Real numbers and the Stern-Brocot tree

TonesInDeepFreeze

never being able to complete the computation involving infinite strings — keystone

Exactly.

Meanwhile, with the other common definitions, we do define addition and multiplication of real numbers and that is not blocked by the fact that computations do not accept infinite sequences as inputs.

what is a point if not a node on the tree? — keystone

Unless instructed otherwise, I would take 'point' and 'node' as synonymous in this context.

Sqrt(2) does not converge towards any node the tree. — keystone

If we are redefining 'is a real number' as 'is a path in S-B' (I would prefer 'is a sequence of nodes on a path in S-B'), then of course such a sequence does not converge to sqrt(2), since sqrt(2) is a sequence and not a node.

it appears to converge to a node that exists at 'row infinity' — keystone

Right, it doesn't converge to any node on the tree. But with the definition

a real number is a path on the S-B tree

convergence is no longer relevant in this context, since a real number is path and not a node to which a sequence converges.

And about "row infinity":

Of course, there is no 'row infinity' which is why I relate it to a mirage. — keystone

Exactly, you don't have a mathematical basis, so you resort to merely figurative, undefined, subjective language. One should not mind figurative language used to convey intuitions about mathematics, to help us get a "mind's eye" grasp of certain concepts. But when the figurative language ends up not backed by actual mathematics then it is fitting to respond, "Okay, that might be interesting, so get back to us when you've worked out the math."

There is an inconsistency in claiming that both (1) the real number line exists and (2) 'row infinity' does not exist. — keystone

No there is not. You're lying. Inconsistency is the derivability of the conjunction of a statement and its negation. You can't show any such derivation.

The real line is constructed in our mathematical theory. And also, in our mathematical theory, the S-B tree does not have a "row infinity". If you claim that is inconsistent, then PROVE it. And if you can't, then you should desist from lying about it.

Your speciousness and intellectual dishonesty here is similar to the previous threads with you.

And it's even WORSE in this thread, because in the other threads, the discussion was about thought experiments, which are informal analogies about mathematics, while in this thread, we are talking about an exact mathematical object.

TonesInDeepFreeze

↪keystone

Keeping track of where the discussion stands:

I wrote:

It is not problematic that it is the limit of a sequence of rationals but is not one of the entries in that sequence. — TonesInDeepFreeze

That still stands, notwithstanding your reply (which ends with a lie).

keystone

Exactly. Meanwhile, with the other common definitions, we do define addition and multiplication of real numbers and that is not blocked by the fact that computations do not accept infinite sequences as inputs. — TonesInDeepFreeze

All I said was that the computation when the inputs were infinite strings wouldn't complete. The SB algorithm can certainly accept infinite strings as inputs though, it would just absorb the digits little by little and emit output digits little by little, never actually completing the output. In other words, I wasn't presenting a limitation of the algorithm, I was conveying my disbelief in supertasks. I would relate this to computing the digits of pi. The pi generating algorithm works, it would just never actually output all digits of pi.

sqrt(2) is a real number. A real number doesn't converge. A sequence converges. — TonesInDeepFreeze

As we've informally agreed, irrationals (e.g. sqrt(2)) are unending paths on the SB tree. In an informal sense, it is reasonable to say that paths converge. The paths LR and RL both converge to the node 1. When I mentioned sqrt(2) not converging to a point, I was referring to the path sqrt(2) not converging to any node (in that sense). Would you rather I use another term than converge?

The difference between

And it's even WORSE in this thread, because in the other threads, the discussion was about thought experiments, which are informal analogies about mathematics, while in this thread, we are talking about an exact mathematical object. — TonesInDeepFreeze

The difference here is that we have a mathematical object (SB tree) which is very simple but yet still (it appears to) sufficiently captures to topic of concern. But unlike with our other thread, you cannot defend your position by making your arguments more and more complex. I, on the other hand, am defending my position by making my augments more and more simple. I suppose one way for you to end this would be by explaining why the SB tree (and my argument) oversimplifies the issue. If you don't like my description of the real number line as it relates to the SB tree, can you propose a better description using the SB tree?

For example, the square root of 2 does not remind me of a mirage. It is not problematic that it is the limit of a sequence of rationals but is not one of the entries in that sequence. — TonesInDeepFreeze

It does not bother me that there are (unending) paths on the SB tree having no destination node. I also think there's value in saying that such a path is approaching a 'virtual' node. Calculus demonstrates the utility of such a belief. But I wouldn't accept that virtual nodes and actual nodes are the same thing. I think when we say rationals and irrationals are both numbers, we are essentially doing that.

TonesInDeepFreeze

the computation — keystone

There are two separate matters:

(1) The definitions of the operations.

For addition, this is of the form:

x=y = z <-> P
where P is a formula in which only x, y and z occur free and such that we have the theorem:

AxyE!zP

If you propose the tree as a basis for the real numbers, then you have to provide such a definition.

Please tell me whether you understand that. Actually, don't bother, because I already know you know nothing about mathematical definitions.

(2) Computations. That's not what I have been talking about.

As we've informally agreed, irrationals (e.g. sqrt(2)) are unending paths on the SB tree. — keystone

I don't take it as merely an informal proposal. If it's going to hold up, then it will be formalized.

sqrt(2) is a real number. A real number doesn't converge. A sequence converges. — TonesInDeepFreeze

I had edited that to:

If we are redefining 'is a real number' as 'is a path in S-B' (I would prefer 'is a sequence of nodes on a path in S-B'), then of course such a sequence does not converge to sqrt(2), since sqrt(2) is a sequence and not a node. — TonesInDeepFreeze

unlike with our other thread, you cannot defend your position by making your arguments more and more complex — keystone

Not only are you lying about there being an inconsistency, but you're moving on to mischaracterize what I posted in other threads. And indeed you even more obnoxious, because what I did in the other threads was to generously give you increasingly detailed explanations of things you're ignorant about. As you continued not to grasp the basic mathematics, I generously explained it to you in yet more detail. That is not me resorting to defending my arguments by making them more complex. You are a real piece of work.

I, on the other hand, am defending my position by making my augments more and more simple. — keystone

Yes, it's a simple lie that the existence of the real line is inconsistent with the existence of the S-B tree.

Again, you have evaded the point:

Since you claim there is an inconsistency, then PROVE it.

I suppose one way for you to end this would be by explaining why the SB tree (and my argument) oversimplifies the issue. — keystone

What? Why should I do that? I don't claim that the S-B tree "oversimplifies" anything. Now, you're also resorting to strawman.

If you don't like my description of the real number line as it relates to the SB tree — keystone

What I don't like, because it's a lie, is your claim that the existence of the real line contradicts the existence of the S-B tree.

can you propose a better description using the SB tree? — keystone

What? I have no problem with saying that set of real numbers is the set of paths in the S-B tree, with whatever other finer qualifications need to be given to make it all work out rigorously. And then defining an ordering to define the continuum. I said explicitly that I can imagine it all working out.

But your lie is saying that there is contradiction between the existence of the continuum and the existence of the S-B tree.

For example, the square root of 2 does not remind me of a mirage. It is not problematic that it is the limit of a sequence of rationals but is not one of the entries in that sequence.
— TonesInDeepFreeze

It does not bother me that there are (unending) paths on the SB tree having no destination node. I also think there's value in saying that such a path is approaching a 'virtual' node. — keystone

(1) You are evading my point. I'll say it again (whether regarding ordinary analyis or an S-B proposal):

It is not problematic that a real is the limit of a sequence of rationals but is not one of the entries in that sequence.

(2) "virtual node". More undefined figurative language. The last refuge of the crank.

TonesInDeepFreeze

The mathematical axioms prove the existence of the S-B tree.

The mathematical axioms also prove the existence of a complete ordered field with the carrier set being the set of Dedekind cuts.

The mathematical axioms also prove the existence of a complete ordered field with the carrier set being the set of equivalence classes of Cauchy sequences of rationals.

Those two complete ordered fields are isomorphic.

We can also try to figure out showing that we have a complete ordered field with the carrier set as the set of paths in the S-B tree. And, since all complete ordered fields are isomorphic with one another, the one based on the S-B tree would be isomorphic with the others too.

The set of paths in the S-B tree is not part of the S-B tree. But that doesn't contradict anything else in the mathematical theory from the axioms. The S-B tree exists, and lots of things other than the S-B also exist. The set {p | p is a path in the S-B tree} is one of them.

I don't know, but maybe you are thinking that there is ONLY the S-B tree and anything other than the S-B tree doesn't exist (except as a "mirage" or whatever crank dodge concept you invoke)?

