I don't know. First you would need to define "speed in all infinite steps". — TonesInDeepFreeze

I can define the speed in $i$th step as follows: $v_i=\frac{(\Delta x)_i}{(\Delta t)_i}$ where $(\Delta x)_i$ is the length of $i$th interval and $(\Delta t)_i$ is the time duration it takes the runner (I am referring to Dichotomy paradox) to move $i$th interval. The series however has infinite steps so I cannot define the speed in all infinite steps since $i$ is a natural number.

My point was that he didn't ask for a definition of 'the continuum'. The takeaway for you is to not conflate 'the continuum' with 'continuous'. — TonesInDeepFreeze

Continuum is a continuous series. He understands what continuous is if he understands what continuum is.

I didn't say that he needs to read a book. I said the definition is in chapter 1 of such books. — TonesInDeepFreeze

Thanks. @fishery gave a definition for a continuum from wiki: "Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two distinct elements there is another (and hence infinitely many others), and complete, i.e., which "lacks gaps" in the sense that every nonempty subset with an upper bound has a least upper bound."

The arrow paradox says each is zero, as in time "points". Yet there is still the forward motion of the action, driven by energy — Gregory

I am not sure whether he was familiar with the concept of speed or not. But, the average speed in the interval $i$ can be calculated as $v_i=\frac{(\Delta x)_i}{(\Delta t)_i}$ where the $(\Delta x)_i$ is the length of $i$th interval and $(\Delta t)_i$ is the time duration it takes the arrow to move $i$th interval. So everything is clear for now. The problem is however with the index $i$ which cannot be infinite since it is a natural number yet we know that infinite steps exist.

The speed of Achilles is 10meters/1second. The speed of Tortoise is 1meter/1000seconds. — TonesInDeepFreeze

Could you calculate the speed in all infinite steps?

Correct meaning you understand that the rationals are dense but not continuous? — fishfry

If by dense you mean that there exists a point between two arbitrary points then I understand that applies to the set of rational numbers. The link you provide is technical for me and I have to put more effort into understanding it.

Haven't we been doing that all along? Not sure what you mean. — fishfry

By that, I mean that there exists a point between two arbitrary points in which the between is defined as the geometrical mean.

The set of standard real numbers, as you yourself have defined it since the first post in this thread, when claiming it doesn't exist. I believe you've now come around to accepting that it does exist. So that's the mathematical continuum. The real numbers.

ps -- Technically, what I've described is a linear continuum.

Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two distinct elements there is another (and hence infinitely many others), and complete, i.e., which "lacks gaps" in the sense that every nonempty subset with an upper bound has a least upper bound.
— Wikipedia — fishfry

Ok, that definition seems good and simple for @tim wood. Thanks for providing the definition.

And the Zeno paradox does not threaten mathematics. — jgill

How could you index an infinite set of steps?

I haven't seen a conceptual analysis that concludes it is discrete, but my impression is that it's typically assumed to be continuous. — Relativist

Well, if space is continuous then it means an infinite number of steps exists yet we cannot complete them. The same applies to time in the example of the infinite staircase.

Is it your opinion, as a physicist, that chaotic systems are not (in principle) reducible to deterministic laws of physics? My impression is that the math related to chaotic systems is pertains to identifying functional patterns to make predictions. That, at least, seems to be the nature of weather forecasts - it's not that the movement of air molecules is fundamentally indeterminstic, rather it's that it's that the quantity of data that would be needed to identify the locations and trajectory of each molecule is orders of magnitude too large to be practical to compute. — Relativist

The laws of physics are deterministic but that does not mean that the chaotic behavior does not exist. It means that any error in the calculation of physical variables leads to a significant deviation from what we observe and what the calculation provides. The source of the error in the case of weather forecast is twofold: (1) The error in the estimate of physical variables in the initial point and (2) Using a discrete approach to solve a set of continuous equations.

We've been considering it at least fifty times already in this thread. What about it do you want to say? — TonesInDeepFreeze

I want to say that you could sweep all points of the continuum using that definition.

He didn't ask for a definition of 'the continuum'. 'the continuum' is a noun. He asked for the distinction between 'continuous' and 'discrete'. 'continuous' and 'discrete' are adjectives.

'the continuum' has been defined at least three times already in this thread.

'continuous function' is the defined as usual in chapter 1 of any Calculus 1 textbook.

