It probably doesn't make sense to roll an infinite sided dice in the first place — flannel jesus

The limitation of possible representation is still an issue here; there must be a finite maximum magnitude for the game. Infinity cannot be a stipulation. — Nils Loc

You could fix the problem with doing away the limit of natural numbers and including the negative integers to infinity. — Nils Loc

Maybe it's undefined for the natural numbers also. — Nils Loc

Just as a polygon with infinite sides of equal length would be empirically indiscernible form a perfect circle—this irrespective of how close one observes it, say with a quantum microscope—so too would a die with infinite sides of equal length be empirically indiscernible from a perfect sphere, even when analyzed with a quantum microscope. — javra

There would be no way to read the numbers because any of them could appear to be unlimited and by the physical limitations of the universe would therefore be impossible. — Nils Loc

"Infinite" is not a number as an amount of sides but is that which cannot be gotten to since it never completes, but to play along, the die is ever becoming more of a higher and higher resolution smoother sphere. — PoeticUniverse

It gives your paradox a much easier angle of angle of attack than the unexpected hanging has. You can point out the error in the formulation. That "an adjacent pair of real numbers" doesn't exist. — fdrake

I also wouldn't like to call it a sequel to the unexpected hanging because your riddle isn't (at least at face value) anything to do with induction (the mathematical reasoning concept) or self reference. — fdrake

So you can't detect the exact points a step change in a function ([0,1]->red,green) would occur by putting in any countable list - at least your guesses have probability 0 of being right if God's not been too human about it. — fdrake

In that respect the kind of ambiguities in the framing are already solved problems, and it's a riddle about something completely different from the unexpected hanging. — fdrake

Well, I don't think the "paradox" even makes sense here. If he manages to find a moment when the guillotine is going through his neck (different example), the light should shine green. — Lionino

Besides, it is not a paradox. — Lionino

I think you're wrong to call it a sequel to the hanging paradox. It's conceptually nothing like the hanging paradox, other than the fact that both word problems both involve a hanging - which is just a superficial similarity. Conceptually it has more in common with zenos paradox of motion, which also has to do with problems of infinites and infinitely divisible units. — flannel jesus

Hold on a second! What is a "pair of adjacent real numbers"? — Pierre-Normand

Consider also that most statements of the paradox have the prisoner ignorant of the day on which they will be hung. The days are discrete and hence don't span a continuum. — Pierre-Normand

I didn't say that I'll only consider formalizations. I have been interested in the earlier proposals though not formalized. Rather, I said that I'm not inclined now to study your latest revisions. — TonesInDeepFreeze

Somehow, I don't believe you. To formalize you'd have to know what formalization IS. Be honest: Learning what goes into an axiomatic formulation is not a goal for you. — TonesInDeepFreeze

So, hopefully, you understand now that there's no "cake and eating it too" about the S-B tree and Cauchy sequences in set theory, or generally in set theory having both finite algorithms and infinite sets. — TonesInDeepFreeze

No, set theory shows how the paradoxes with the naive notion of sets are avoided. — TonesInDeepFreeze

Set theory is quite intuitive to me. I listed the axioms for you. I find each of them to be eminently intuitive. — TonesInDeepFreeze

Then you added more apparatus that doesn't seem to me to improve the more basic and original goal that was not being addressed. Then you went further about "higher dimensions". I'm not sufficiently interested in whatever that's about to invest time and energy on it, while instead my curiosity is with the original questions of defining ordering and the operations. — TonesInDeepFreeze

So while the mathematician is still in the pre-formalized stage, deepening and extending the intuitions, she is putting herself into a kind of "intellectual debt". That is, the mathematician eventually is going to have to "pay" for the intuitive commitments with the hard cash of formalizing them. — TonesInDeepFreeze

So, in set theory, there is both the tree that doesn't have a final row or "row infinity" and the continuum. This is not having our cake and eating it too. Whatever we have comes from proofs from the axioms. The axioms are productive enough to proof the existence of many things including: the continuum, the S-B tree, finite algorithms, etc. — TonesInDeepFreeze

You are in an intuition stage. If you ever followed through to write some mathematics, then you would confront the debt you're accumulating and pay it off with rigorous formulations. But, in the meantime, one still needs discipline to not just mouth a bunch of incoherent mental picture stories. Even with intuitions, one would like not to commit to informal contradictions (unless one wants to base the proposal in a paraconsistent logic). Which is to say, crankery is a dead end. — TonesInDeepFreeze

You seem to have a notion that we have to distinguish numbers. No, every object is a set. — TonesInDeepFreeze

I don't know the purpose of this exercise. — TonesInDeepFreeze

So what? Lots of things use similar language, but say RADICALLY different things. — TonesInDeepFreeze

A continuum requires that the ordering is complete, meaning that for every bounded set, there is a least upper bound. — TonesInDeepFreeze

You don't get to incorrectly say what I'm willing to imagine. I don't even have notion of "points are fundamental", let alone that I won't imagine that it's not the case….
Please don't make pronouncements about what I am willing to imagine. — TonesInDeepFreeze