If that is the case, then dump that thought. The S-B tree exists; the Dedekind cut real numbers exist: the Cauchy sequence real numbers exist; and the set of paths in the S-B tree exists (and possibly it too as a basis for a complete ordered field).

keystone

If you propose the tree as a basis for the real numbers, then you have to provide such a definition. — TonesInDeepFreeze

Fair enough. The author of the paper proved the algorithm to work specifically for the rationals (not reals). I imagine the proof is much more difficult (impossible?) to extend to reals. You're right, I don't have the knowledge to produce a formal definition, let alone prove it. My take was based on my experience coding and utilizing the alorithm.

of course such a sequence does not converge to sqrt(2), since sqrt(2) is a sequence and not a node. — TonesInDeepFreeze

(RL) converges to the node corresponding to 1. RL (phi) does not converge to any node.

what I did in the other threads was to generously give you increasingly detailed explanations — TonesInDeepFreeze

You brought the conversation to a level of complexity/formality that I wasn't comfortable with so the conversation ended. I don't believe it, but if that level of complexity/formality is required to discuss infinity then so be it.

What I don't like, because it's a lie, is your claim that the existence of the real line contradicts the existence of the S-B tree. — TonesInDeepFreeze

You are evading my point. — TonesInDeepFreeze

Let's table this for a moment as I better understand your interpretation of real numbers on the SB tree.

I have no problem with saying that set of real numbers is the set of paths in the S-B tree — TonesInDeepFreeze

Do you think the real number RL (phi) is the path itself or the limit of the path?

And, since all complete ordered fields are isomorphic with one another, the one based on the S-B tree would be isomorphic with the others too. — TonesInDeepFreeze

I agree with this. The reason why I prefer the S-B tree view is that it's more understandable to amateurs like myself. What I want to reiterate is that I'm not trying to abandon the reals or the existence of continua like the real line. Instead, I'm trying to understand what the real line is from the perspective of the S-B tree.

The set of paths in the S-B tree is not part of the S-B tree. — TonesInDeepFreeze

I understand that there are no sets in the tree, but the paths do exist in the tree, right? The tree is composed of both the paths and the nodes, right?

TonesInDeepFreeze

What I don't like, because it's a lie, is your claim that the existence of the real line contradicts the existence of the S-B tree.
— TonesInDeepFreeze
You are evading my point.
— TonesInDeepFreeze

Let's table this for a moment as I better understand your interpretation of real numbers on the SB tree. — keystone

I'm not tabling it for anything. It is flat out incorrect that

There is an inconsistency in claiming that both (1) the real number line exists and (2) 'row infinity' does not exist. — keystone

The S-B tree is clear enough by the ostensive definition we have of it. (Eventually though we should have a rigorous definition.)

And we agree that there is no row of the S-B tree that is not indexed by a natural number. (I.e., we agree that there is no "row infinity" of the S-B tree.)

And I understand the notion of the irrational reals being the denumerable paths of the S-B tree.

So we don't have to wait to see what my interpretation is.

These are consistent with one another:

The S-B tree exists.

The set of finite paths of the S-B tree exists and there is a 1-1 correspondence between the set of finite paths and the set of rational numbers. So the proposal is to take the rational real numbers as the finite paths.

The set of denumerable paths of the S-B tree exists. And there is a 1-1 correspondence between the set of denumerable paths and the set of irrational real numbers. So the proposal is to take the irrational real numbers as the denumerable paths.

There is no row that is not indexed by a natural number. (I.e. there is no "row infinity".) Therefore, no irrational real number has a path with a final node.

Now, you claim there is an inconsistency there. So PROVE that there is an inconsistency there. Otherwise, you are making the claim utterly without basis; you are fabricating, which is to say you are lying.

/

Do you think the real number RL (phi) is the path itself or the limit of the path? — keystone

I'm accepting whatever coherent proposal YOU are making. You want to make phi the path. I've said that that is fine with me.

Just to be clear on what we're talking about:

'graph' is defined different ways. I use the definition by which a graph is a certain kind of triple. It follows from the definition that:

Every graph is a triple <V E f>. V is the set of vertices (aka 'the nodes'), E is the set of edges, and f is a function on E that assigns to each edge either an unordered pair or ordered pair of vertices.

Watch out now, here comes something quite pedantic:

<V E f> =
<<V E> f> =
{{<V E>} {<V E> f}} =
{{{{V} {V E}}} {{{V} {V E}} f}}

So the only two things that are literally in the graph (i.e., are members of the graph) are {{{V} {V E}}} and {{{V} {V E}} f}.

I mention that to explain why I personally don't like to say "the paths are in the tree" but rather "the paths are of the tree".

But it would be unduly pedantic indeed to disallow that conversationally we use the word "in" more loosely (i.e., not just for membership), and that most mathematicians don't say "of" but say things like:

"The nodes are in the graph", "the edges are in the graph", the "paths are in the graph" and "the rows" in the graph".

So I'll go along with that usage.

Now the bullet points:

Every path is a sequence of edges.

A path is NOT a sequence of nodes.

Every tree is a graph.

The S-B tree has both finite and denumerable paths.

As I understand the proposal, the finite paths are the rational real numbers, and the infinite paths are the irrational real numbers. (I suggest that it might be better to make the rational reals the finite sequences of nodes and the irrational numbers the denumerable sequences of nodes. I.e. "eliminate the middleman".)

of course such a sequence does not converge to sqrt(2), since sqrt(2) is a sequence and not a node.
— TonesInDeepFreeze

(RL) converges to the node corresponding to 1. — keystone

Actually 1/1 is the one exception. 1/1 is not represented by a path but by a node, since there is no path leading to the node 1/1.

RL (phi) does not converge to any node. — keystone

At least for right now, I won't quibble with the notion of a path converging to a node. But I don't see what your reply has to to do with my point that sqrt(2) is a sequence not a node, except that you give phi as another example.

You brought the conversation to a level of complexity/formality that I wasn't comfortable — keystone

And that does not entail that I did that as something similar to a "No true Scotsman" ploy (as I surmise you were suggesting).

The reason why I prefer the S-B tree view is that it's more understandable to amateurs like myself. — keystone

There are attractions in using the S-B tree:

Every fraction is in lowest terms. (But we will still have to deal with justifying, e.g., 2/4 = 1/2. There is no node 2/4, so there is no path to 2/4, so 2/4 is not a rational number. So what is it? This is why the standard approach takes rationals to be equivalence classes.)

It's intuitive that the rational reals would be the finite paths and the irrational reals the denumerable paths. And that also means that, e.g., 1/3 doesn't need to be mentioned as represented by a denumerable sequence.

Real numbers can be sequences and not equivalence classes of sequences.

I'm trying to understand what the real line is from the perspective of the S-B tree. — keystone

The continuum is <R S> where R is the set of real numbers and S is the standard 'less than' ordering on the set of real numbers. There are some details to work out in taking the set of real numbers to be the set of paths (but tossing in 1 also, as 1/1 has no associated path), but it seems to be a plausible idea. But then you have to define the 'less than' ordering.

But you are also trying to impugn the standard theory, which you have objected to for being infinitistic. But the S-B approach is no less infinitistic. Then, irrationally, and even defeating your own purpose ("cut your nose to spite your face") of using the S-B tree, you say there is an inconsistency. And, aside from S-B, it's fine to propose a finitistic or constructive alternative or computable analysis or non-standard analysis or even a theory with a paraconsistent logic. But that doesn't permit propounding the lie that the standard theory is inconsistent or inconsistent with the S-B tree (indeed, the S-B tree itself is within the standard theory).

keystone

Thanks for providing a detailed and thoughtful response.

I'm accepting whatever coherent proposal YOU are making. — TonesInDeepFreeze

Let me re-share this image of the S-B tree (https://imgur.com/vWBO6U9) because it shows how I see the connection between the S-B tree and the real number line. The nodes are projected down and at the limit we have nodes for all real numbers forming the continuous real number line in totality. Granted, the limit is not a part of the tree, after all, there are no nodes for irrational numbers in the tree. However, the limit can still be a property of the tree just as an irrational number can be a property of a Cauchy sequence. So from the perspective of the Stern-Brocot tree, the real line is like a mirage. It's not actually a part of the tree, but it's a valid property of the tree nonetheless. With this view, the objects of concern are the nodes (not the paths). But since some paths are not actually linked with nodes, I proposed we say that they converge to virtual nodes. Or in other words, the real number line is composed of points and virtual points.

While it may be convenient to think in terms of paths, there is no analogous pictorial connection between paths and the real number line. I also think paths are more like the Cauchy sequences themselves while the nodes are more like the limits of Cauchy sequences.

Now, you claim there is an inconsistency there. So PROVE that there is an inconsistency there. Otherwise, you are making the claim utterly without basis; you are fabricating, which is to say you are lying. — TonesInDeepFreeze

With my view now outlined above, I am arguing that it is inconsistent to say that the real number line is composed entirely of actual points. That would only be possible if there was a row infinity in the tree, which we agree that there is not. Instead, the real number line can only exist at the limit and must be composed of both points and virtual points.