Other senses of 'continuous' depend on context. And definitions of 'discrete' depend on context. — TonesInDeepFreeze

He asked for a definition of continuous and discrete in plain English. Could you please provide the definition in plain language without referring him to read a Calculus book?

Step (the verb) = the act of setting ones foot onto the next step (the noun; a thing).

The set of actions maps to the set of things.
The stairway consists of the set of steps, which we're stipulating as being infinite. Unlike the staircase, the acts of stepping don't exist (they are actions). — Relativist

Ok, I see what you mean and I agree.

Ok. The surface of a table-top. Discrete or continuous? A sandy beach? Or the surface of a liquid? — tim wood

You are talking about physical objects that have extensions in space so their location is not definable unless you talk about their center of mass. Do you know what the center of mass is? If not think of an ice cube. The center of an ice cube is its center of mass. The center of mass of the ice cube is definable though hence you can define the location of the center of mass of the ice cube. Now, you can move the ice cube along a line. This means that its center of mass moves from one point to another point along the line. So, by now you have a definition of a point, the center of mass of the ice cube, and a line, its motion along the line.

Certainly by your definition the number line continuous, but made up of discrete points - how can that be? — tim wood

Mathematicians work on abstract objects like points and lines all the time. They define a line as a set of dimensionless points and show that things are consistent. Whether these objects are real or not is subject to discussion.

It would seem that "discrete" and "continuous" are abstract convenient fictions their utility depending on usage in context. Thus when misused you might bet on the tortoise, but I'll bet on Achilles every time. — tim wood

Well, the Zeno paradox certainly threatens mathematics, especially the continuum concept. I also bet on Achilles since my common sense tells me he will win.

Many important metaphysical questions have implications for the physical world. Metaphysics tries to figure things out with conceptual analysis (which can include math and logic) and intuition. In this case, it appears the process can't reach a definitive conclusion. — Relativist

What about the conclusion that spacetime is discrete?

But I wonder: is it really hopeless for physics? You said that treating spacetime as discrete would lead to errors if it's actually continuous. Couldn't this be tested? — Relativist

Yes, weather forecast for example. Any chaotic system in general. Even nonchaotic systems show the error in the long term.

I wasn't claiming it disproved the existence of infinitely many stairs, but it proves that an infinite number of steps cannot be completely traversed in a sequence of of steps of finite temporal duration. — Relativist

Correct. So we are on the same page.

This is in spite of the fact that the set of steps (the activity) maps 1:1 to the set of physical steps that comprise the stairway. — Relativist

Isn't the set of steps the set of physical steps? If yes why do you use a one-to-one map?

The more important conclusion is that there's a logical disconnect between this logical mapping and the analogous temporal process; IOW, the mapping doesn't fully describe the temporal process; something is missing - and it would be worthwhile to develop a mathematics that accounted for this. — Relativist

I cannot figure out what you are trying to say here. Do you mind elaborating?

Correct. How about considering the point between two arbitrary points, namely a and b, to be mean, namely (a+b)/2? If not, could you please define the continuum for @tim wood in plain English?

Although it's a metaphysical question, it pertains to the physical world. — Relativist

Yes, it has an implication. I think it means that spacetime is discrete.

There are an infinite number of steps but we cannot complete them. The fact that we cannot complete the steps does not mean that they do not exist.

Consider a set of points. We say that the set is continuous if there is a point between any arbitrary pair of points. We say that the set is discrete if there is a minimal distance between a pair of points. In other words, there is no point between such a pair. The example of a continuum is the real numbers and the example of a discrete is the natural numbers (the minimal distance between a pair of points is 1).

OK, so we risk introducing error if we treat spacetime as discrete, — Relativist

Yes, if the spacetime is continuous and we treat it as discrete then we are introducing error.

but if it IS discrete, we introduce no errors by treating it (mathematically) as continuous. — Relativist

If spacetime is discrete we introduce error by treating it as continuous. We however might not be able to observe the error if our measurement devices are not precise enough.

So treat it as continuous and use the math. Problem solved, right? — Relativist

Yes, we can use a continuous model as far as our measurement devices are not precise enough. Otherwise, we have to use a discrete model.

