Then what term would you use to describe having to pass the 0.5m mark before the 1m mark, the 0.25m mark before the 0.5m mark, the 0.2m mark before the 0.25m mark, and so on? “Consecutive” maybe? Regardless of the term used to describe it, it’s the same logic as having to count to 0.5 before 1, 0.25 before 0.5, 0.2 before 0.25, and so on. It’s a series of events with no start and so cannot be started.

Then what term would you use to describe having to pass the 0.5m mark before the 1m mark, the 0.25m mark before the 0.5m mark, the 0.2m mark before the 0.25m mark, and so on? — Michael

"Ordered." Rational and real numbers can be ordered, just in the way that you describe - indeed, that is how they are usually ordered. But they cannot be put into a sequence in that order.

It’s a series of events with no start and so cannot be started. — Michael

'Series', like 'sequence' is a technical mathematical term. Usually it is used to describe the sequence of partial sums of a sequence, although sometimes it is used just as a synonym for 'sequence'. Either way, as we observed above that since the set of events S is not a sequence, under the natural order, neither is it a series.

All we can say is that it is a totally ordered set. To be a series or sequence, and to have a 'start' it would have to be a 'well-ordered set'.

But totally-ordered sets are not all well-ordered, and sets of the form S are examples of that.

I would say that the object O passes the points in S in order. Intuitively, we feel that that implies that there must be a first passing, a 'start'. But that is only an intuition, not a logical consequence of any of the properties of the objects being considered. Sometimes our intuitions lead us astray, and this is such a case.

That's why the Zeno paradox is veridical rather than falsidical. It conflicts with our intuitions, but not with logic. Hence we have to conclude that, in this case, our intuitions are wrong.

Let me get this straight, make sure I understand.

Zeno's paradox can be stated as follows 'In order to travel a distance of 1 meter, you must first travel 0.5 meters, in order to travel a distance of 0.5 meters, you must travel 0.25 meters. For any distance you can travel, you must first travel half that distance. Thus you cannot travel the distance.'

Fleshing out the paradox entails fleshing out the relationship between the thought exercise of division and the impossibility of travelling the distance. Michael's version seems to be:

(1) If journey did not have a beginning, it could not have occurred.

Your response to this is:

(1A) For the purposes of the paradox, a journey is a sequence of distances which must be travelled. This is $\big[\frac{1}{2n}\big]_{n=\infty\rightarrow 1}$ (abusing notation but I think it makes sense). The beginning of the journey would be the least element of the set - the smallest distance travelled - since this set has no least element in the ordering described, the usual notion that a journey must have a beginning (the journey presumably being the sequence of distances travelled in their usual ordering) is not in play. In effect, this is a confusion of two distinct concepts - the well ordering of travelled locations in typical journeys, and the mere total ordering of journeys constructed through the thought experiment. The same distances are considered, but under different orderings. IE

$\big[\frac{1}{2n}\big]_{n=\infty\rightarrow 1}\neq\big[\frac{1}{2n}\big]_{n=1\rightarrow \infty}$
even though they are equal as raw sets. Equality of sets does not imply equality of ordered sets.

Do you think your response also addresses the case where we replace (1) with (2):

(2) The number of distances travelled is infinite, and we cannot do an infinite task.

?

I baulk at the first sentence of (1A). Describing the journey as a 'sequence' is inaccurate and almost presupposes Zeno's conclusion.
Also - and I'm not sure if this pedantic or crucial - I would say the journey does have a beginning, and it is the spacetime event that is the location A at the time that is the infimum of all times at which the object O is anywhere in Y - {A,B} (recall that Y is the locus of the object O). Similarly, the end of the journey is location B at the supremum of that set of times.

In that sense, the journey has a well-defined beginning.
It is the set of passings of the waypoints in S that has no beginning. But that set of passings is not the whole journey.

the usual notion that a journey must have a beginning (the journey presumably being the sequence of distances travelled in their usual ordering) is not in play. — fdrake

Yes, if 'journey' is used to refer only to the passings of waypoints in S, rather than the usual meaning of the whole path Y, that is central to where Zeno goes wrong.

Do you think your response also addresses the case where we replace (1) with (2):

(2) The number of distances travelled is infinite, and we cannot do an infinite task. — fdrake

I think this has even more problems than (1). The term 'task' is dragged up out of nowhere, with no clear meaning or relation to the problem. Nor is any support provided for the claim that we cannot do an infinite task - a claim that seems very unintuitive to me.

But if motion is continuous then there isn't a first position. — Michael

If space (or the number line) is continuous, and motion is analogously continuous, then there shouldn't be a first position. Our inability to define the value of a first position is what we should expect. This should prove that continuous motion is possible rather than impossible.

That's a lot clearer to me, thanks.

If space (or the number line) is continuous, and motion is analogously continuous, then there shouldn't be a first position. — Luke

Even if you want to talk about the movement from A to B being continuous the half-way point between them is a discrete point that actually exists and passing through it is a discrete event that actually happens.

Does that somehow make motion impossible?

Consider that the ancient greeks identified numbers with visual geometric constructions obtained using a compass and a straight edge.

If our concept of numbers is derived from geometrical intuition, then any attempt to address Zeno's paradox via appeal to the resulting mathematics is equivalent to trying to make sense of the A series of time via the B series.

Not only does this way of thinking make Zeno's paradox even more bewildering and intractable, it isn't acceptable from the point of view of the intuitionist, who identifies the construction of numbers not with sketches on paper, but with the very phenomenal passage of time.

From this perspective, first-position events are constructed by the very act of starting to count. Zeno's paradox can instead be interpreted as a paradox concerning the conceptual issue of what it means to distinguish spatial positions that represent different times, such as the hands of a clock.

Quantified measurements of position information appeal to machinery, and hence to Computable Analysis that recognizes only a countable set of numbers, namely the computable numbers which represent equivalent-classes of Turing computable total functions.

Computable analysis reveals that the ordering of any two computable numbers is decidable provided the numbers are in fact different. But due to the negative result of the Halting Problem, there is no universal algorithmic test for deciding whether two computable numbers are in fact different or equal. Consequently the very notion of either difference or equality with regards to nearby positions or times is not a mathematically meaningful a priori notion for the constructivist.