I mention that to explain why I personally don't like to say "the paths are in the tree" but rather "the paths are of the tree". — TonesInDeepFreeze

Might we also be allowed to say that the limit is of the tree?

(I suggest that it might be better to make the rational reals the finite sequences of nodes and the irrational numbers the infinite sequences of nodes — TonesInDeepFreeze

Can we say that the rational/irrational numbers are the limits of their corresponding paths.

And that does not entail that I did that as something similar to a "No true Scotsman" ploy (as I surmise you were suggesting). — TonesInDeepFreeze

I followed my statement with "I don't believe it, but if that level of complexity/formality is required to discuss infinity then so be it." In other words, you made the conversation too complex for me. Whether it was too complex for the topic is not for me to decide.

so there is no path to 2/4, so 2/4 is not a rational number. So what is it? — TonesInDeepFreeze

I suppose you can use the S-B tree to compute 2 divided by 4 and you'd get the rational 1/2. But I see your point.

But you are also trying to impugn the standard theory, which you have objected to for being infinitistic. But the S-B approach is no less infinitistic. — TonesInDeepFreeze

One can write a finite (but complete) computer program to create the entire S-B tree. My point is simply that that program cannot actually be executed to completion. As such, the object of study should not be the complete output of the program (which cannot be generated) but instead the program itself whose execution is potentially infinite.

So while I don't believe in 'the set of all natural number', I don't mind a computer program that prints all natural numbers. And when 'we' work with infinite sets, 'we' don't actually work with the infinite sets themselves, 'we' work with the 'algorithms' used to generate them. If my view were to be proven true, standard theory wouldn't fall apart. It would just be a philosophical shift from actually infinite objects to potentially infinite algorithms/programs. For example, in set theory, it could be as small as moving from saying 'there exists a set' to 'construct a set'.

TonesInDeepFreeze

We'll go with our continuing lead story:

Now, you claim there is an inconsistency there. So PROVE that there is an inconsistency there. Otherwise, you are making the claim utterly without basis; you are fabricating, which is to say you are lying.
— TonesInDeepFreeze

With my view now outlined above, I am arguing that it is inconsistent to say that the real number line is composed entirely of actual points. — keystone

That's yet more dishonest obfuscation by you.

For about the 1000th time: Inconsistency is having both a statement and its negation as theorems.

"actual" is not a terminology of the theory.

So there is no "inconsistency" regarding it.

You need to stop using the word 'inconsistency' with your own private meanings. Inconsistency is an exact rigorous notion in mathematics. When you appropriate the term to use it with your own private and undefined meaning, you disservice honest and coherent discussion.

"Just remember it's not a lie if you believe it".

Look up 'lie' in Merriam-Webster.

he nodes are projected down and at the limit we have nodes for all real numbers forming the continuous real number line in totality. — keystone

You mangle terminology. Choose a definition and stick with it.

First you said the reals are the paths. Now you say they are nodes.

And you use "limit" in an undefined way.

Your insouciance in not making definitions and sticking to them invites confusion and is annoying.

Can we say that the rational/irrational numbers are the limits of their corresponding paths. — keystone

(1) What is a limit of a path? I can see what the limit of a sequence of nodes is, but you would need to define what you mean by a limit of a sequence of edges.

(2) You said you wanted the reals to be the paths. Now you want to switch to something else. Let us know when you've reached a stable decision.

(3) Your notion doesn't even make sense, as follows: You are trying to define 'is a real number' per the S-B tree. But you need also to prove the existence of the objects that meet that definition. You haven't proven the existence of whatever the limits are supposed to be.

Look, you have the existence of the paths. And you have the existence of the sequences of nodes. But you don't have the existence of whatever you think are going to serve as "the limits". If you want to have the things that are going to serve as the limits, then you need to prove they exist. And not "mirages"'. Or give a mathematical definition of 'mirage'.

One can write a finite (but complete) computer program to create the entire S-B tree. — keystone

To be clear, since you write ambiguously "create the entire tree". Yes, there is a program such that, for any n, the program will generate up to and including the nth row and stop. But there is no program that generates all rows and stops. I take it that you agree.

One can write a finite (but complete) computer program to create the entire S-B tree. — keystone

A program can generate any arbitrary finite part down the tree, but it can't create "the entire" tree.

Again, the tree is an infinite object. And if you consider it to be the limit of the finite stages, then it still has to exist to BE that limit!

Meanwhile, you seem now to regard the reals as limits of successive finite sequences of finite paths. I don't see a problem in that. But, again, it entails that those limits are THEMSEVLES objects - existing. Indeed, they are the denumerable paths. You can't say that the denumerable paths are limits but that they don't exist.

The notion of limits in this context is infinitistic. The limits of sucessive finite sequences of finite paths are the denumerable paths. If you want them to be our real numbers, then they exist and it's just jejune to say [paraphrase], "They don't "actually" exist but they exist as "mirages"". That's just undefined verbiage, nice for poetry but it's not mathematics. Please don't insult intelligence that way.

Look at sqrt(2) in standard mathematics. It is the limit of a sequence of rationals, but it has to exist ITSELF to BE there to be the limit. We can't coherently say, "There are just the finite approximations and the sqrt(2) is the mirage (something that does not exist but only seems to exist) at the end of those approximations." No, for the sqrt(2) to be a real number, it has to exist.

Now, you can say that there are only successive finite sequences of rationals getting closer to one another progressively in succession. You could cook up a theory in which it works that way. But then you cannot say that there is a limit that is the sqrt(2).

As such, the object of study should not be the complete output of the program (which cannot be generated) but instead the program itself whose execution is potentially infinite. — keystone

Ah, so you don't want the S-B tree after all. You want instead a program that generates successive finite number of rows.

But does the tree exist for you or not? Please don't answer with yet more wiffle waffle undefined terminology. Please just say whether it exists or not.

I understand the notion of the entire tree as a "limit" of the finite sequences of accumulations of rows. But a limit in mathematics is itself an existing object. It's not a "mirage". You can't have it both ways: You can't say BOTH (1) there is not an entire tree but only finite approximations and (2) the entire tree is the limit of those approximations. To be the limit of a sequence is to already exist to be the limit. Sequences don't converge on something that doesn't itself exist.

If the paths don't exist in their entirety, then those real numbers that are supposed to be those paths don't exist, or whatever you think are supposed to be the "limit" nodes don't exist.

But let's say the object of study is the program itself. Okay, but then pray tell how do you extract from that study of the program real analysis for the mathematics for the sciences? I think there are rigorous theories (I guess they can be made fully rigorous?) that do that kind of thing. But you don't provide a clue how you would do it.

Actually, this is a huge bait and switch by you. You said that the real numbers are to be the paths in the tree. But now you don't want to have the existence of those paths, so you switch to saying "study the program". I was game for talking about the initial proposal, but now you've switched to something undefined to the point of nebulousness.

I don't mind a computer program that prints all natural numbers. — keystone

To be clear, you mean a program that at any stage generates up to the nth natural number for some n and always goes to the next stage and never halts. That is called a 'recursive enumeration'.

'we' don't actually work with the infinite sets themselves, 'we' work with the 'algorithms' used to generate them — keystone

There is mathematics formulated along those lines.

But that doesn't entail that there is inconsistency in the standard approach!

Do your thing, whatever it is, but you should lay off spreading disinformation that there is inconsistency in infinitistic mathematics.

jgill

SB is a fairly deep mathematical subject, as seen here.

TonesInDeepFreeze

↪jgill

It is a fascinating subject. I wish I were up to speed to absorb and appreciate that paper.

Count Timothy von Icarus

↪TonesInDeepFreeze

Thanks for the detailed explanation. That makes sense, I was thinking the resolution was somewhat along those lines.

jgill

↪TonesInDeepFreeze

Wiki puts SB in the category of number theory. Perhaps it's analytic number theory since limits appear. Apart from the simple continued fraction expansion it's beyond me. :cool:

keystone

I want to characterize all paths on the S-B tree by their destination. Infinite paths do not actually have destinations but neither does 1/x have a value at x=infinity. After all, x never is infinity and there is no row infinity on the S-B tree. But there is value in saying that 1/x approaches 0 as x approaches infinity. Similarly, there is value in saying that the path RL approaches 1 as we descend the tree and approach 'row infinity'. In both of these examples, infinity is a useful fiction. But it's important to acknowledge that it is a fiction, however you want to phrase it.

