In that thread they state that "any set of sentences can be a set of axioms." I want to distinguish between what is (i.e. actual) and what can be (i.e. potential). It is tempting to actualize everything and declare that there are uncountably many mathematical truths. However, I would argue that these truths are contingent on a computer constructing them. When I speak of finite necessary truths I'm referring to the rules of logic itself. — keystone

Mercifully short. Thank you muchly.

Logic isn't constrained to computability.

"However, I would argue that these truths are contingent on a computer constructing them."

Argue that all you like. I can't engage, since I'm not a constructivist. AND, having tried to learn constructivism from time to time, it just doesn't resonate with me. But nevermind the other thread then.

I'm trying to establish a view of calculus which is founded on principles that are restricted to computability (i.e. absent of actual infinities). You don't have to abandon your view of actual infinities to entertain a more restricted view. Perhaps we can set aside the more philosophical topics and return to the beef. — keystone

I have answered this several times already.

(1) Constructivism is fine, you should study it.

(2) I'm the wrong person to discuss this with.I have no affinity for constructivism despite trying over the years.

The term 'line' comes loaded with meaning so to start with a clean slate I'll use 'k-line' to refer to objects of continuous breadthless length (in the spirit of Euclid). I'll use <a,b> to denote the k-line between a and b excluding ends and <<a,b>> to denote the k-line between a and b including ends. If b=a, then <<a,b>> corresponds to a degenerate k-line, which I'll call a k-point and often abbreviate <<a,a>> as "a". I'll call the notation <a,b> and <<a,b>> k-intervals. — keystone

Ok. I feel like we're about to go through this same exposition again. At least the notation's less confusing.

The systems always start with a single k-line described by a single k-interval (e.g. <-∞,+∞>). A computer can choose to cut the k-line arbitrarily many time to actualize k-points. For example, after one cut at 42, the new system becomes <-∞,42> U 42 U <42,+∞>.

The order relation comes from the infinite complete trees.

Are we at a place where we can we move forward? — keystone

I certainly hope so.

Ok. I feel like we're about to go through this same exposition again. At least the notation's less confusing. — fishfry

I'll try to do a better job this time. But first, one other area of confusion has been the distinction between infinite and arbitrary as it relates to an algorithm's design vs. it's execution. A Turing algorithm for constructing N, is designed to output a set of m elements, where m can be arbitrarily large. By this I mean that the algorithm itself sets no limit on the size of its output; rather, the size of the output depends on the execution (i.e., the chosen 'precision' based on available resources). Please note that I'm not saying that m is a particular number, nor is it infinity. Instead, when talking about the algorithm itself, m serves as a placeholder for a value that is determined only upon execution of the algorithm. Upon execution, m is replaced with a natural number and the output is a finite set. In a similar vein, when I speak of ε being arbitrarily small, I am using it as a placeholder to describe an algorithm. Upon executing that algorithm, ε is replaced with a positive rational number that is small, but by no means the smallest. Is this clear?

I'll try to do a better job this time. But first, one other area of confusion has been the distinction between infinite and arbitrary as it relates to an algorithm's design vs. it's execution. A Turing algorithm for constructing N, is designed to output a set of m elements, where m can be arbitrarily large. By this I mean that the algorithm itself sets no limit on the size of its output; rather, the size of the output depends on the execution (i.e., the chosen 'precision' based on available resources). Please note that I'm not saying that m is a particular number, nor is it infinity. Instead, when talking about the algorithm itself, m serves as a placeholder for a value that is determined only upon execution of the algorithm. Upon execution, m is replaced with a natural number and the output is a finite set. In a similar vein, when I speak of ε being arbitrarily small, I am using it as a placeholder to describe an algorithm. Upon executing that algorithm, ε is replaced with a positive rational number that is small, but by no means the smallest. Is this clear? — keystone

You know, that's a very interesting point.

One difference is that rationals get arbitrarily close to 0. But I'm not sure it's all that different. Maybe you have a good analogy. I am not 100% sure what I think about this yet.

I believe you're saying this:

You can always find a rational interval small enough to suit the needs of any computation you do, by analogy with always being able to find a suitably large but finite natural number when you need it for a computation.

Is that a fair understanding of your point?

Also yes, that's arbitrary. You can always find as many as you like, but always a finite number.

We call that "finite but unbounded." There are computations that need 1, that need 2, etc. There's no upper limit to how large a number you can use."