Why do we have to deal with Zeno's paradox? Is there some problem in physics where it makes a difference, or are you like the rest of us navel-gazers around here - and just curious the logical implications? — Relativist

Zeno paradox is a metaphysical problem rather than a physical one. It tells us something about reality without a need for any measurements.

The infinite series entails an unending series of steps. So a final step is logically impossible. — Relativist

That is not a solution but the point of Zeno. If the final step is logically impossible then you cannot complete an infinite series of finite steps therefore you cannot finish the task.

Suppose spacetime is continuous. We still can't distinguish spatial measurements that differ less than a planck length, nor time measurements less than a planck time. — Relativist

I think if spacetime is discrete and our capacity to measure spacetime interval is much higher than Planck length and time then we can treat spacetime continuously, hence we can use the continuous physical models that describe reality well. We however still have to deal with Zeno's and infinite staircases paradoxes.

This suggests to me that we will make no errors by treating space and time as discrete, even if it is continuous. What's your thoughts? — Relativist

If spacetime is continuous then we are dealing with an error in treating space and time as discrete. There are numerical methods that allow us to minimize the error but at the end of the day, we cannot avoid the error at all. In most cases, we are safe if we discretize space and use good numerical methods. In the case of time, however, the error accumulates over time so we can find significant errors in our calculation in the long term. This error in the predicted variables can be catastrophic over time if the system is chaotic, such as the weather processes.

I agree with Sime, and I also gave a solution in that thread that is similar to his. — Relativist

What is your solution to the paradox? Could you explain @sime's solution to me?

More or less. Both demonstrate the fact that limits don't correspond to the completion of an infinite series of finite steps. — Relativist

Exactly right! So what are the options in this situation: (1) Spacetime is discrete or (2) Spacetime is continuous. In the first case, we don't have the problem of infinite division so there are no conceptual problems or paradoxes such as the one of Zeno. The rule of mathematics, Leibniz's calculus is to help us easily calculate things, such as differential and integral, so it is just a useful tool. In the second case, we however need a mathematical formulation that allows us to directly deal with infinity, for example, we should be able to set n equal to infinity, if not we cannot complete an infinite series, so we are dealing with the paradoxes such as the Zeno's or infinite staircases. In simple words, we cannot move and time cannot pass. So let's wait for mathematicians to see if they have a solution for the second case. If there is no solution for (2) then we are left to (1)!

Thank you very much for your post. Unfortunately, my country is under sanction and I cannot purchase any book from Amazon. I am retired and living in the countryside so I don't have access to any library as well. So, I am out of luck when it comes to reading books unless I find a PDF file online.

Yes, there is. But if you want to discuss it, use that thread and tag me. — Relativist

That paradox is nothing more than Zeno's paradox. It simply replaces the distance in Zeno's paradox by time. There is however a problem when you want to discuss Zeno's paradox by standard analysis. To discuss this further, let's consider the following sequence:
f(1)=1/2
f(2)=f(1)/2
...
f(n+1)=f(n)/2
...
where n is a natural number. This sequence is nothing but the sequence that was first mentioned by Zeno. The sum of the sequence then can be found and it is s(n)=1-1/2^n. Although we can use standard analysis to calculate the limit of s(n) when n tends to infinity we are not allowed to consider n to be infinity. That is because n is a member of the natural numbers and all members of natural numbers are finite. @sime in this post discusses the paradox and it seems that he resolved it. It seems to me that he uses nonstandard analysis to resolve the paradox. I am however pretty ignorant of nonstandard analysis so I cannot tell how he resolves the paradox. To my understanding, motion is real and Zeno's paradox is invalid. I however don't have the mathematical tools to discuss Zeno's paradox since my knowledge of analysis is limited to standard analysis. Perhaps, other mathematicians @sime, @fishfry, and @TonesInDeepFreeze can offer us a solution to the problem.

There is no paradox there. The total time which is needed to get to the top is 60 seconds. You might find this link useful.

(1) In open forums like this, there is usually more disinformation and confusion about mathematics than there is information and clarity. Instead, prolific cranks dominate, or discussions center on a few reasonable people trying to get a prolific crank to come to the table of reason. — TonesInDeepFreeze

You can find all sorts of people in any forum. I agree that the number of knowledgeable individuals may vary from one forum to another.

(2) There are no set theory experts in this thread (or, to my knowledge, posting in this forum). — TonesInDeepFreeze

That is all right. You are enough good to teach me a few things in set theory.

