness. Do you now retract that? Or have a private definition? — fishfry

If executed, such a program will eventually output the same number over and over, until its computing resources run out. You are factually wrong and I hope you can see why. — fishfry

Even so, there is no smallest positive real number, and you have not provided an argument. — fishfry

• 1) Assume that any computer running the program will eventually overflow and stop printing numbers.
• 2) If we limit our scope to computers that print at least two numbers, for any given computer there will always be a last number printed (let's assign this property of a given computer the label e).
• 3) For a computer with few resources, e might equal $2^{-5}$.
• 4) For a computer with more resources, e might equal $2^{-1,000,000}$.
• 5) In our observable universe, there is a smallest possible value for e.
• 6) However, it is possible that in an existence outside of our observable universe e can be even smaller.
• 7) Instead of placing a finite upper bound on the size of a computer, it is convenient to assume that a computer can be arbitrarily large but must necessarily be finite. To be clear, I'm not claiming that there is an infinite computer. I'm simply remaining agnostic on the bounds of finite computers.
• 8) Thus, e can be arbitrarily small but must necessarily be a positive number. To be clear, I'm not claiming that there is a smallest positive number. I'm simply remaining agnostic on the bounds of finite computers.
• 9) e is not a property of the program, but instead a property of the computer executing the program.
• 10) To describe the program, we must talk in general terms. Let ε be a possible value for e.
• 11) ε necessarily lies on the open interval (0,1).
• 12) I illustrate this description of the program as follows:

But even so. I have repeatedly asked you to give me the big picture. Give me something. — fishfry

• 1) Get you to agree to my use of ε in the computer example (including understand the illustration).
• 2) Get you to agree to my use of ε in the pi example (including understand the illustration).
• 3) Progress to 2D, where the Cartesian Coordinate system is replaced with a top-down alternative, and the zeros of y=x^2-2 have a very different meaning. Illustrations become important here which is why we need to get past 1) and 2) first.
• 4) Top-down interpretation of calculus.

What is an isolated real number? — jgill

Show us elementary calculus from the top down. I am curious. — jgill

As I said earlier, I've got the points bundled into a continuous line, but not all of the points can be isolated. So if by 'completeness' you mean a line without gaps then my line is complete. However, if by 'completeness' you mean a line that can be described as the disjoint union of infinite points/numbers then my line is incomplete. — keystone

What you are essentially saying is that a turing machine cannot operate on an infinite memory tape since such a tape cannot exist in a finite world. Ok, you're right. — keystone

I largely agree but I would phrase it as 'there is no smallest possible positive number'. This distinction is important if numbers are emergent but it's not worth discussing at this time. — keystone

Let me rephrase my argument to address these points you've made.
...
Is that more clear now? — keystone

As always, I'm grateful for this discussion and I'm certainly not complaining, but I hope you see that I have to walk a very thin line with you. — keystone

I can't talk too high level as you will ask for the beef, I can't show figures as they will make your eyes glaze over, I can't use analogies because my analogies don't stick, and when I try to talk technical you often skip over or misunderstand my ideas. — keystone

Of course, it doesn't help that I'm not a trained mathematician. Again, I'm extremely grateful for this discussion, just trying to put things in perspective. — keystone

1) Get you to agree to my use of ε in the computer example (including understand the illustration).
2) Get you to agree to my use of ε in the pi example (including understand the illustration).
3) Progress to 2D, where the Cartesian Coordinate system is replaced with a top-down alternative, and the zeros of y=x^2-2 have a very different meaning. Illustrations become important here which is why we need to get past 1) and 2) first.
4) Top-down interpretation of calculus. — keystone

Hopefully this plan will at least give you confidence that I'm heading somewhere with all of this... — keystone

It's ok, I have my second wind I think. Especially now that I know you're just doing computer arithmetic, fixed or floating point. — fishfry

Well if we're doing computer arithmetic and some variant of discrete calculus, that's interesting to know. What do you think? — fishfry

• π is the familiar irrational number
• ε1 and ε2 are placeholders for positive numbers which can be as small as your computer allows
• (π-ε1,π+ε2) describes a line with rational upper and lower bounds. This line can be as small as your computer allows. Ultimately, I want to call this line pi. To distinguish it from the point/number, π, I'll call this line π.

• With the top-down view, the plan and it's execution are distinct steps such that π remains a line, no matter how powerful your computer is.
• With the bottom-up view, the plan and it's execution are equivalent such that π collapses to a point. I believe this is an unacceptable and an unnecessary leap of thought akin to claiming that there is a last term in a Cauchy Sequence.
• Although step 2 is incredibly useful for applied mathematics, that's not what I'm concerned with. I'm solely concerned with step 1 and I believe step 1 is what is of interest to pure mathematicians.
• Counter to standard belief, I believe calculus is about plans (not their execution) and I believe it's unknowingly been this way all along.
• For example, when a mathematician describes π they always describe the algorithm, they rarely talk about the algorithm's execution...unless referring to a Pi Recitation Contest...

I believe step 1 is what is of interest to pure mathematicians. — keystone

So, you replace pi with a tiny line segment whose length depends upon a computer. So, changing computer affects this small interval. — jgill

I am one of those and I doubt your claim, but there may be others who find it of interest. I don't see anything of substance here so far, but I may be missing the point. — jgill

Things get much more interesting in 2D — keystone

A point of clarity. Thanks. Calculus started with discrete, then moved to infinitesimal, then with technology back to discrete in some sense. — jgill

Discrete calculus is certainly important to my view but it's not what I'm talking about. — keystone

There are two steps to a cut:
Step 1: Planning the cut with an algorithm
Step 2: Executing the plan by completing the algorithm — keystone

ε1 and ε2 are placeholders for positive numbers which can be as small as your computer allows — keystone

(π-ε1,π+ε2) describes a line with rational upper and lower bounds. This line can be as small as your computer allows. Ultimately, I want to call this line pi. To distinguish it from the point/number, π, I'll call this line π. — keystone

With the top-down view, the plan and it's execution are distinct steps such that π remains a line, no matter how powerful your computer is.
With the bottom-up view, the plan and it's execution are equivalent such that π collapses to a point. I believe this is an unacceptable and an unnecessary leap of thought akin to claiming that there is a last term in a Cauchy Sequence.
Although step 2 is incredibly useful for applied mathematics, that's not what I'm concerned with. I'm solely concerned with step 1 and I believe step 1 is what is of interest to pure mathematicians.
Counter to standard belief, I believe calculus is about plans (not their execution) and I believe it's unknowingly been this way all along.
For example, when a mathematician describes π they always describe the algorithm, they rarely talk about the algorithm's execution...unless referring to a Pi Recitation Contest... — keystone

What about computer arithmetic, fixed and floating point representations, smallest and largest possible values?...So you are doing normal math except within the limits of a finite computational space. If not fixed/floating point, something else. But computer arithmetic regardless. — fishfry

• Step 1: Manipulation of real numbers
• Step 2: Computation based on rational numbers (approximations)

You cannot telescope down to pi on computer-limited representations of numbers. If you mean that your number pi is actually a little interval around pi with approximation bounds given by the limitations of your computer representation, I'm fine with that. — fishfry

Don't see the point though. — fishfry

If you reject the noncomputable reals, what you have is the constructive real numbers, and the calculus based on them is called constructive real analysis. — fishfry

I'm genuinely sorry I can't be of more help. — fishfry