I don't see a need for disagreement here. You can notate as you wish; and I can say why I also use that notation but like to point out that 'inf' is dispensable when we unpack, which has a pedagogical purpose: Many people, such as in Internet threads, are clueless about axiomatic, rigorous mathematics, so they have the misconception that 'inf' must name a certain entity. So it is instructive to explain that it is merely contextual notation that does not invoke any entities other than real numbers and a function.

Limiting ourselves for the sake of convenience to just Achilles and the tortoise, what exactly is there to debate about? — tim wood

What should be the payoff if you bet 1000 euros on Achilles.

Personally, I don't take the first one as involving the extended reals.

I take 'inf' in

lim[x -> inf] 1/x = 0

as notation that unpacks as:

0 = the limit of the function 1/x where x ranges over the positive natural numbers

So there is no need to involve an object named 'inf'. It's a more parsimonious approach.

I'm not sure offhand how I would unpack lim[x -> 0] 1/x = inf; but my hunch is that I could do it.

Back to matters at hand:

Hopefully you might now at least be beginning to understand the answers to your question:

There is the denumerable sequence that pairs each step in the staircase with a natural number. But there is also a proper superset of that sequence that has the set of natural numbers (w) itself as an argument and you may define whatever you wish to be the value of the sequence at that argument.

For example (starting with 1 instead of 0):

1 maps to 1
2 maps to 1/2
3 maps to 1/4
...
w maps to 0

I take from @fishfry (as an analogy in my thinking) that the extended reals have a point of positive infinity (inf) that is like w in this sense: inf is not itself an integer, but it is an extension that comes after all the integers. And w is not itself a natural number, but it is an ordinal that comes after all the natural numbers.

ave correctly said that the poster is ignorant and confused about mathematics.
— TonesInDeepFreeze — Metaphysician Undercover

The quintessential crank* Metaphysician Undercover misquotes me. I didn't post any sentence that begins with "ave". But this is a true sentence I did post:

After many posts where I made no personal comments, I have correctly said that the poster is ignorant and confused about mathematics. — TonesInDeepFreeze

If I have an "MO" it is providing an ignorant and confused poster with information and explanations (the poster himself this time makes a point of repeatedly saying that the information and explanations have been helpful to him) but also remarking on the emerging pertinent point that the poster persists in ignorance and confusion.

But now the crank, himself ignorant and confused about mathematics, emerges with a false, pointless, substanceless, sophomoric, clueless attempt at wit, including even botching his quote of me. He has nothing to say other than to spit an inane, gratuitous attempt at a snipe. Actually, it's good that he doesn't say anything about the subject of discussion, since virtually everything he spouts about mathematics is irrational disinformation.

* Being a crank includes sophistry and more.

Bull. You spout misinformation. Then you get explanations and ignore them, evidenced by asking questions that were already answered as you just go right ahead to re-spout. And you refuse to grasp the very simple fact that any explanation given you relies on previous concepts so that both the logical way and only downright practical way to know any of this is for you to start at page 1. If not trolling literally, it is trolling de facto. But do carry on ... it is amusing as pathetic comedy.

After many posts where I made no personal comments, I have correctly said that the poster is ignorant and confused about mathematics. And that is not gratuitous, especially as he persists to ignore the information and explanations given him. That is not analogous to your nearly immediately taking umbrage at corrections or suggestions for sharper formulation and then turning that to one of your sophomoric snipes. And, in this case, instead of responding to my substantive points about the terminology.

The formulation you gave doesn't mention that the ordering includes the standard ordering on w; it only mentions that every member of w is greater than -inf and less than inf.
— TonesInDeepFreeze

LOL. Pedants 'Я' us. — fishfry

That is not pedantic. The formuation you gave is literally very incorrect. I simply offered a correct formulation. For that, you reply petulantly (though not pedantically). Maybe, eventually, you'll get past nonsense like that.

Your obsessive pedantry is leading you astry. — fishfry

[emphasis added]

Your penchant for making discussions personal leads you astray.

It's not pedantic but it is pedagogical.

The key idea is sequences. And it is clear and concise to say:

The domain is w+1 = wu{w} (or, taking liberties (w\{0})u{w}).

But to go through cutting the real numbers down to positive integers and then adding points of infinity (moreover raising the pedagogical question of what exactly are "points of infinity") is an unnecessary detour and distracts from the insight of using sequences. Indeed, such things as dividing infinitely in a discussion such as this involves explicitly sequences. So the insight is that we have sequences not just on the ordinal w but also on ordinals such as w+1. Indeed that highlights that the range of such sequences may have other than ordinals (such as natural numbers) but that the domains are ordinals.

You may use the lemniscate however you wish. But I wouldn't use it for w. Because it is a typical that uninformed people (I mean other posters, not you) are prone to use "infinity" as a catch-all for infinite sets. Indeed a point of infinity in the extended reals may be any set whatsoever except a real; doesn't even have to be an infinite set. That is, if we want to instantiate the extended real system, then we only need to choose two sets that are not themselves real numbers, then define the ordering and operations in the usual manner.

But uninformed people don't distinguish between inf ('inf' read as the lemniscate) and the set of natural numbers. They regularly conflate inf ("infinity") with the set of natural numbers, which is an infinite set but not necessarily a point of infinity in the extended reals. So it is good pedagogically to use notation that doesn't lead to that conflation.

On the other hand, making an analogy between w+1 and the positive extended reals is a nice idea. I would state it as only an analogy though.

But of course, you are free to stipulate whatever definitions you like.

You mean the ordering:

L u {<-inf n> | n in w} u {<n inf}> | n in w}, where L is the standard ordering on the natural numbers.

The formulation you gave doesn't mention that the ordering includes the standard ordering on w; it only mentions that every member of w is greater than -inf and less than inf.