To be actually infinite is far stronger. It's like putting out a good but rational approximation to a real number, versus "printing it all out at once" as it were. Having not just as many digits of pi as you need; but rather all of them at once.

That's the magic of the axiom of infinity. The difference between unbounded, or arbitrary, and infinite.

You can always find a rational interval small enough to suit the needs of any computation you do, by analogy with always being able to find a suitably large but finite natural number when you need it for a computation.....We call that "finite but unbounded."....Is that a fair understanding of your point? — fishfry

That is a fair understanding of my point but I do want to highlight one thing: it's not always about the computation. If I want to focus on algorithm design (and not execution), I can keep ε's floating around. The ε's only need to be replaced when I execute the algorithm and perform the computation. Fair?

To be actually infinite is far stronger. It's like putting out a good but rational approximation to a real number, versus "printing it all out at once" as it were. Having not just as many digits of pi as you need; but rather all of them at once. — fishfry

Yes, actual infinities are beyond computation.

That's the magic of the axiom of infinity. — fishfry

It does seem to be a bit magical. I'd like to avoid magical thinking if at all possible.

That is a fair understanding of my point but I do want to highlight one thing: it's not always about the computation. If I want to focus on algorithm design (and not execution), I can keep ε's floating around. The ε's only need to be replaced when I execute the algorithm and perform the computation. Fair? — keystone

Yes fair. Though I am not entirely sure I ever understood the distinction you're making. Of course I do understand the difference between an algorithm written on paper and an execution of the algorithm in a digital computer; a physical process requiring time, space, and energy; and outputting heat. That's something a guy I worked with said. The only observable output of a cpu is heat.

Anyway. Yes I understand the difference. No I don't understand what POINT you are making about the difference.

Yes, actual infinities are beyond computation. — keystone

Yes but indirectly IMO. Once you get a countably infinite infinity, you immediately get from the powerset axiom an uncountable infinity. But here are only countably many ways to talk about things. Leaving most of the world inexpressible.

Now the constructivists are entirely missing that much of the world, and imagining it doesn't matter. I think it matters. So I have a philosophy about his. I think constructivists and computationalists of all kinds: simulation theorists, mind uploading theorists, philosophers who claim mind is computational, etc -- I think all these people are missing something really important about the world. So I think I'm a bit of an anti-constructivist!!

It does seem to be a bit magical. I'd like to avoid magical thinking if at all possible. — keystone

The magic is the best part. I love the mathematics of infinity. I would never deny it from my world. I am a great devotee of the axiom of infinity.

Anyway. Yes I understand the difference. No I don't understand what POINT you are making about the difference.

I haven't made a point yet. I just wanted to clarify this as I previously found it a stumbling block in our conversation.

simulation theorists, mind uploading theorists, "mind is computational," etc -- I think all these people are missing something really important about the world. So I think I'm a bit of an anti-construcivist!! — fishfry

I'm a computational fluid dynamics analyst, so I naturally approach things from a simulation perspective.

Once you get a countably infinite infinity, you immediately get from the powerset axiom an uncountable infinity. — fishfry

In the context of a potentially infinite complete tree, to me it makes sense to talk about potentially (countably) infinite nodes and potentially (uncountably) infinite paths. In this sense, the paths have more potential than the nodes. I don't think any beauty is lost in reducing infinity to a potential.

But here are only countably many ways to talk about things. Leaving most of the world inexpressible. Now the constructivists are entirely missing that much of the world... — fishfry

As QM suggests, something funny happens when we're not observing the world. I consider this to be the magic of potential. Anyway, this is fluffy talk about potential...let me get to the beef.

I haven't made a point yet. I just wanted to clarify this as I previously found it a stumbling block in our conversation. — keystone

Really? I thought you made a pretty good point. You got me to understand what epsilon is.

I'm a computational fluid dynamics analyst, so I naturally approach things from a simulation perspective. — keystone

Ahhhhhhhh ... now I understand your point of view. You use computers to study continuous flows. This makes perfect sense now.

In the context of a potentially infinite complete tree, to me it makes sense to talk about potentially (countably) infinite nodes and potentially (uncountably) infinite paths. In this sense, the paths have more potential than the nodes. I don't think any beauty is lost in reducing infinity to a potential. — keystone

I'm on record as disagreeing. The axiom of infinity adds a whole beautiful new world. AND is indispensable to developing the very approximation techniques that you use!