Cauchy sequences have a limit. But if we somehow defined the limit of an algorithm, then that would be infinitistic — TonesInDeepFreeze

our use of 'line' is only a figure of speech. It's not a line. It has nothing on it; it's a placeholder only - as YOU said. It's not a line in the sense of geometry or analytic geometry. — TonesInDeepFreeze

And at no output does the cutting remotely resemble the continuum. First, at every output, there are only finitely many cuts and thus only finitely many rationals described. Second, there are no irrationals described. That is VERY different from the continuum that has both rational reals and irrational reals and altogether not just finitely many, but uncountably many, and proving a continuum. — TonesInDeepFreeze

Your "line", the k-line, has NOTHING on it, as YOU said. So 'continuous' is not even applicable. And there is no infinite set of cuts on the k-line that comes after all the rows. You just now admitted that. — TonesInDeepFreeze

Length is the absolute value of a difference. Even without irrationals, we have length with just rationals. Uncountability is not required to define length. Sheesh! — TonesInDeepFreeze

But even if it worked out, calculus needs more than just lengths. It seems you don't know what calculus is. Do you? — TonesInDeepFreeze

You keep wanting to have both only finite objects but also objects that exist only as provided as an end of an infinite process, while refusing in different forms to recognize that there is no such end hence no such objects. — TonesInDeepFreeze

We don't need the k-line. It is extraneous to capturing the information we want. We can just say a row is the set of cuts.

We don't need cuts. They are extraneous to capturing the information we want. We can just mention the fractions and their ordering.

I think the reason you want all that is to give the illusion that it amounts to a kind of pseudo-"continuum". But it doesn't. Essentially it's a big red herring. Toss out the red herring and simplify as I showed you, which is basically what you proposed yesterday. — TonesInDeepFreeze

I was asking whether you understand the post, which includes the various aspects of its explanations. Knowing your answer would let me know how much communication is taking place here. — TonesInDeepFreeze

I get to say, "Phi is the limit", because set theory proves there IS such a limit. You do not, because your framework PRECLUDES that infinitistic limit. — TonesInDeepFreeze

• A point in the context of my view is a k-point
• A line in the context of my view is a k-line
• An interval in the context of my view is a k-interval
• An algorithm which takes as input any natural number (corresponding to the row) and always outputs a k-interval (in finite time) is a k-algorithm
• A function which takes as input rational number(s) and outputs rational number(s) is a k-function
• A string of L's and R's (and underlines to represent repeats) is a k-string
• A k-line which has no gaps is k-continuous.
• ..and so on.

Meanwhile, I've asked you three times now whether you understand this post:
https://thephilosophyforum.com/discussion/comment/806060
But you still say not a word about it. — TonesInDeepFreeze

Bringing in a concept of an initial object that is determined solely by -inf and +inf and then pseudo-intervals is extraneous. — TonesInDeepFreeze

• y=3x^2+2
• y=0
• x=1
• x=0

Are my proposed algorithms that different from Cauchy sequences?
— keystone
Indeed they are! I EXPLAINED this. — TonesInDeepFreeze

Your use of 'line' is only a figure of speech. It's not a line. It has nothing on it — TonesInDeepFreeze

So what? It used infinitisitic methods. Set theory provided axioms to make those methods rigorously derived from axioms. — TonesInDeepFreeze

There is an algorithm, call it the 'k-S-B algorithm', that generates rows, starting with the base row, then to the next row that is row 0, ad infinitum. The k-S-B algorithm recursively exhausts all "turn decisions" of R and L. — TonesInDeepFreeze

I really should not continue to reply when you so obnoxiously continue to apply the same fallacy, clothed differently, each time though I have explained it over and over and over and even asked you whether you understand, yet you don't reply even to that question itself. — TonesInDeepFreeze

You STILL don't get it. You just keep putting new clothes on an old pig. Every time, you reformulate but you retain the essential fallacy.

There is no single "vanishingly small pseudo-interval". There are only successively smaller pseudo-intervals on successive rows. — TonesInDeepFreeze

Except that I don't see that it improves the more simple approach — TonesInDeepFreeze

But it is the standard calculus depends on the completeness of the reals. — TonesInDeepFreeze

There are no points on this line.
— keystone
[bold ORIGINAL]

You seem to have no compunctions about insulting intelligence. — TonesInDeepFreeze

Changed from standard analysis? No, your proposal is radically different from standard analysis, from the start: Standard analysis uses infinite sets, you disclaim infinite sets. Standard analysis has a continuum; you don't. Standard analysis has uncountably many reals. With you, it's not clear how many comp-reals there are, since there are denumerably many real-ithms but you disclaim that there are denumerable sets (but maybe you could argue that there is not a set of all real-ithms but instead a program that itself generates real-ithms). — TonesInDeepFreeze

Then, define '<', '+' and '*' on real-ithms, and you'd be on our way to something.