Now, given that paths (as described by infinite sequences of rational numbers) are analogous to Cauchy sequences, it doesn't seem like a big jump to say that the limit of a SB path is analogous to the limit of a Cauchy sequence, or in other words, that the limit of infinite SB paths are nodes corresponding to real numbers. But these nodes only exist at row infinity (which is a fiction) and so they too are useful fictions. Of course, it's obvious that they're fictions since, once again, infinite paths do not end at any node (in a similar way that Cauchy sequences do not end at any rational number).

As we agreed earlier, nodes are analogous to points. The issue then is that in the S-B tree, we only have nodes corresponding to the rational numbers. The nodes corresponding to the real numbers are fictional. Comparatively, with the real number line, we do not distinguish between the rational and irrational points on the real number line. They are of the same essence. This disagreement between the S-B tree and the real number line is what I'm trying to highlight.

You need to stop using the word 'inconsistency' with your own private meanings. — TonesInDeepFreeze

Granted, the branches of the S-B tree never actually intersect, but if we're trying to illustrate the real line as the limiting row of the tree, one approach we might be able to use is projective geometry. With this approach, the parallel lines of 0/1 and 1/0 (and all the branches in between) meet at a single node at infinity. So in a sense, maybe one can say that at infinity (if it were not a fiction) all of the real numbers would be equal. Or analogously, going back to the 1/x example, at x = infinity we have a singularity. Perhaps all numbers being equal is more in line with a standard notion of inconsistency. However, this is all 'handwavy'. For example, what if 0/1 and 1/0 were instead diverging lines? Anyway, until I can support arguments like this with something more concrete, I'll refrain from using the term inconsistency.

"Just remember it's not a lie if you believe it".
Look up 'lie' in Merriam-Webster. — TonesInDeepFreeze

Lie - to make an untrue statement with intent to deceive.
Perhaps cranks are deluded, but our intentions are pure.

Choose a definition and stick with it.
First you said the reals are the paths. Now you say they are nodes. — TonesInDeepFreeze

I think it is a virtue to be able to adjust one's view upon new evidence. I wanted to treat the real numbers as unending journeys along a path but you were inclined to treat them as the paths themselves. While I went along with this for a while I realized that it doesn't align with my views regarding infinity. As such, I have adjusted my view to something that seems more agreeable to both of us - limits. We can both speak of limits while in our heads, you are imagining an actually infinite object (path) and I am imaging a potentially infinite process (journey).

Your insouciance in not making definitions and sticking to them invites confusion and is annoying. — TonesInDeepFreeze

Sorry, I can see how this can frustrate someone.

If you want to have the things that are going to serve as the limits, then you need to prove they exist. — TonesInDeepFreeze

In the 1/x analogy above, does one have to prove that infinity is a 'thing' to talk about x approaching it?

To be clear, since you write ambiguously "create the entire tree". Yes, there is a program such that, for any n, the program will generate up to and including the nth row and stop. But there is no program that generates all rows and stops. I take it that you agree. — TonesInDeepFreeze

There is a difference between writing a program and executing it. I can write a program that generates all rows and stops. However, I cannot execute it. The program is no less of a program just because the output doesn't exist. The output is a fiction. The execution of the program is a potentially infinite process, the output of the program is an actually infinite object.

But, again, it entails that those limits are THEMSEVLES objects - existing. — TonesInDeepFreeze

Analogously, the output of the program doesn't have to exist for us to use it to describe the program and its execution. The [fictitious] output is used to describe the program/execution (not the other way around).

But does the tree exist for you or not? Please don't answer with yet more wiffle waffle undefined terminology. Please just say whether it exists or not. — TonesInDeepFreeze

FOR ME - The program to create the tree exists, and it can be written with finite characters. But the program cannot be executed to completion (it doesn't halt - i.e. it is potentially infinite) so the output of the program does not exist. There is no actually infinite object corresponding to the S-B tree. Nobody has ever seen the actually infinite object with their minds eye. What we see with our minds eye is the program.

But let's say the object of study is the program itself. Okay, but then pray tell how do you extract from that study of the program real analysis for the mathematics for the sciences? — TonesInDeepFreeze

I believe that reformulating calculus using limits did just that. In my view, calculus doesn't need any reformulation. Consider that I can write a program to perform Newton's method but if I'm hunting for an irrational solution it will never halt. The Cauchy sequence of intermediate rational 'approximations' describes the program/execution, not the solution. There is no solution.

Actually, this is a huge bait and switch by you. You said that the real numbers are to be the paths in the tree. But now you don't want to have the existence of those paths, so you switch to saying "study the program". I was game for talking about the initial proposal, but now you've switched to something undefined to the point of nebulousness. — TonesInDeepFreeze

This is not a bait and switch. Infinity is inseparably tied with calculus. If one denies infinite objects one has to then accept infinite processes otherwise calculus doesn't get off the ground. And the simplest way to talk about processes is to treat them as programs being executed. It is unreasonable for you to disallow programs from the discussion, especially given that computer science is so closely tied with mathematics.

EDIT: You're not disallowing programs from the discussion so my last paragraph should be retracted. Instead, perhaps I should have acknowledged that my ideas and method of communication are evolving. However, I wouldn't call this a bait and switch since I'm not trying to trick you.

TonesInDeepFreeze

↪keystone

Four posts to follow. Untangling your confusions and lies.

Unfortunately, it's likely that you'll reply with even more confusions and lies.

TonesInDeepFreeze

↪keystone

For reference, these are the approaches that have been discussed here:

STANDARD

(1) Reals are Dedekind cuts.

Advantages:

* Easy to visualize.

* 'less than', addition and multiplication have been defined.

Disadvantage:

* Unintuitive that a real is the entire set of all the rationals less than the real.

(2) Reals are equivalence classes of Cauchy sequences of rationals.

Advantages:

* Intuitive that sequences rather than entire "less than sets" are involved.

* 'less than', addition and multipication are handily defined.

Disadvantage:

* Unintuitive that a real is an entire set of equivalent Cauchy sequences of rationals.

(3) Reals are sequences of rationals.

I have heard about this, but don't know how it works as a definition or how 'less than', addition and multiplication are defined. Also, I don't know why we couldn't take reals in (0 1) to be the ordinary binary(or decimal, whatever) expansion (excluding sequences of all 1s after some index, or all 9s after some index, whatever). But since that is not one of the two standard methods - (1) or (2) - I can only guess that there's a complication in this method.

Advantage:

* Intuitive that a real is a particular sequence.

S-B TREE

(4) Reals are paths.

This seems plausible to me.

Advantage:

* Intuitive that rational reals are a finite path and irrational reals are a denumerable path.

Disadvantages:

* What about fractions not in lowest terms? I guess division would be defined in a special way. And I think we might end up with having to deploy equivalence classes of integers, which is what we do anyway in the standard definition of the rationals.

* Need to define the negative reals.

* Need to define 'less than', addition and multipication.

* Sequences of nodes are slightly more intutive (familiar) than sequences of edges.

(5) Reals are sequences of nodes.

This seems plausible to me and better than (4).

Advantage:

* Intuitive that rational reals are finite sequences and irrational reals are denumerable sequences.

* Sequences of nodes are slightly more intutive (familiar) than sequences of edges.

Disadvantages:

* What about fractions not in lowest terms? I guess that division would be defined in a special way. And I think we might end up with having to deploy equivalence classes of integers, which is what we do anyway in the standard definition of the rationals.

* Need to define the negative reals.

* Need to define 'less than', addition and multipication.

(6) Reals are limits of paths.

This is NONSENSE. "limit of path" is not defined. If paths converge to something that is a real number, then those things they converge to must already be defined and proven to exist.

No Advantage, because it is NONSENSE.

Disadvantage:

* It isNONSENSE.

(7) Reals are programs.

I think there is mathematics for this, but I don't know enough about the details - either the definition of 'is a real' or the definition of 'less than', addition and multiplication.

Advantage:

* Perhaps it's more friendly to finitism, but I don't know that infinite objects can be actually dispensed with.

Disadvantages:

* Unintuitive that numbers are programs.

* Might be very complicated to formalize.

* I guess it would only account for the computable real numbers. So it might be complicated to formalize calculus.

TonesInDeepFreeze

↪keystone

FALSE TELLERS

"Just remember it's not a lie if you believe it".
Look up 'lie' in Merriam-Webster.
— TonesInDeepFreeze

Lie - to make an untrue statement with intent to deceive.
Perhaps cranks are deluded, but our intentions are pure. — keystone

Now you are even lying about lying. I mentioned a dictionary definition, and instead of reporting back the whole definition, you truncated it to just the sense you like while leaving off the senses that dispute you:

The two other senses you left off:

": an untrue or inaccurate statement that may or may not be believed true by the speaker or writer

: something that misleads or deceives"

It is not uncommon for people to believe their own lies. People convince themselves of untrue claims, and over time they entrench themselves deeper and deeper into the falsehoods. And when they are confronted with refutations of those falsehoods, they respond by entrenching themselves even deeper. They intentionally ignore all clear evidence and irrefragable arguments contrary to their false beliefs so that they can continue to propagate the falsehoods. That's not just lying; it is systematic lying. And when a person makes a ludicrous claim but has no basis for it, even if they irrationally believe it, we may still call it a lie. "There are transmitters in vaccines that let the government know where you are and who you are with at all times". Even if a person believes it, it's a lie, not a mere mistake.