As QM suggests, something funny happens when we're not observing the world. I consider this to be the magic of potential. Anyway, this is fluffy talk about potential...let me get to the beef. — keystone

ok

I'd like to distinguish between a fraction and a real. The fraction description is finite (e.g. $\frac{1}{1}$), whereas the real description is infinite (e.g. $1.\overline{0}\$ which can be represented as an algorithm that generates the Cauchy sequence of fractional intervals: $(\frac{9}{10}, \frac{11}{10}), (\frac{99}{100}, \frac{101}{100}), (\frac{999}{1000}, \frac{1001}{1000}), (\frac{9999}{10000}, \frac{10001}{10000}), \ldots$).

Because fraction descriptions are finite, a cut at a fraction can be planned and executed all in one go. A cut of $(-\infty, +\infty)$ at the fraction $\frac{1}{1}$ results in: $(-\infty, 1) \cup \{1\} \cup (1, +\infty)$.

Because real descriptions are infinite, a cut at a real must be planned and executed separately.

The algorithm to cut $(-\infty, +\infty)$ at the real $1.\overline{0}\$ is generalized as: $(-\infty, 1-\epsilon) \cup \{1-\epsilon\} \cup (1-\epsilon, 1+\epsilon) \cup \{1+\epsilon\} \cup (1+\epsilon, +\infty)$ where $\epsilon$ can be an arbitrarily small positive number.

In the spirit of Turing, the execution of the cut of $(-\infty, +\infty)$ at the real $1.\overline{0}$ could be: $(-\infty, \frac{9}{10}) \cup \{\frac{9}{10}\} \cup (\frac{9}{10}, \frac{11}{10}) \cup \{\frac{11}{10}\} \cup (\frac{11}{10}, +\infty)$, where $\epsilon$ in this case is replaced by $\frac{1}{10}$.

eyes glazed?

I'd like to distinguish between a fraction and a real. The fraction description is finite (e.g. 11 — keystone

I'm curious to know what that notation 1/1 means. In abstract algebra class I learned how to construct the rational numbers as the field of quotients of the integers. That's as bottom-up as you can be.

So what is this 1/1 you speak of?


 

which can be represented as an algorithm that generates the Cauchy sequence of fractional intervals: (910,1110),(99100,101100),(9991000,10011000),(999910000,1000110000),… — keystone

If you fixed your notational issues I could quote your markup. Can you figure out why your ChatGPT output is doing that? And as I said before, stop using ChatGPT. The purpose of AI is to make everyone stupid.

(
1
,
+
∞
)
. — keystone

I can't read any of this when I'm replying to your post, without going back to the original.

eyes glazed? — keystone

Do you not know what I'm talking about? Quote one of your own posts.

But just tell me what 1/1 means. Start with "1". What's that?

I'm a computational fluid dynamics analyst — keystone

I've wondered about that. Thanks for illuminating.

I'm curious to know what that notation 1/1 means. In abstract algebra class I learned how to construct the rational numbers as the field of quotients of the integers. That's as bottom-up as you can be. So what is this 1/1 you speak of? — fishfry

Please allow me to respond in the context of the SB-tree. Fractions correspond to nodes. Reals correspond to arbitrarily long paths (well, almost but providing clarifying details would bloat this post). There's no point to introduce natural numbers, integers, or rational numbers as disagreement would ensue. I would say they are all fractions but you would likely say they are all reals.

The cut at fraction 1/1 is fully captured at row 1 of the tree. The algorithm corresponding to the cut at real 1.0 generalizes how the cut would be captured at any arbitrary row beyond row 1 (well, to be precise I should really use ε_left and ε_right instead of just ε). Finally, the execution of the cut at 1.0 happens on a particular row once the computer chooses values for the ε's. What should be clear is that none of this happens at the bottom of the tree. This is an entirely top-down approach.

If you fixed your notational issues I could quote your markup. — fishfry

I think it's just that Latex does not get used properly in quotes. I'm rewriting my last post in plain text and using the notation I recently proposed.
---------------------------------------------------------------------------
I'd like to distinguish between a fraction and a real. The fraction description is finite (e.g. 1/1), whereas the real description is infinite (e.g. 1.0, which can be represented as an algorithm that generates the Cauchy sequence of fractional k-intervals: <9/10, 11/10>, <99/100, 101/100>, <999/1000, 1001/1000>, <9999/10000, 10001/10000>, ...

Because fraction descriptions are finite, a cut at a fraction can be planned and executed all in one go. A cut of <-∞, +∞> at the fraction 1 results in: <-∞, 1> ∪ 1 ∪ <1, +∞>.

