The PSA statement (that there is a step that reaches the goal) directly violates the premise that any given step gets only halfway to the goal. — noAxioms

Or the PSA is correct, and the goal can't be met.

Relativist: "Simply denying a final step is necessary doesn't make it so."
Simply asserting that such a step is necessary doesn't make it so — noAxioms

I'm not merely asserting it. You have to agree that a final step is necessary for completion when there are finitely many steps. Why would it matter if the number of steps is infinite?

Most importantly: What does it even mean for a kinematic process to be infinite? My answer: it means the process continues forever and does not end. What's your answer?

Relativist: "you have to explain why it's not necessary for a kinetic task to require a final step in order to be completed."
I don't know how the task being 'kinetic' changes the argument. — noAxioms

Here's how: the infinity is manifested as a never-ending kinetic process.

Points on a number line exist concurrently (in effect). Steps in a kinetic process do not: they occur sequentially, separated by durations of time.

I'm going to defer commenting on the Achilles/tortoise problem. It just clouds the issue with the stairway supertask.

Doing successive steps does not get you past the tortoise unless the passing of the tortoise is done by one of the steps. That's the same as suggesting a final step, which suggests that infinity is a number. — noAxioms

Yes, the PSA entails taking a final step. We agree infinity is not a number, so there is no final step.

I cannot buy into that PSA statement.

Show the PSA is false.

But I'm making the stronger claim that it is logically impossible.
— Relativist
I'm trying to get a justification of that claim without the addition of the necessity of a final step, which would by definition be contradictory. — noAxioms

Why? The claim is indeed justified by the necessity of a final step for completion. Simply denying a final step is necessary doesn't make it so - you have to explain why it's not necessary for a kinetic task to require a final step in order to be completed.

In the case of Achilles, we know that the task can be completed, but it is presented to us in a form in which it cannot be completed. I mean that we know that Achilles will pass the tortoise and even calculate when with simple arithmetic (no infinities required). — Ludwig V

It depends on how the race is framed. It CAN be described as a supertask, wherein Achilles runs to a series of destinations, each established by where the tortoise is located when he begins each leg of the race. In that case, Achilles never actually reaches the turtle, he just gets increasingly closer. If you frame it in terms of constant speeds by both, then it's not a supertask - it's a different kind of puzzle.

Countably infinite means that any step can be assigned a number. It does not in any way mean that there is a meaningful count of steps. — noAxioms

We can assign those numbers as we take each step. That's counting, and it's perfectly meaningful.

Perhaps you mean there's no way to say we can meaningfully complete the counting of all the steps. That's true, but it seems to be contradicted by the fact that the infinite process completes before the 1 minute mark. .

Physical (fixed size) stairs are of infinite length, and such a distance cannot be traversed in finite time. If the stairs get smaller as we go, then we get into the physical problem of matter being discreet, not continuous. Hence the steps have a minimum size. That's what I mean about physical stairs not qualifying as a supertask. — noAxioms

OK, but speed of light limitations put a physical limit on how fast the stairs can be descended, so that it eventually becomes physically impossible to descend a step in the prescribed period of time. The minimum size limitation also relates to a physical impossibility. But I'm making the stronger claim that it is logically impossible.

Relativist:"The mathematical series completes, but this is an abstract, mathematical completion. The kinetic activity of descending the stairs does not complete."
Again, the stairs is utterly abstract. There's no kinematics to it. — noAxioms

The entire exercise is abstract, but the scenario is written in terms of the kinematic (not abstract) process of descending stairs: each step is a motion, taking place in a finite amount of time.

PSA:The performance of a sequence of successive acts does not complete a particular task unless it is completed by the performance of one of the acts in the sequence.

I cannot parse this. What is an 'act' that is distinct from a 'task'? The word 'sequence' seems to refer to the entire collection.
A 'task' (what, one of the steps??) is not completed by a performance unless 'it' (what, the performance?, the task?) is completed I cannot follow it at all. — noAxioms

Taking a single step is an act. The acts are performed in a sequence (from step n to step n+1). The term (sequence) is not referring to the entire collection. The task is to reach the bottom of the stairs (as stated in the description in the first post of this thread). Perhaps you can already see that it's trivial: it's actually impossible to reach the bottom of the stairs, since there is no bottom to a staircase with infinitely many stairs.

Am I right to think that you are not saying that all the stairs can be counted, even though any stair could be included in a counting sequence? — Ludwig V

Correct.