And you know who epitomizes all of that? Cranks.

It is wonderful for you to have just provided such an ironic example where you deceptively left off the part of the definition of the word 'lie' itself!

And if you were to reply that you didn't mean to deceive but that you forgot to read the rest of the definition or whatever, then you're still lying about the definition, because failing even a modicum of diligence to get the facts right but nevertheless spouting egregious falsehoods is another form of lying.

But anyone can make a mistake, right? Sure, of course, people get things wrong and sometimes because they were not thorough in checking the facts or whatever, and that doesn't ordinarily deserve to be called 'lying'. But when it is a pattern, over and over and over, such as with cranks, then it deserves to be said that it is lying.

So, no, cranks do not have "pure" intentions.

I think it is a virtue to be able to adjust one's view upon new evidence. — keystone

Of course. But what is annoying with you is that you don't clearly say that you've changed your plan.

First you talked of reals as paths. Then you talked of reals as "limits" of paths (though that is not defined). Then you said actually there are only programs. But the way you casually slip around among those plans makes it difficult to follow and know really at each juncture what it is you actually claim. If you are to keep changing, then it would be helpful and just the least of consideration to say something like "Previously my approach was X. But now I see that X doesn't work. So from now on my approach is Y". Otherwise, it's an undo strain to follow your continual swerves.

This is not a bait and switch. — keystone

First it is was paths, then suddenly and briefly nodes, then paths again briefly, then programs.

It is unreasonable for you to disallow programs from the discussion, especially given that computer science is so closely tied with mathematics — keystone

As you added in an edit, I don't. I don't disallow computable analysis or any other rigorous mathematics. But I also don't disallow myself from commenting in full when you're spouting bull.

I should have acknowledged that my ideas and method of communication are evolving. However, I wouldn't call this a bait and switch since I'm not trying to trick you. — keystone

Regarding moving from paths to nodes to programs, it's not a question of tricking. Rather, it's that you are so confused and sloppy that you shift from one half-baked idea to another instead of organizing a coherent proposal or even point of view. That careless shifting adds up to misdirection even if not intended. And there is your continual disinformation about mathematics, some of it so egregiously produced as to be lies.

And there is this whopper lie:

I wanted to treat the real numbers as unending journeys along a path but you were inclined to treat them as the paths themselves. — keystone

I'll address that in another post.

"INCONSISTENT"

You need to stop using the word 'inconsistency' with your own private meanings.
— TonesInDeepFreeze

Granted, the branches of the S-B tree never actually intersect [...] — keystone

That and the rest of your paragraph have nothing to do with my point, which is (AGAIN) that 'inconsistency' has an exact mathematical definition. A theory is not inconsistent because it doesn't comport with your hand waving! A theory is inconsistent if and only if it there is an S and its negation that are both theorems of the theory.

until I can support arguments like this with something more concrete, I'll refrain from using the term inconsistency. — keystone

Not just "something more concrete". You need to prove that there is a sentence such that both it and its negation are theorems of the mathematics in which both the S-B tree and the continuum exist. You can't fudge it. It's flat out the case that NOTHING else justifies claiming inconsistency.

PHILOSOPHY

There is no actually infinite object corresponding to the S-B tree. Nobody has ever seen the actually infinite object with their minds eye. What we see with our minds eye is the program. — keystone

Oh no no no, you don't get away with that.

First of all, it's ludicrous to say "we" see a program when considering the S-B tree. Who all are the people that constitute the "we" who say the tree doesn't exist but there is a program held in their mind?

Of course, EVERY mathematical object in this context is abstract. No one has seen the number 0 nor any other number. No one has ever seen a line or plane or space. No one has ever seen Euclidean space or any other space, finite or infinite.

To say what I mean by saying "exists", I merely need to say that there is such and such an existence theorem. Of course, one may have philosophical views that certain things exist independently of formalizations. But all that is required at minimum are theorems. So when I say "exist", without opining on philosophical notions, I mean at least there is an existence theorem.

Now, it is true that, at least in principle, one can picture all the members of a finite set together but not of an infinite set. Moreover, aside from capability in principle, there are finite sets that humans can't picture - take any sufficiently large finite set, not even just abstractions, but concretes. It doesn't even have to be very large. Try closing your eyes and picture seventeen similar looking but different objects as distinct and all at once.

More importantly, one can also understand the meaning of a predicate such as 'is a natural number' and then take the set of natural numbers to be the abstract mathematical object that "stands for" that property.

Anyway, this is a diversion from the mathematical subject we've been talking about. The subject is not philosophical senses of 'exist'. That's a fine subject, but nothing in it alters that when one says "the S-B tree and the continuum exist" one may reduce that to the fact that the mathematical axioms prove that statement. So, for the record, notwithstanding any other philsophical bents, when I say "X exists" I mean that, at least, the axioms prove the theorem "X exists".

TonesInDeepFreeze

↪keystone

DID SOMEBODY SAY 'PATHS'?

I wanted to treat the real numbers as unending journeys along a path but you were inclined to treat them as the paths themselves. — keystone

You are SUCH a liar.

To be clear, my point here is not to claim that one should be discouraged from revising one's position or proposal, but rather that you're lying that it was my idea, prior to yours, to take the reals as paths.

From the very first post in this thread:

A number on the tree can be described by the right/left turns needed to get there from the top.
For example:
R = 2/1
RL = 3/2
RLR = 5/3

If we continue down the tree with this alternating pattern RLRLRLRLRLRL... we approach the Golden Ratio.

Is there anything wrong with completing this tree and saying that the infinite digit RL[...] is the Golden Ratio? — keystone

You were asking, obviously basically rhetorically, about an irrational number being a denumerable* sequence. First you mentioned a sense in which a denumerable sequence of rationals (each rational is a node coded itself as a binary finite sequence of Rs and Ls) approaches a real, but then you asked about just taking the denumerable sequence itself to be that real. There was no mention of "journey". As you wrote it, you were suggesting taking reals to be the denumerable sequences themselves.

* Denumerability is indicated by '...', which is ordinary informal notation for a denumerable sequence.

The notion of reals being infinite sequences of S-B branching came from YOU not me.

might irrationals be all the infinite strings which do not end in R_repeated or L_repeated? — keystone

Right there, you suggested reals as infinite strings (i.e. denumerable sequences). Literally, "infinite strings" (not "journeys") is what you said.

The notion of reals being infinite sequences came from YOU not me.

https://thephilosophyforum.com/discussion/comment/803226

There I mistakenly rejected your idea. I mentioned the standard approach, which is fine to do, but also I misconstrued that you meant that an irrational could be a node. Ironically though, even though at that juncture you weren't suggesting irrationals as nodes, later you advocated for irrationals as "mirage"-like limits. In the link above, I mistakenly took you to mean nodes, and I unnecessarily explained that there are no such nodes. But later I also explained that there are no "mirage"-like limits either. Or if the real is to be taken as a limit of such a sequence, then the object that is that limit must have been proven already to exist.

The notion of reals being infinite sequences came from YOU not me.

Why can't we say that (non-repeating) infinite decimals are journeys that are described by unending processes (e.g. limits) and not 'destinations' (numbers)? — keystone

"journey" and "destination" had not been given mathematical definitions by you - you had not stated a difference between a path and a "journey" or a difference between a node and a "destination". The crux of your point could only be understood to be that a real is an infinite string as that is what you literally suggested in the previous quote.

While irrationals do not correspond to any node in your tree, they do describe a paths on that tree (from the top all the way down), no? — keystone

There it is, right there: Associating reals with paths. Now, explicitly 'paths'.

The notion of reals being paths came from YOU not me. You are lying when you say it is the reverse.

No, the paths are not real numbers. First, a path is a sequence of edges, not a sequence of nodes. Second, a sequence of nodes is not a real number. Rather the limit of the sequence is a real number. — TonesInDeepFreeze

There, I was still adhering to the standard approach, in which it turns out that every real is a limit of a sequence of rationals. So clearly I was not pressing you to say that reals are paths, which you had just done yourself anyway. I hadn't yet realized the plausibility of the notion, and I was mistakenly RESISTING that notion, not pressing you to adopt it.

You have it backwards: It was not I who first suggested reals as paths; it was YOU who first suggested it. You're lying when you say it is the reverse.