(3) Picking up bits and pieces of mathematics, hodge podge, is not an effective, not even a coherent, way to understand concepts that are built from starting assumptions and definitions. This thread itself is evidence of that. — TonesInDeepFreeze

Quite oppositely I learned a few things in this thread. Thank you very much for your time and patience.

That you have limited time for mathematics is all the more reason for not wasting that limited time in routes that lead to dead ends, misinformation and confusion. — TonesInDeepFreeze

I have to disagree. I have a wide range of interests. One of the main reasons that I signed up in this forum was the very good quality of knowledge of posters in this forum such as you. I am an expert in a few fields as well, such as physics, epidemiology, philosophy of mind, and the like. The idea is to share the knowledge that one accumulated over decades with others through discussion in the forums so that all individuals can benefit from it. In this way, one can save lots of time in understanding a topic through discussion with experts and decide where to focus on a topic and how to manage the valuable time.

Ok, I will try to get the book you suggested in this thread.

If your question is whether spacetime is continuous then I have to say that there is an ongoing debate on the topic. I am not an expert on the topics of loop quantum gravity and string theory. I searched on the net a while ago and I found this manuscript which you might like to read. The manuscript is however old, 2003, so it does not reflect the current state of debate.

Thanks for the writing. I will see if I can find any time in the future to study the filter. I have other interests as well rather than mathematics.

Ok, thanks for the clarification.

Thanks for writing. On my reading schedule.

Again, thank you very much for providing the argument and extensive writing. I will read through the argument and try to understand it. I will look for proof of the theorems on the net by consulting ChatBot or reading through books or math forums.

I didn't ask for a proof that shows that aleph_0 is the least infinity but to show that X is the least infinity namely aleph_0.

A mistake on my part and I am sorry for that. I should have written: "To avoid confusion, assume that the cardinality of the set of natural (I wrote real instead of natural) numbers is X. How could one show that X is the least infinity namely aleph_0?"

Ok, that I agree. How about whether the real number represents reality or not? Do you know any references that claim that reality is not continuous? I am currently reading a manuscript about loop quantum gravity and discrete time. I am a condensed matter physicist but I studied particle physics and cosmology to a good depth. I am somehow ignorant of loop quantum gravity and string theory though.

I am aware of that. To avoid confusion, assume that the cardinality of the set of real numbers is X. How could one show that X is the least infinity namely aleph_0?

Are you suggesting this proves real numbers are logically impossible, or are you arguing that there is no valid 1:1 mapping from the set of real numbers to the actual world? — Relativist

I directly attacked the continuum in the mathematical sense. The discussion is ongoing but it seems that the classical continuum exists but suffers from problems which are discussed here. The post is very technical and I have problems understanding it though.

I ask, because it fails to do the former. — Relativist

Do you mean later (instead of former)? If yes, it would be nice of you to elaborate.

Please accept my sincere apology for not replying to your post earlier. I read your post a few times in the past and had a difficult time understanding it. Unfortunately, I forgot to respond to your post. I recalled forgetting to reply to this post when I saw your second post in my thread. I read this post more than ten times trying to understand what you are trying to say but I have to say I am still ignorant in many parts. Please find my reply in the following.

Formally, the classical continuum "exists" in the sense that that it is possible to axiomatically define connected and compact sets of dimensionless points that possesses a model that is unique up to isomorphism thanks to the categoricity of second order logic. — sime

So classical continuum exists. I had to read about second-order logic, first-order logic, and zeroth-order logic trying to make sense of what you are trying to say here. Unfortunately, I don't understand what you are trying to say in the bolded part, the last part of the paragraph. Do you mind elaborating?

But the definition isn't constructive and is extensionally unintelligible for some of the reasons you pointed out in the OP. — sime

Do you mind elaborating on what problem you specifically have in your mind?

Notably, Dedekind didn't believe in the reality of cuts of the continuum at irrational numbers and only in the completeness of the uninterpreted formal definition of a cut. — sime

Do you by cut mean the exact position of an irrational number for example? What do you mean by the bolded part?

Furthermore, Weyl, Brouwer, Poincare and Peirce all objected to discrete conceptions of the continuum that attempted to derive continuity from discreteness. For those mathematicians and philosophers, the meaning of "continuum" cannot be represented by the modern definition that is in terms of connected and compact sets of dimensionless points. — sime

Ok, thanks for the reference.