It's not a quirk. It's odd terminology, as far as I know; and the context here is not just points at infinity but sequences, and sequences are functions whose domain is an ordinal (or at least a set with a well ordering).

The standard ordering on the real numbers is not a well ordering. So talking about points of infinity as in the domain of a sequence is confusing at best.

The sequence is (I'm taking the liberty of starting at 1 rather than at 0):

{<1 step-1> <2 step-2> ... <w final-state>}

You can call them whatever you want.

(I see that there is a Wikipedia article that does use the terminology though. I don't usually reference the unreliable and haphazardly edited Wikipedia for mathematics.)

But sequences are defined as having ordinals as the domain. That clarifies and can be widely referenced in the literature. And the extended reals of course is also a common notion. But the points of infinity in the extended reals are not ordinarily (if at all) understood as ordinals that are the domains of sequences.

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? — MoK

What's to elaborate? I gave you the definitions of 'the continuum' and 'continuous function'. Meanwhile, a definition of 'a continuum' is not needed in this context, only 'the continuum'. Moreover there are different definitions of 'a continuum' in different contexts and they are way too technical for you. For purposes of the discussion at this juncture, 'the continuum' is sufficient. You ask for elaboration on concepts that depend on previous concepts that you refuse to learn. You won't even find out for yourself what a function is but you want people to explain mathematics to you.

I wouldn't, but suit yourself.

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

Yes exactly. — fishfry

Except that w is not called an "extended natural number".

Not "extended natural number".

Rather the ordinal w+1 = wu{w}. That has been explained to you probably at least five times already.

And there is no "problem of indexing". You're using the word 'indexing' incorrectly, as I've explained to you twice already.

Any set can be indexed by being the range of a function. Indeed, trivially, any set is indexed by being the range of the identity function on the set.

Your question might be whether a certain set can be indexed by w. By definition, any denumerable set can be indexed by w.

But your question might go further: Can a set with an ordering on the set be isomorphic with w and the usual ordering on w. Of course, there are sets and orderings that are not isomorphic with one another. For example, in the current discussion, consider a set ordered

1 2 3 ...

That is isomorphic with the standard ordering on w.

Consider a set ordered

1 2 3 ... x

That is not isomorphic with the standard ordering on w

But it is isomorphic with the standard ordering on w+1.

But these explanations are wasted on you because you won't actually look at a beginning reference on this subject so that your claims and questions aren't based on your confusions and ignorance.

My, argument here was for Dichotomy paradox. — MoK

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

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.

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?

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'.

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.

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.

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.

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?

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".

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?

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.

the Zeno paradox certainly threatens mathematics — MoK

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

I applied mathematics to determine that, in the 100 meter race, Achilles will cross the finish line in 10 seconds and that Tortoise will cross the finish line in 100000 seconds.

Then I called my bookie Zeus "The Moose" to place my 1000 euro bet on Achilles (favored 1000000000 to 1) and turned on the TV to watch the race on MSPN (the Mythic Sports Programming Network). After Achilles won, I called The Moose to collect. He started to say that my payoff is infinitesimal, so there's no way he can pay me; but I corrected him by telling him that there are no infinitesimals in the reals and that though the payoff is small, it is not infinitesimal. So he said he'd that he'd apply the fraction of a cent to my account. Meanwhile, there were a lot of Parmenideans who lost their togas betting on Tortoise.

Seems math did a pretty good job. Maybe math threatens Zeno's paradox.

We say that the set is continuous if there is a point between any arbitrary pair of points. — MoK

Who is "we"? Other than you? What does "between" mean for sets in general? Or do you mean for real intervals? Usually, 'continuous' refers to functions. Perhaps there is an even more general notion of 'continuous sets', but we'd have to see it mathematically defined, in which case it's not going to be "there is a point between any two points".

Why don't you look up some mathematics rather than just making up claims about it?

How about considering the point between two arbitrary points, namely a and b, to be mean, namely (a+b)/2? — MoK

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

please define the continuum for @tim wood in plain English? — MoK

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.

I am not a mathematician.
Proof:

Let Mx <-> x is a mathematician
Let Rx <-> x produces results in mathematics
Let t = TonesInDeepFreeze

1. Ax(Mx -> Rx)
2. Mt -> Rt {1}
3. ~Rt
4. ~Mt {1 3}
QED CIA FBI DHS MLB NBA NFL NBC CBS ABC JFK LBJ FDR ETC

It's fine that you're talking about a question that's different from the one I replied to.

I have a good grasp of the some of the basics of set theory, but I am not very knowledgeable beyond those basics.

Anyway, the idea of someone, who doesn't understand that the set of natural numbers is not a member of itself, trying to grapple with how ultrafilters play into proving the existence of hyperreals is ridiculous.

No one said anything about ZF.

But if it is taken that there only finitely many things in what is designated as 'the real world', and it is regarded that there is no injection of an infinite set into a finite set, then the question is thereby settled, regardless of ZF; also, as far as I can tell, the other poster's call to Zeno's paradox or other supertask paradoxes would be unneeded.

(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.

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

(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.

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.

What 1:1 map are you referring to? A 1:1 map from a real interval into points in space? A 1:1 map from a real interval into points of time? A 1:1 map from a real interval into a set of particles?

I don't have much to say about those vis-a-vis Zeno's paradox. I'm only asking what your argument is that Zeno's paradox entails that there is not a map.

Please please please do not refer to AI bots for math proofs. They are so often incorrect. I've tried it a few times, and the bot gives clearly incorrect arguments - petitio principii - using as an assumption what it claims to be proving.

And looking around at forums is also a disorganized and extremely poor way to learn mathematics.

Get a book if you actually want to understand the material.