If RL[...] looks like the golden ratio and it behaves like the golden ratio, why do you not say that it is the golden ratio? — keystone

There it is again, right there: You were arguing that reals may be taken to BE paths.

The notion of reals being paths came from YOU not me. You are lying when you say it is the reverse.

I think we both agree that RL[...] corresponds to a specific path on the Stern-Brocot tree, not a node. If the algorithm treats RL[...] as the golden ratio, then it seems reasonable to say that the golden ratio (and all real numbers) are paths on the Stern-Brocot tree. — keystone

There it is again, right there in plain quotes and explicitly 'paths': "(all the real numbers) are PATHS" [emphasis added]

The notion of reals being paths came from YOU not me. You are lying when you say it is the reverse.

I take it that by 'RL', you mean the particular denumerable path. — TonesInDeepFreeze

So by that time, based on all you posted, I said I take it that you mean the reals are paths.

I didn't first suggest that idea. YOU did. You are lying that it was initially my idea and not yours.

CORRECTION:

I initially misconstrued you. I was not reading carefully enough. I made the point that no irrational is a node on the tree. That is true, but not relevant, since your point (which I failed to read correctly) is that irrationals may be certain paths (not nodes). — TonesInDeepFreeze

There I even corrected myself to align with YOU that reals are paths, as I had earlier misconstrued you to be taking reals to be nodes.

I didn't first suggest that idea. YOU did. I was even mistakenly arguing against the plausibility of it. You are lying that it was initially my idea and not yours.

The SB tree might offer something here since it appears that each real number has a single path which can correspond to a sequence of rationals [as you hinted]. — keystone

Again, paths.

The notion of reals being paths came from YOU not me. You are lying when you say it is the reverse.

I think it's clear that every decimal number can be captured by a SB string (of L's and R's) but that is no proof. — keystone

Again, paths.

The notion of reals being paths came from YOU not me. You are lying when you say it is the reverse.

Every path (whether finite or infinite) leads to a different number. Finite paths lead to rational numbers. Infinite paths lead to irrational numbers. Or I think you'd be more comfortable saying that infinite paths that don't end in R or L lead to irrational numbers. I think of the limit of the tree as the real number line as depicted here: — keystone

There you slipped to undefined "leads to". And "limit of the tree", while there is no such thing defined in this discussion.

what is a point if not a node on the tree? Sqrt(2) does not converge towards any node on the tree. However, it appears to converge to a node that exists at 'row infinity'. Of course, there is no 'row infinity' which is why I relate it to a mirage. — keystone

Right there you switched from saying a real is a path to saying a real is a node at "row infinity" which is a "mirage".

First, as I've explained in spades, that's nonsense.

Second, this is where it would have helped if you had clearly said, "I am abandoning reals as paths and I am switching to reals as mirage nodes."

If we are redefining 'is a real number' as 'is a path in S-B' (I would prefer 'is a sequence of nodes on a path in S-B') — TonesInDeepFreeze

Again, I was following YOUR view that the reals are paths, as it was not clear to me that you had actually abandoned that proposal.

I think it's clear that every decimal number can be captured by a SB string (of L's and R's) but that is no proof. — keystone

'string' again. If not a paths, then what other sequences? Later, I gave reasons why sequences of nodes would be better, but, if I recall correctly, you ignored that, so I kept on with paths. So, by that time you were suggesting two proposals at the same time:

reals are paths
reals are "mirage" nodes that the paths converge to

The first notion is plausible.
The second notion is nonsense.

In any case a path is not a "mirage" node, so it is incoherent to suggest that reals are both paths and "mirage" nodes.

Every path (whether finite or infinite) leads to a different number. Finite paths lead to rational numbers. Infinite paths lead to irrational numbers. Or I think you'd be more comfortable saying that infinite paths that don't end in R or L lead to irrational numbers. I think of the limit of the tree as the real number line as depicted here: — keystone

No, I am UNcomfortable with that. I don't use the word 'leads'. It is not mathematically defined here.

And you need to mathematically define what you mean by the continuum being a "limit" of the tree, and prove that it is.

As we've informally agreed, irrationals (e.g. sqrt(2)) are unending paths on the SB tree. — keystone

Putting aside in what sense this is formal or not, there you again explicitly said that reals are paths. LITERALLY "irrationals are unending paths".

With this view, the objects of concern are the nodes (not the paths). — keystone

So it was paths, then "mirage" nodes, then paths again, then nodes again.

Can we say that the rational/irrational numbers are the limits of their corresponding paths. — keystone

"limit" is UNDEFINED by you in this context. I guess you're still talking about your nonsense "mirages".

As such, the object of study should not be the complete output of the program (which cannot be generated) but instead the program itself whose execution is potentially infinite. — keystone

And there you moved from reals being limits (after they were previously paths) to reals being programs.

So you started by taking reals as paths, and I eventually went along, then very recently you quickly and briefly switched to nodes. Then paths again, and now they're not paths or nodes but programs. Then you lie by saying it was my notion, not yours, that reals are paths.

You are such a piece of work.

TonesInDeepFreeze

↪keystone

VARIOUS

Infinite paths do not actually have destinations but neither does 1/x have a value at x=infinity. — keystone

There are two different notations:

(1)

lim 1/x [x = 1 to inf] = 0

There 'inf' does not stand for an object named 'inf'. Rather, it is an informal placeholder as we may replace it with a formulation that does not invoke 'inf':

Let f = {<x 1/x> | x is a natural number greater than 0}.

lim f = 0

No mention of 'inf' there. Saying "to inf" is merely a figure of speech and does not imply that there is an object named "inf".

(2)

An extended system where there are the objects -inf and +inf.

Note that those may be any objects other than real numbers. They don't have to actually be infinite sets.

Ordinarily the stipulative (and I stress 'stipulative') definition of the division operation yields:

1/inf = 0

But there is value in saying that 1/x approaches 0 as x approaches infinity. — keystone

We don't just "say" it. We prove it.

there is value in saying that the path R L[...] approaches 1 as we descend the tree and approach 'row infinity'. — keystone

(1) You've cited two different versions of the S-B tree. Please choose one and stick with it.

In one version, 1/1 is the initial node and it is not obtained by any path. In the other version, 0/1 and 1/0 are the initial nodes, and 1/1 is the sole rational that is obtained twice by two different path, viz. R from 0/1 and L from 1/0.

In any case, 1/1 is not reached by by any finite or denumerable sequence R L[...].

And again, please choose one of the two versions and stick with it.

(2) Please stop saying "row infinity". There IS NO row infinity. We PROVE that there is no row infinity. It is not coherent mathematics to keep saying "row infinity" no matter that you put single scare quotes around it or call it a "mirage" or "fiction".

In both of these examples, infinity is a useful fiction. — keystone

Wrong. In the case of standard mathematics, we PROVE the existence of whatever objects we use. But in your case, you just hand wave that somehow there are "fictions" not defined even as abstract mathematical objects that explain your arbitrary claims.

however you want to phrase [the notion of "row infinity"]. — keystone

I don't phrase it in any way. Because I don't have the intellectual mathematical dishonesty of trying to get by with nebulous undefined hand waving terminology.

paths (as described by infinite sequences of rational numbers) — keystone

Just to be clear, I still don't know whether you understand that paths are not sequences of nodes.

it doesn't seem like a big jump to say that the limit of a SB path is analogous to the limit of a Cauchy sequence, or in other words, that the limit of infinite SB paths are nodes corresponding to real numbers. — keystone

For crying out loud, I gave you DETAILED explanation why that is incoherent.

One more time: With Cauchy sequences, there IS an object that is the limit. But with your bull, there is no object (except you preposterously and egregiously hand wave that these objects are "mirages" or "fictions").

I gave the example of sqrt(2), but you SKIPPED.

AGAIN, you can't just take a sequence and say that there is a limit; it's not enough to say that the terms of the sequence get closer to each the next - you have to PROVE that there is an object such that the terms get arbitrarily close to it.

When you just say that there is a "limit" of the sequence of paths (or now it's nodes, you keep switching). You have to PROVE that there is such a limit.

In the rational numbers, there is no limit to the sequence 1, 1.4, 1.41, 1.414, 1.4142 ... But we prove that in the reals there is a limit, viz. the least upper bound of the range of the sequence. And that least upper bound is the square root of 2.

That is, we prove that, in the reals, every bounded set has a least upper bound. {x | x^2 < 2} is bounded. So it has a least upper bound. Then we prove that the least upper bound of {x | x^2 < 2} squared is 2, thus the least upper bound of {x | x^2 < 2} = sqrt(2) . And the sequence of rationals that are approximations converges to the limit, which is sqrt(2).

You can't just say, "The terms of the sequence get closer, so PRESTO POOF OF MAGIC, there's this mystical, fictional "mirage" thing that I say is the limit!" You have to PROVE that there is such a limit.