In other words, for arbitrary real-ithms G and R:
G < R <-> [fill in definiens]
G+R = [fill in definiens]
G*R = [fill in definiens] — TonesInDeepFreeze

You want only finite objects, but you also want real numbers — TonesInDeepFreeze

By contrast, in a language with lazy semantics such as Haskell, terms can be used and passed around in partially evaluated form. This means that real numbers can exist in the sense of partially-evaluated "infinite lists" consisting of an evaluated prefix and an unevaluated tail. These lazy languages allow runtime conditions to decide what rational value is used in place of a term of real-number type, which is allowed to vary during the course of computation and which corresponds more closely to the notion of "potential infinity". — sime

Do you consider this a valid function?
— keystone

I haven't the foggiest. — TonesInDeepFreeze

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

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

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

• 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

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

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

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

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

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

I can only take your word for it that you've satisfactorily worked out that arithmetic. Don't forget that you have to manage not just finite sequences but infinite ones too. — 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. But some people just can't grok the idea of the entries of a sequence getting arbitrarily close to a point but that point is not itself an entry in the sequence. — TonesInDeepFreeze

I'm not talking about that tree in that context. I was talking about the three competing definitions of 'is a real number' and how easy or difficult it is to define the operations for real numbers based on those definitions. — TonesInDeepFreeze

so if 'is a real number' would be defined as just one particular Cauchy sequence of rationals, then which of the infinitely many should it be? — TonesInDeepFreeze

'is a path on the left side of the SB tree' as a fourth competing definiens? It would be of 'is a real number between 0 and 1 inclusive'. — TonesInDeepFreeze

And are you sure that every irrational number is one of the denumerable paths? And that the sequence of nodes of every denumerable path converges to an irrational number? — TonesInDeepFreeze

Aren't there denumerable paths that stay constant on a single rational number? — TonesInDeepFreeze

But 2=1.9. If your method entails that that is not the case, then I doubt that your method actually provides a complete ordered field. — TonesInDeepFreeze

That's a strawman. He didn't say that all discourse has to be at the level of a mathematics journal. — TonesInDeepFreeze

Here's what you need to provide for your SB proposal: rigorous definition...We also don't yet have a rigorous (not just ostensive) definitions of the SB tree, 'R' and 'L'. But I don't doubt that there are ones, though complicated they probably are, so we could at least provisionally work with the ostensive definitions we know. Also, you might want to consider taking reals not as paths but as sequences of nodes on paths. Perhaps it's easier to talk about sequences of nodes rather than sequences of edges, or at least it's more familiar. — TonesInDeepFreeze

Anyway, I am interested in the idea of SB used for defining the reals, as another poster has proposed, but I'd like to see that notion developed beyond mere handwaving. — TonesInDeepFreeze

He may have devised the continued fraction expansion of the equation — jgill

Who are you trying to convince here? Philosophers who consider definitions optional? — jgill

every real has a decimal representation, it is not common to define 'is a real number' that way. — TonesInDeepFreeze

Also, I don't know how easy are the definitions of addition and multiplication compared with the definitions of those operations with equivalence classes of Cauchy sequences or Dedekind cuts. — TonesInDeepFreeze

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

A description is a linguistic object. A description is not a real number. — TonesInDeepFreeze

To answer that question, we would need a mathematical definition of 'describe'.

Meanwhile, GR [Phi], like any real number, is the limit of a Cauchy sequence of rationals. — TonesInDeepFreeze

Real numbers are not sequences. Real numbers are equivalence classes of Cauchy sequences of rationals. And a real number is the limit of a Cauchy sequence of rationals. — TonesInDeepFreeze

So, you can't just magically add Phi as a node to this tree. — TonesInDeepFreeze

We could simplify by taking the complete infinite binary tree....But there is no irrational number represented by a node in that tree. Period. — TonesInDeepFreeze

What is your definition of 'completely described'? — TonesInDeepFreeze

I don't think an infinite series can encode an irrational number while remaining itself rational — Count Timothy von Icarus

I'm saying a decimal number is just an encoding of a pattern/abstraction, it isn't identical with it. — Count Timothy von Icarus

Pi is the ratio of circumference to diameter of a circle. The Golden Ratio can be defined as the ratio of a particular line segment to another - you can look it up on Wikipedia. Other irrationals, have at it. — jgill

Since these expansions are non-ending they do not completely describe the mathematical entities they represent. — jgill

Since these expansions are non-ending they do not completely describe the mathematical entities they represent. — jgill

the Golden Ratio is irrational. As such, how are we going to describe it with the positive rationals? There is no "last" set of turns that can describe the GR, since there will always be more to describe. — Count Timothy von Icarus

However, if we could fully describe one irrational with the rationals than it stands to reason that we should be able to do this with others through a different series of turns. However, that can't be the case given the aforementioned. — Count Timothy von Icarus

It is not an abstract object, but rather a property of abstract objects, ratios being necessarily relational. — Count Timothy von Icarus

The complete series of decimals or turns wouldn't be the ratio though, even if such a series was finitely possible. — Count Timothy von Icarus

How would you use infinite decimal digits to define the GR? — jgill