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.

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?

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.

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.

And the Zeno paradox does not threaten mathematics. — jgill

How could you index an infinite set of 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.

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

Could you calculate the speed in all infinite steps?

Could you calculate the speed in all infinite steps — MoK

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

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? — MoK

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'.

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

'continuous function' is a mathematical notion, and best understood in its mathematical formulation, which is not complicated. But for informal explanations, one can do an Internet search on 'continuous function'. Such explanations include such mentions even as illustrative as "can draw the graph without lifting your pencil", which takes quite a bit of liberty from rigor but at least gives one a kind of mental picture.

I mentioned that 'discrete' depends on context.

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

What definition of what? And what does "sweep" mean?

Could you calculate the speed in all infinite steps? — MoK

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

How could you index an infinite set of steps? — MoK

What does that mean? Ordinarily, "to index" means to make a set the range of a function, as the domain is the index set. The domain is the indexing set and the range is the indexed set.

If there are denumerably many steps, then the steps may be indexed by the set of natural numbers. The set of natural numbers is the index set and the set of steps is the indexed set.

What is the point of your question?

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

(x+y)/2 is the arithmetical mean of {x y}, not the geometrical mean.

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. — MoK

You're still conflating 'continuum' with 'continuous'. They are closely related concepts, but not the same concept. Also, the least upper bound property of the continuum already had been mentioned several times in this thread, so you had that information all along anyway.

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 arrow paradox is that the arrow does not move but that it moves.

/

Average speed is distance/time. In Zenos's paradox, both are finite.

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."

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.

What does that mean? — TonesInDeepFreeze

I mean you cannot give indexes to all members of an infinite series.

(x+y)/2 is the arithmetical mean of {x y}, not the geometrical mean. — TonesInDeepFreeze

Thanks for the correction.

The arrow paradox is that the arrow does not move but that it moves.

/

Average speed is distance/time. In Zenos's paradox, both are finite. — TonesInDeepFreeze

Please accept my apology. My, argument here was for Dichotomy paradox. You need to replace the arrow in that post with the runner, Atalanta.

I mean you cannot give indexes to all members of an infinite series. — MoK

1, 2, 3, 4, 5, 6, ...

Is that not an infinite sequence? (You mean sequence. A series is a sum)

It it not indexed by the natural numbers?

Take the sequence 1/2, 1/4, 1/8, 1/16, ...

That's an infinite sequence. It's also indexed by the natural numbers 1, 2, 3, ...

In fact every infinite sequence is indexed by the natural numbers, by definition.

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. — MoK

I agree that the Wiki article could be more clear.

The point is that we say that a linearly ordered set is dense if between any two elements, there is a third strictly between those two.

Another definition is that between any two elements are are infinitely many distinct elements between the two.

These two definitions are equivalent. The argument is exactly the one that you originally gave: that you just keep taking midpoints.

1, 2, 3, 4, 5, 6, ...

Is that not an infinite sequence? — fishfry

That is an infinite sequence. I am however interested in the sequence first mentioned by Zeno in Dichotomy Paradox in which the infinite member exists. Each member of the above sequence is finite, so you cannot use the above sequence to give indexes to all members of the sequence in Dichotomy Paradox since the infinite member exists.

You mean sequence. A series is a sum. — fishfry

Thanks for the correction.

That is an infinite sequence. I am however interested in the sequence first mentioned by Zeno in Dichotomy Paradox in which the infinite member exists. Each member of the above sequence is finite, so you cannot use the above sequence to give indexes to all members of the sequence in Dichotomy Paradox since the infinite member exists. — MoK

Do you know what a limit is? The sequence 1/2, 1/4, 1/8, ... has 0 as a limit.

If you read through the supertask thread that's been referenced elsewhere in this thread, I explained that you can view the index of the limit as the ordinal number $\omega$ or as a hypothetical "point at infinity," just as plus/minus infinity are hypothetical points at each end of the real number line in the extended real numbers.

https://en.wikipedia.org/wiki/Extended_real_number_line

Continuum is a continuous series — MoK

Wrong. A series is a certain kind of function. The continuum is not a function.

He understands what continuous is if he understands what continuum is. — MoK

Wrong. I explained the difference between them. Knowing the definition of 'the continuum' does not provide knowing the definition of 'continuous'.

you cannot give indexes to all members of an infinite series. — MoK

Wrong. A series is a certain kind of function. Since it is a function, the range of the function is indexed by the domain of the function.

Again, you're using mathematical terminology without a clue as to what it means. But that's okay. After, all, what is an Internet forum such as this good for if not to provide a platform for people who don't know what they're talking about to prolifically shoot their mouth off about it anyway?

1, 2, 3, 4, 5, 6, ...

Is that not an infinite sequence?
— fishfry
That is an infinite sequence. I am however interested in the sequence first mentioned by Zeno in Dichotomy Paradox in which the infinite member exists. Each member of the above sequence is finite, so you cannot use the above sequence to give indexes to all members of the sequence in Dichotomy Paradox since the infinite member exists. — MoK

You're very confused and resistant to the explanations given you to cure your chronic confusion.

My, argument here was for Dichotomy paradox. — MoK

Dichotomy schmicotomy. You mentioned 'average speed' and I gave you the formula.

Thanks. So you simply extend the natural number to the extended natural number and resolve the problem of indexing.

Wrong. I explained the difference between them. Knowing the definition of 'the continuum' does not provide knowing the definition of 'continuous'. — TonesInDeepFreeze

Do you mind elaborating?