Because real descriptions are infinite, a cut at a real must be planned and executed separately.

The algorithm to cut <-∞, +∞> at the real 1.0 is generalized as: <-∞, 1-ε> ∪ 1-ε ∪ <1-ε, 1+ε> ∪ 1+ε ∪ <1+ε, +∞> where ε can be an arbitrarily small positive number.

In the spirit of Turing, the execution of the cut of <-∞, +∞> at the real 1.0 could have us replace ε with 1/10 as follows: <-∞, 9/10> ∪ 9/10 ∪ <9/10, 11/10> ∪ 11/10 ∪ <11/10, +∞>
---------------------------------------------------------------------------
One thing that I've failed to get across is that I'm not outlining a procedure which will be used to construct infinite numbers. These systems I'm outlining, such as <-∞, 1> ∪ 1 ∪ <1, +∞>, are valid systems in and of themselves. Finite systems such as these are all we can ever construct in the top-down view.

Please allow me to respond in the context of the SB-tree. Fractions correspond to nodes. Reals correspond to arbitrarily long paths (well, almost but providing clarifying details would bloat this post). There's no point to introduce natural numbers, integers, or rational numbers as disagreement would ensue. I would say they are all fractions but you would likely say they are all reals. — keystone

I'm afraid that in the absence of a bottom-up approach, I have no idea what are fractions or reals.

Perhaps your entire approach is pre-axiomatic, in which case we have to accept a lot of things we can't formalize.

The cut at fraction 1/1 is fully captured at row 1 of the tree. — keystone

You missed the point of my asking you what the notation 1/1 means, in the absence of building up the rationals from the integers, the integers from the naturals, and the naturals from the axioms of set theory. Or even PA if you can do that.

You have to ask me to imagine I know what those things are, hoping that you yourself are not committing errors by declining to define your own notation.

The algorithm corresponding to the cut at real 1.0 generalizes how the cut would be captured at any arbitrary row beyond row 1 (well, to be precise I should really use ε_left and ε_right instead of just ε). Finally, the execution of the cut at 1.0 happens on a particular row once the computer chooses values for the ε's. What should be clear is that none of this happens at the bottom of the tree. This is an entirely top-down approach. — keystone

I still don't know what 1/1 signifies unless you are secretly assuming all the bottom-up stuff you pretend to reject.

I think it's just that Latex does not get used properly in quotes. — keystone

I didn't check that but I'm not sure you're right. Maybe you are. I'll check that here.

$e^{i \pi} + 1 = 0$

I'll commit my post then quote it and see what I get.

I'm rewriting my last post in plain text and using the notation I recently proposed.
---------------------------------------------------------------------------
I'd like to distinguish between a fraction and a real. The fraction description is finite (e.g. 1/1), whereas the real description is infinite (e.g. 1.0, which can be represented as an algorithm that generates the Cauchy sequence of fractional k-intervals: <9/10, 11/10>, <99/100, 101/100>, <999/1000, 1001/1000>, <9999/10000, 10001/10000>, ...

Because fraction descriptions are finite, a cut at a fraction can be planned and executed all in one go. A cut of <-∞, +∞> at the fraction 1 results in: <-∞, 1> ∪ 1 ∪ <1, +∞>.

Because real descriptions are infinite, a cut at a real must be planned and executed separately.

The algorithm to cut <-∞, +∞> at the real 1.0 is generalized as: <-∞, 1-ε> ∪ 1-ε ∪ <1-ε, 1+ε> ∪ 1+ε ∪ <1+ε, +∞> where ε can be an arbitrarily small positive number.

In the spirit of Turing, the execution of the cut of <-∞, +∞> at the real 1.0 could have us replace ε with 1/10 as follows: <-∞, 9/10> ∪ 9/10 ∪ <9/10, 11/10> ∪ 11/10 ∪ <11/10, +∞>
---------------------------------------------------------------------------
One thing that I've failed to get across is that I'm not outlining a procedure which will be used to construct infinite numbers. These systems I'm outlining, such as <-∞, 1> ∪ 1 ∪ <1, +∞>, are valid systems in and of themselves. Finite systems such as these are all we can ever construct in the top-down view. — keystone

Ok. You start by assuming we all know what these symbols like "1" are, while you reject the standard math definitions, but secretly use them anyway. Ok as far as it goes, but I have to suspend disbelief.