The bottom line being that if your magical mystical mirages are merely fictions, not proven mathematical objects, then you don't have any math about them!

I don't know how I can make this any clearer for you.

it doesn't seem like a big jump to say that the limit of a SB path is analogous to the limit of a Cauchy sequence — keystone

It is a HUGE jump, because 'limit of a Cauchy sequence' has an exact mathematical definition, while 'limit of a path' in this context is undefined and your notions about it are INCOHERENT. See above

infinite paths do not end at any node (in a similar way that Cauchy sequences do not end at any rational number — keystone

There is a similarity only if we ignore, as you do, the decisive difference that the Cauchy sequences DO converge to objects that we DO prove to exist, as we don't rely on magic wand waving as you do.

And you need to drop the word "end". The Cauchy sequences do not end at a limit. The sequences do not have an end. Rather they CONVERGE. Every time you say things like "end" you only entrench in your mind a basic misconception and then add it to the landfill of misinformation on the Internet.

The issue then is that in the S-B tree, we only have nodes corresponding to the rational numbers. — keystone

First you said the rationals are the finite paths. In that case each node is the terminal node on a finite path. So a rational is a finite path and has its corresponding terminal node.

The nodes corresponding to the real numbers are fictional. — keystone

Your "fictional" is meaningless. Anyone can say, "Here are my objects that are solutions. They're fictional objects that I made up. I like them. So there ... my math! By the way, I have an answer to Goldbach's conjecture! There is an even number that is not the sum of two primes. It's a fictional number I made up. I call it 'the Emerald number'. Behold my math!" It's bull.

Comparatively, with the real number line, we do not distinguish between the rational and irrational points on the real number line. They are of the same essence. — keystone

Oh, yes, "essence". Yeah right, everyone knows what an essence is in mathematics.

Meanwhile, I'll inform you:

A rigorous formulation is that rational numbers are equivalence classes of integers. Then real numbers are equivalence classes of Cauchy sequences of rational numbers. So, strictly speaking, the set of rationals is disjoint from the set of reals. But there is an embedding of the rationals in the reals. So, in another sense of 'rationals', we take the rationals to be the image of that embedding.

Why is it important to correct you on this? Because garbargy undefined rubrics like 'essence', 'mirage', etc. only invite fallacious inferences based not on math but on mere suggestibility.

This disagreement between the S-B tree and the real number line is what I'm trying to highlight. — keystone

You're only highlighting your ignorance, confusion and dishonesty.

I've explained more than once that there is NO "disagreement" between the S-B tree and the continuum. They are both proven to exist from the mathematical axioms without inconsistency.

Indeed, if we countenance a proposal to take real numbers as the paths in the S-B tree, then presumably we get a number system isomorphic with the standard treatment. There's not a quandary about this.

I can write a program that generates all rows and stops — keystone

No, you can't.

However, I cannot execute it. The program is no less of a program just because the output doesn't exist. — keystone

If someone says I "wrote a program to do X but it doesn't do X", I wouldn't know what that would mean.

Ordinarily, by "program that does X" we mean that it does X.

The [fictitious] output is used to describe the program/execution (not the other way around). — keystone

There you go again with your "fictitious" stuff. It's bull.

A program is not specified by an outcome. A program is specified by instructions, and those instructions entail outcomes upon inputs. One such algorithm provides for printing natural numbers increasing in size and never stopping. But there is no algorithm that provides for printing every one of the natural numbers and then stopping. That is, there is a program such that for any natural number, the output will write that natural number; but there is no program that will finish writing all the natural numbers. That is, the set of natural numbers is computably enumerable but it is not finitely enumerable.

TonesInDeepFreeze

PS. As I mentioned, in my first few posts I mistook your initial proposal. In those posts I have now added edits in brackets, and marked as 'Edit', to note my mistake.

keystone

@TonesInDeepFreeze

As always, I appreciate your detailed responses.

I believe in what I'm saying but IF it's untrue then you're right - according to Merriam-Webster, my statements would be considered lies. I don't want to argue with you about my intentions. I was wrong to claim that you came up with the path definition. I will not claim something is 'inconsistent' unless I can prove that both a statement and its negations are theorems of the mathematics of concern. Sorry. Overall, I've been sloppy with my communication. I intend to be more clear. I ask that we set aside my earlier comments in this thread and I'll start afresh to better capture my position as I respond to your last set of messages. I'll need some time to digest all of your comments and formulate my response.

TonesInDeepFreeze

↪keystone

Probably, the idea of reals as programs has been written about. I wonder whether it's covered in the subject of computable analysis.

And it seems interesting to apply the notion of programs to the S-B tree. I can see the basic idea: Each program would generate sequences of Rs and Ls. A real number would be a program. For a rational real, execution of the program would halt. For an irrational real, execution of the program would not halt.

But we have to keep in mind that this only accounts for the computable reals. So we'd have to explain how to formalize calculus with only the computable reals. I think that has been written about, though I don't know enough about it.

keystone

Consider the follow Python function:

def endless_loop()
	while True:
	    print("Looping indefinitely...")
	Return 1

Do you consider this a valid function? It certainly obeys Python syntax rules and can be interpreted by a computer program. Does it make sense to say that it is designed to return 1 even though it never actually could? I see in your recent post that you can see the basic idea of reals being programs so perhaps these questions don't need answering.

THE EXTENDED S-B TREE

Moving forward, let's use this extended version of the S-B tree so that all real numbers are covered.

At every row we have a blue line corresponding to the real number line.
The white nodes indicate that the blue lines corresponds to the open interval (-inf,+inf).
One can travel from the 0/1 node to any rational node in the tree by a unique sequence of left and right turns as described by S-B strings. Some examples of S-B strings are RL (1/2) and LL (-2/1).
There are two ways to interpret infinite journeys: (1) as completed processes, whereby they correspond to the entire path travelled and (2) as incomplete unending processes. I prefer the latter but until it becomes an issue, let's use journey ambiguously without saying whether it is completed or not.
Row infinity does not exist. From now on, when I say 'approaching row infinity' it will be an informal placeholder corresponding to an unending journey down the tree.
I have an issue with infinite sets but I see value in the concept of equivalence classes. Instead of discussing what an equivalence class might mean in the absence of infinite sets, please grant me slack to use this concept loosely. While each node only contains the reduced form of a rational number, it actually correspond to an equivalence class of integer pairs.
As we progress down the tree, nodes are placed directly on top of their corresponding point on the real number line. For example, the node corresponding to 2/3 lies directly on top of the point corresponding to 2/3. Although the nodes are illustrated as large circles, let's imagine that they are actually points. So while the blue line gets hidden from the illustration as the nodes become more numerous, let's imagine that the real number line is still present in its entirety at every row.

Who all are the people that constitute the "we" who say the tree doesn't exist but there is a program held in their mind? — TonesInDeepFreeze

When I imagine the Stern-Brocot tree, I visualize the first few rows of the tree and then internally mutter 'and so on as governed by the rules for constructing it'. That 'and so on' is not a formal program, but it refers to an algorithm which we can fit into our finite brains and be comprehended.

AGAIN, you can't just take a sequence and say that there is a limit; it's not enough to say that the terms of the sequence get closer to each the next - you have to PROVE that there is an object such that the terms get arbitrarily close to it. — TonesInDeepFreeze

With the real number line now embedded in the tree, hopefully it is reasonable to say that as we approach row infinity, the journey corresponding to RRL converges to the point corresponding to the golden ratio.

So when I say "exist", without opining on philosophical notions, I mean at least there is an existence theorem. — TonesInDeepFreeze

Given this, I'll stop using 'exists' since my view of existence extends beyond mathematics. I know you don't like it when I make up new terms, but hopefully you'll give me slack here as I'm trying not to hijack 'exists'. Instead, I'll use 'actualized' for what I mean. In my view, an object is actualized if it is present in the memory of a 'computer'. I am actualized because I am present in the 'computer of the universe'. I'm thinking of a purple cow so that purple cow is actualized because it is present in the 'computer of my mind'. This statement is actualized as I type because it is present in the memory of my laptop. When a memory of an object is flushed, it becomes 'potentialized'.

But we have to keep in mind that this covers only the computable reals. So we'd have to explain how to formalize calculus with only the computable reals. — TonesInDeepFreeze

The infinite decimal corresponding to pi is forever potentialized.
Leibniz's formula for pi/4 is currently actualized because it is present in my mind.
If I understand correctly, non-computable reals cannot be present in the memory of any computer so they are forever potentialized.

Let's say we color the necessarily potentialized points on the real number line red and all possible actualized points green. Given that the computable reals are countable, they have measure 0 so the real number line is entirely red. But we still have the line and can perform calculus as usual. I don't understand why we would need to adjust anything in our formalization of calculus.

keystone

@TonesInDeepFreeze

I just realized I incorrectly described the black lines in the S-B tree image. They do not correspond to paths from 0/1 to all the rational numbers. They link any node to its two parent nodes. Doh! It was a while ago when I made that figure...

