never being able to complete the computation involving infinite strings — keystone
what is a point if not a node on the tree? — keystone
Sqrt(2) does not converge towards any node the tree. — keystone
it appears to converge to a node that exists at 'row infinity' — keystone
Of course, there is no 'row infinity' which is why I relate it to a mirage. — keystone
There is an inconsistency in claiming that both (1) the real number line exists and (2) 'row infinity' does not exist. — keystone
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
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
sqrt(2) is a real number. A real number doesn't converge. A sequence converges. — TonesInDeepFreeze
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
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
the computation — keystone
As we've informally agreed, irrationals (e.g. sqrt(2)) are unending paths on the SB tree. — keystone
sqrt(2) is a real number. A real number doesn't converge. A sequence converges. — TonesInDeepFreeze
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
I, on the other hand, am defending my position by making my augments more and more simple. — keystone
I suppose one way for you to end this would be by explaining why the SB tree (and my argument) oversimplifies the issue. — keystone
If you don't like my description of the real number line as it relates to the SB tree — keystone
can you propose a better description using the SB tree? — keystone
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
If you propose the tree as a basis for the real numbers, then you have to provide such a definition. — TonesInDeepFreeze
of course such a sequence does not converge to sqrt(2), since sqrt(2) is a sequence and not a node. — TonesInDeepFreeze
what I did in the other threads was to generously give you increasingly detailed explanations — 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
I have no problem with saying that set of real numbers is the set of paths in the S-B tree — TonesInDeepFreeze
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
The set of paths in the S-B tree is not part of the S-B tree. — 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
There is an inconsistency in claiming that both (1) the real number line exists and (2) 'row infinity' does not exist. — keystone
Do you think the real number RL (phi) is the path itself or the limit of the path? — keystone
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
RL (phi) does not converge to any node. — keystone
You brought the conversation to a level of complexity/formality that I wasn't comfortable — keystone
The reason why I prefer the S-B tree view is that it's more understandable to amateurs like myself. — keystone
I'm trying to understand what the real line is from the perspective of the S-B tree. — keystone
I'm accepting whatever coherent proposal YOU are making. — TonesInDeepFreeze
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
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
(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
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
so there is no path to 2/4, so 2/4 is not a rational number. So what is it? — TonesInDeepFreeze
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
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
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
Can we say that the rational/irrational numbers are the limits of their corresponding paths. — keystone
One can write a finite (but complete) computer program to create the entire S-B tree. — keystone
One can write a finite (but complete) computer program to create the entire S-B tree. — keystone
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
I don't mind a computer program that prints all natural numbers. — keystone
'we' don't actually work with the infinite sets themselves, 'we' work with the 'algorithms' used to generate them — keystone
You need to stop using the word 'inconsistency' with your own private meanings. — TonesInDeepFreeze
"Just remember it's not a lie if you believe it".
Look up 'lie' in Merriam-Webster. — TonesInDeepFreeze
Choose a definition and stick with it.
First you said the reals are the paths. Now you say they are nodes. — TonesInDeepFreeze
Your insouciance in not making definitions and sticking to them invites confusion and is annoying. — TonesInDeepFreeze
If you want to have the things that are going to serve as the limits, then you need to prove they exist. — TonesInDeepFreeze
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
But, again, it entails that those limits are THEMSEVLES objects - existing. — TonesInDeepFreeze
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
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
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
"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
I think it is a virtue to be able to adjust one's view upon new evidence. — keystone
This is not a bait and switch. — keystone
It is unreasonable for you to disallow programs from the discussion, especially given that computer science is so closely tied with mathematics — keystone
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
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 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
until I can support arguments like this with something more concrete, I'll refrain from using the term inconsistency. — keystone
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
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
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
might irrationals be all the infinite strings which do not end in R_repeated or L_repeated? — keystone
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
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
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
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
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
I take it that by 'RL', you mean the particular denumerable path. — TonesInDeepFreeze
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
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
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
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
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
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
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
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
As we've informally agreed, irrationals (e.g. sqrt(2)) are unending paths on the SB tree. — keystone
With this view, the objects of concern are the nodes (not the paths). — keystone
Can we say that the rational/irrational numbers are the limits of their corresponding paths. — keystone
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
Infinite paths do not actually have destinations but neither does 1/x have a value at x=infinity. — keystone
But there is value in saying that 1/x approaches 0 as x approaches infinity. — keystone
there is value in saying that the path R L[...] approaches 1 as we descend the tree and approach 'row infinity'. — keystone
In both of these examples, infinity is a useful fiction. — keystone
however you want to phrase [the notion of "row infinity"]. — keystone
paths (as described by infinite sequences of rational numbers) — keystone
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
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
infinite paths do not end at any node (in a similar way that Cauchy sequences do not end at any rational number — keystone
The issue then is that in the S-B tree, we only have nodes corresponding to the rational numbers. — keystone
The nodes corresponding to the real numbers are fictional. — keystone
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
This disagreement between the S-B tree and the real number line is what I'm trying to highlight. — keystone
I can write a program that generates all rows and stops — keystone
However, I cannot execute it. The program is no less of a program just because the output doesn't exist. — keystone
The [fictitious] output is used to describe the program/execution (not the other way around). — keystone
def endless_loop() while True: print("Looping indefinitely...") Return 1
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
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
So when I say "exist", without opining on philosophical notions, I mean at least there is an existence theorem. — TonesInDeepFreeze
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
At every row we have a blue line corresponding to the real number line. — keystone
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
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
As we progress down the tree — keystone
let's imagine that the real number line is still present in its entirety at every row. — keystone
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
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
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
Let's say we color the potentialized points on the real number line red and the actualized points green. — keystone
I don't understand why we would need to adjust anything in our formalization of calculus. — keystone
as we approach row infinity, the journey corresponding to RRL converges to the point corresponding to the golden ratio. — keystone
There you go again, invoking an infinitistic object (the real number line) that you claim doesn't exist. — TonesInDeepFreeze
Consider the follow Python function:
— keystone
def endless_loop() while True: print("Looping indefinitely...") Return 1
Do you consider this a valid function? — keystone
A little knowledge (from Wikipedia) can be a dangerous thing. — jgill
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