As far as the sense of what you're doing, it eludes me. Are you building the constructive real line? Lost on me.

Ok I posted this, then quoted my LaTeX and got

eiπ+1=0 — fishfry

so clearly that's not right, on the one hand, but not a column of 1-character lines as you get.

Please allow me to respond in the context of the SB-tree. Fractions correspond to nodes. — keystone

Each row of the tree involves medians, which require ratios of integers and arithmetic of these ratios. So, your top down approach always involves bottom up procedures. You cannot correlate rational numbers with nodes without using expressions like a/b. Instead of simplifying, you are complicating something you assume. Just my opinion.

EDIT: After posting this I realized that there might be some confusion about Niqui Arithmetic. I have since posted another message entitled NIQUI ARITHMETIC. Please read that first.

You missed the point of my asking you what the notation 1/1 means, in the absence of building up the rationals from the integers, the integers from the naturals, and the naturals from the axioms of set theory. Or even PA if you can do that. — fishfry

Peano arithmetic can be formalized in Coq. Similarly, Niqui arithmetic on the SB tree, which builds on Peano arithmetic, has been proven in Coq. There's an unquestionable structure to natural numbers and fractions that we both agree on. What we disagree on is the ontology related to these necessary truths. You believe that Peano arithmetic applies to infinite natural numbers, whereas I believe it applies to arbitrary natural numbers. By this, I mean that Peano arithmetic corresponds to an algorithm designed to take as input any arbitrary pair of natural numbers and output the expected natural number. My ontology does not require the existence of any number. I only need numbers when I want to execute the algorithm, and I only need two numbers at that, not an infinite set.

Although the above focuses on Peano arithmetic, the same applies to Niqui arithmetic. While the actual computations of Niqui arithmetic involve the manipulation of symbols or electrical signals, an elusive structure emerges in our mind when studying the algorithm—the SB-tree. Nobody has ever envisioned the complete tree, but we have seen the top part, and when I say 1/1 occupies a particular node, that top part is all I need to see. I don't need to assert the existence of an unseen complete tree; after all, it is merely an illusion that helps us understand the underlying algorithm (Niqui arithmetic).

As far as the sense of what you're doing, it eludes me. Are you building the constructive real line? Lost on me. — fishfry

I'm trying to establish parallel ontologies: Actual vs. real. At this point, we have actual numbers (fractions) and real numbers. We have actual points (k-points corresponding to fractions) and real points (k-lines corresponding to real numbers). This distincting is rather bland in 1D but it becomes much more consequential in 2D when establishing a foundational framework for geometry and calculus.

eiπ+1=0 — fishfry

The Philosophy Forum appears to be quirky. I tried quoting this multiple times, sometimes including the spaces surrounding it, sometimes not, and about half the time it puts a column of 1-character lines.

Each row of the tree involves medians, which require ratios of integers and arithmetic of these ratios. So, your top down approach always involves bottom up procedures. You cannot correlate rational numbers with nodes without using expressions like a/b. Instead of simplifying, you are complicating something you assume. Just my opinion. — jgill

As I was suggesting to fishfry, in the context of Stern-Brocot, what fundamentally exists is the algorithm for Niqui arithmetic. The illusion of the SB-tree emerges from our contemplation of this algorithm. When we examine the illusion, we notice interesting mediant relations between the nodes, but all of these relations stem from Niqui arithmetic. What I'm proposing is that Niqui arithmetic is more fundamental than the SB-tree. Interestingly, the SB-tree was discovered before Niqui arithmetic, but mathematics often reveals its truths in unexpected ways.

And perhaps you'll argue that almost no one views the SB-tree as a result of explicitly considering Niqui arithmetic. I would counter that by saying that when looking out the window no one consciously processes the mathematical properties of the light waves hitting their eyes, yet that is precisely what we are doing subconsciously.

NIQUI ARITHMETIC

Niqui arithmetic doesn't start by populating the nodes of a tree with fractions using the mediant operation. Rather, it doesn't assign any number to any node. It simply takes a pair of arbitrary nodes and outputs a resulting node based on the chosen arithmetic operation. It's entirely location-based. As an afterthought, we can observe the behavior of these nodes and realize that their behavior corresponds perfectly to the familiar arithmetic, provided we label the nodes according to the SB-tree's labeling. And when we label the nodes in this manner, a peculiar property related to mediants becomes apparent. But in no way does Niqui arithmetic rely on the mediant operation.