E.g, Peirce thought that there shouldn't be an upper bound on the number of points that a continuum can be said to divide into, — sime

My understanding is that there is no upper bound on the number of points that a continuum can be divided into. I however don't understand whether he agrees with the classical notion of continuum or not. If not, what is his point?

whereas for Brouwer the continuum referred not to a set of ideal points, but to a linearly ordered set of potentially infinite but empirically meaningful choice sequences that can never be finished. — sime

Do you mind discussing his results further?

The classical continuum is unredeemable, in that weakening the definition of the reals to allow infinitesimals by removing the second-order least-upper bound principle, does not help if the underlying first-order logic remains classical, since it leads to the same paradoxes of continuity appearing at the level of infinitesimals, resulting in the need for infinitesimal infinitesimals and so on, ad infinitum.... whatever model of the axioms is chosen. — sime

So, one cannot define infinitesimal in the classical continuum. Is that what you are trying to say?

Alternatively, allowing points to have positions that are undecidable, resolves, or rather dissolves, the problem of 'gaps' existing between dimensionless points, in that it is no longer generally the case that points are either separated or not separated, meaning that most of the constructively valid cuts of the continuum occur at imprecise locations for which meta-mathematical extensional antimonies cannot be derived. — sime

What do you mean by antimony here?

Nevertheless this constructively valid subset of the classical continuum remains extensionally uninterpretable, for when cut at any location with a decidable value, we still end up with a standard Dedekind Cut such as (-Inf,0) | [0,Inf) , in which all and only the real numbers less than 0 belong to the left fragment, and with all and only the real numbers equal or greater than 0 belonging to the right fragment, which illustrates that a decidable cut isn't located at any real valued position on the continuum. Ultimately it is this inability of the classical continuum to represent the location of a decidable cut, that is referred to when saying that the volume of a point has "Lebesgue measure zero". And so it is tempting to introduce infinitesimals so that points can have infinitesimal non-zero volume, with their associated cuts located infinitesimally close to the location of a real number. — sime

So you are trying to say that introducing infinitesimal can resolve the problem of cut which is problematic for classical continuum.

The cheapest way to allow new locations for cuts is to axiomatize a new infinitesimal directly, that is defined to be non-zero but smaller in magnitude than every real number and whose square equals 0, as is done in smooth infinitesimal analysis, whose resulting continuum behaves much nicer than the classical continuum for purposes of analysis, even if the infinitesimal isn't extensionally meaningful. The resulting smooth continuum at least enforces that every function and its derivatives at every order is continuous, meaning that the continuum is geometrically much better behaved than the classical continuum that allows pathological functions on its domain that are discontinuous, as well as being geometrically better behaved than Brouwer's intuitionistic continuum that in any case is only supposed to be a model of temporal intuition rather than of spatial intuition, which only enforces functions to have uniform continuity. — sime

What do you mean by temporal and spatial intuition here?

The most straightforward way of getting an extensionally meaningful continuum such as a one dimensional line, is to define it directly in terms of a point-free topology, in an analogous manner to Dedekind's approach, but without demanding that it has enough cuts to be a model of the classical continuum. — sime

I am reading about point-free topology right now so I will comment on this part when I figure out what you mean with point-free topology.

E.g, one can simply define a "line" as referring to a filter, so as to ensure that a line can never be divided an absolutely infinite number of times into lines of zero length, and conversely, one can define a collection of "points" as referring to an ideal, so as to ensure that a union of points can never be grown for an absolutely infinite amount of time into having a volume equaling that of the smallest line. This way, lines and points can be kept apart without either being definable in terms of the other, so that one never arrives at the antimonies you raised above. — sime

I did read your links partially but I couldn't figure out what filter and ideal are. I don't understand how they resolve the problem of the classical continuum too but I buy your words on it.

The set of natural numbers is aleph_0. — TonesInDeepFreeze

What do you mean by this? Do you mean that the set of natural numbers is the set of aleph_0? aleph_0 is a number. How could you treat it as a set?

Every set is one-to-one with itself. So the set of natural numbers is one-to-one with alelph_0, so the cardinality of the set of natural numbers is aleph_0. — TonesInDeepFreeze

I don't understand this argument. How could aleph_0 be a number and a set at the same time?