TonesInDeepFreeze

You still systematically resort to sophistry. But I will spend some of my finite supply of patience anyway, though I shouldn't.

First, do you agree that any program you can write can in principle be stated as a Turing machine?

At every row we have a blue line corresponding to the real number line. — keystone

At every row you have a finite set of rational numbers ordered by 'less than'. That's not the real number line.

You say "corresponding to the real number line". There you go again, invoking an infinitistic object (the real number line) that you claim doesn't exist. That's a cheat. You want to develop calculus without infinite sets. So you can't then smuggle in infinite sets. You can't say that something corresponds to something else that you yourself claim does not exist! Please, don't continue to insult my intelligence.

Row infinity does not exist. From now on, when I say 'approaching row infinity' it will be an informal placeholder corresponding to an unending journey down the tree. — keystone

(1) I am wary that you'll smuggle in the existence of a "row infinity" anyway. It would be more clear to just stay away from the threat of such sophistical pits and instead just use defined terminology.

(2) "unending journey". Back to undefined terminology again.

(3) There is no "tree" in this context (in the sense of all the tree).

(4) So you are taking "approaching row infinity" as informal for "unending journey down the tree", so an informal rubric to stand for another informal rubric. As they say in show business, the jokes write themselves.

I have an issue with infinite sets but I see value in the concept of equivalence classes. Instead of discussing what an equivalence class might mean in the absence of infinite sets, please grant me slack to use this concept loosely. While each node only contains the reduced form of a rational number, it actually correspond to an equivalence class of integer pairs. — keystone

In other words, "I don't accept that there are infinite sets, but I'm going use infinite sets anyway".

No, I don't grant you that "slack". The very point is that you propose to do calculus without infinite sets. If you're going to use infinite sets anyway, the all bets are off. You're wasting our time.

As we progress down the tree — keystone

No, in your context, there is no tree. There is only a program. That program never outputs what you are calling "the tree". Even as the program runs with no upper bound of steps, it never outputs what you are calling "the tree". Only, at any given step, the program outputs a finite number of rows. Please stop trying to have your cake and eat it too.

let's imagine that the real number line is still present in its entirety at every row. — keystone

In other words, let's imagine a contradiction. Let's imagine that the real line does not exist but that the real line exists.

Who all are the people that constitute the "we" who say the tree doesn't exist but there is a program held in their mind?
— TonesInDeepFreeze

When I imagine the Stern-Brocot tree, I visualize the first few rows of the tree and then internally mutter 'and so on as governed by the rules for constructing it'. That 'and so on' is not a formal program, but it refers to an algorithm which we can fit into our finite brains and be comprehended. — keystone

(1) You believe the tree does not exist. Yet, earlier in your post, you refer to "the tree", so one must think you are referring to something that does exist.

You need to state some axioms, then definitions and proofs. Otherwise, you're doomed to perpetually post nonsense.

(2) I asked who are the "we". Your reply is that it's you.

With the real number line now embedded in the tree, hopefully it is reasonable to say that as we approach row infinity, the journey corresponding to RRL converges to the point corresponding to the golden ratio. — keystone

What an obnoxious mess that is.

(1) There is no real line in this context. There's only a program that does not output the real line but only successive finite proper subsets of rationals.

(2) "The tree" (as in the full tree) does not exist in this context. Only, at any given stage of output, finitely many rows.

(3) The real line is not "embedded" in the tree in this context. Moreover, 'embedded' has a mathematical definition. But you haven't defined your own sense.

And even in standard mathematics, one would have to say in what way the real line is "embedded" in the three. The real line is the set of points of the form <x 0> where x is a real number, along with the less than ordering. The S-B tree (or your variant) do not have a row that is the real line. And no row has irrational nodes anyway! But, of course, we do prove the existence and define 'the real line' irrespective of the S-B tree anyway.

You propose a program, to avoid committing to infinite objects. Fine. But then you invoke infinite objects anyway. That is stupid and childish.

(4) You said you use 'row infinity' in this sense

"when I say 'approaching row infinity' it will be an informal placeholder corresponding to an unending journey down the tree."

So let's plug it in and see what we get:

as we unending[ly] journey down the tree, the journey corresponding to R RL[...] converges to the point corresponding to the golden ratio.

Aside from, 'journey' undefined, and there being no full tree, and convergence in this context not being defined:

There IS NO point in the tree corresponding to Phi. We've been through this already!

Now we are back to where we were near the beginning of this thread!

Your idea of reals (the computable reals, I think) being programs might be okay. But your latest post does not contribute a coherent extension of the idea nor much hint of making it rigorous. Your latest arguments sputter at the level of nonsense from the start.

I'll use 'actualized' for what I mean. In my view, an object is actualized if it is present in the memory of a 'computer'. — keystone

That's too vague for me to take as mathematics. Let me know if you ever get around to moving past word poetic metaphors and instead present some actual mathematics.

Let's say we color the potentialized points on the real number line red and the actualized points green. — keystone

Better yet, howzabout you give mathematical definitions for 'potentialized' and 'actualized".

I don't understand why we would need to adjust anything in our formalization of calculus. — keystone

Because the standard formalization uses infinite sets. Meanwhile, you don't have formalization.

Get a book on mathematics, why don't you?

TonesInDeepFreeze

↪keystone

Among all the nonsense you posted, this is a good item to especially highlight:

as we approach row infinity, the journey corresponding to RRL converges to the point corresponding to the golden ratio. — keystone

Now we are back to where we were near the beginning of this thread!

Aside from, 'journey' undefined, and there being no full tree, and convergence in this context not being defined: There IS NO point in the tree corresponding to Phi.

After I've posted so much to explain to you why your notes are nonsense, you regress back to your original fallacy!

You are a consummate crank: You cop out with undefined terminology. You assert contradictions but seem not to realize it. And most of all, the very essence of a crank: You SKIP refutations that have been given you and just go back to reasserting what has been refuted as if it was not refuted, not even RECOGNIZING the arguments that refute you, thus putting the conversation in a circle.

jgill

A little knowledge (from Wikipedia) can be a dangerous thing. :roll:

keystone

There you go again, invoking an infinitistic object (the real number line) that you claim doesn't exist. — TonesInDeepFreeze

So that I can respond it will be helpful if you clarify your position on programs by answering these questions I asked in my earlier message to you.

Consider the follow Python function:
— keystone

def endless_loop()
	while True:
	    print("Looping indefinitely...")
	Return 1

Do you consider this a valid function? It certainly obeys Python syntax rules and can be interpreted by a computer program. Does it make sense to say that it is designed to return 1 even though it never actually could?

TonesInDeepFreeze

Do you consider this a valid function? — keystone

I like this better:

(1) Print "Hello". Go to (2).
(2) Go to (1).
(3) Print "Goodbye". Halt.

That's an algorithm. The execution of it successively prints "Hello", and it never prints "Goodbye", and it never halts.

Algorithms may have an instruction that is never executed because executions always loop ahead of the instruction.

That's different from saying that an algorithm executes infinitely but then finally provides an output after executing infinitely.

An algorithm outputs successive finite approximations of Phi. But it never outputs the infinite expansion of Phi. That would be a supertask, while you disavow that there are supertasks! You are again trying to have your cake and eat it too. Grow up. And maybe have the courtesy not to keep looping back to an already refuted starting point in this discussion, which is to say, to not exercise your right as a crank to SKIP whatever refutes you.

Algorithm to emulate conversation between keystone and TonesInDeepFreeze:

(1) Print "keystone says: Phi is a node in the tree." Go to (2).
(2) Print "TonesInDeepFreeze says: Phi is not a node in the tree, because first, in your proposal there is not an infinite tree, and second, even in standard mathematics, there are no nodes in that tree that are irrational numbers, and third, there is no "row infinity" and so no "row infinity" that has nodes for Phi or any other irrational number, and fourth, paths don't "converge" to points the way you incoherently imagine they do." Go to (1).
(3) Print "keystone says: I see now. I should learn some mathematics." Halt.

Instruction (3) is in the program, but never executed.

You said reals are programs. Fine. As far as I can tell, those are computable reals. Fine. So go ahead and write a template for programs that are reals. Then define 'less than', addition and multiplication of programs. Then show how to do calculus with this. But, meanwhile, looping back to junctures that have already been refuted is stupid and obnoxious.

TonesInDeepFreeze

A little knowledge (from Wikipedia) can be a dangerous thing. — jgill

Oh, you said it so well.

Real numbers and the Stern-Brocot tree

Welcome to The Philosophy Forum!

Categories

More Discussions