That's true. What puzzles me is why they are not dismissed out of hand. — Ludwig V

I think it's because they are interesting puzzles, and because they help teach certain concepts.

Wouldn't it be more accurate to say that descriptions of the supertasks are the source of the illusion that there could be a mapping of that mathematical series into the actual kinematic world? — Ludwig V

Yes- that's a better way to describe it.

More than that, surely, there can be a mapping of some mathematical series into the actual kinematic world. Perhaps some similarity between those series is what creates the illusion?

The allure of supertasks is the illusion of being able to complete an infinite process in a finite amount of time. I'm not sure there's anything comparable.

It's more of a description than you've given in this thread. Do you, or don't you, depend on the assumption that "purpose" exists?

This is a thread about the religion Flawlessism. If you actually knew that religion you would know that your argument has no basis because of what I'm referencing to. If you don't know what Flawlessism is then don't comment. — Echogem222

You defined Flawlessism (over here) as a "philosophical religion rooted in the belief that life holds a perfectly good and meaningful purpose."

I guess you would say that "having a meaningful purpose" makes it applicable to us (it's we who have a purpose, apparently), however the mere fact that it would be applicable doesn't establish a purpose as having actual existence.

You go on to say, "Flawlessism encourages rational thinking and critical inquiry. We believe that by seeking wisdom, examining our experiences, and embracing educated critical thinking, we can better understand the nature of the Flawless Good and its implications in our lives."
I have rationally concluded that purpose is not an existent, nor is it a property of any existents. Rather, it is a personification of an intellectional/emotional motivation to achieve something.

I had not mentioned a completion of a count. The supertask is to complete all steps, not to count them, and not to complete a specific step that is nonexistent. — noAxioms

My point is that the stairs are countably infinite. Consequently, they COULD be counted, if we were traversing them.

The series (say the time needed to complete all tasks) converges. The count does not.

Yes, the sequence of defined temporal points (1/2, 1/4, 1/8...) is a series, but the mathematics that identifies the limit does not take into account the kinematics of the task. Supertasks describe a conceptual mapping of the abstract mathematical series into the actual, kinematic world - regardless of whether or not you wished to consider it.

The physical process of descending stairs is not a supertask. — noAxioms

It fits this definition:
"a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time."

Cheap example: You have a bag with a modest quantity of red, blue and yellow marbles in it. The goal is to remove them all. The task is deemed to be complete when the green marble is removed. Such a task cannot be completed by that definition of complete. — noAxioms

The goal of removing all the marbles will therefore never be met if there are at least 2 green marbles, and it will rarely met even if there is only 1. How does this relate to a supertask that allegedly completes?

I notice the SEP article correctly doesn't claim that the last step is taken. — noAxioms

The article discusses the issue:

Max Black (1950) argued that it is nevertheless impossible to complete the Zeno task, since there is no final step in the infinite sequence...
... there is a sense in which this objection equivocates on two different meanings of the word “complete.” On the one hand “complete” can refer to the execution of a final action. ... The two meanings for the word “complete” happen to be equivalent for finite tasks, where most of our intuitions about tasks are developed. But they are not equivalent when it comes to supertasks.

The mathematical series completes, but this is an abstract, mathematical completion. The kinetic activity of descending the stairs does not complete. The SEP article leaves it there, but the implication seems clear: the abstract mathematics does not fully account for the kinetic activity.

Here's a paper in which a philosopher proves it to be impossible to complete infinitely many tasks in a finite time based on the "Principle of Sequential Acts":

PSA:
The performance of a sequence of successive acts does not complete a particular task unless it is completed by the performance of one of the acts in the sequence.

The author argues that those who argue the task completes implicitly deny the PSA, without considering it, and therefore not refuting it. That's what I see going on with the posters who focus only on the mathematical series.

Relativist: "Your preferred perspective ignores this - or pretends there can't be a final step because that introduces a contradiction."
There being a final step leads directly to contradiction, and you say I'm copping out by pretending there isn't a final step? — noAxioms

Yes, it's a cop-out because it ignores the kinematic process. Stating this in terms of the PSA gives you something specific to address, if you want to not cop out.

If your sole purpose was to discuss the math associated with the limit of a series, you'd have been better off avoiding putting it in terms of a supertask.

Relativist: "For the scenario to be coherent, BOTH view of completeness have to be true."
I cannot accept this assertion. I cannot accept a view of completeness that treats infinity as a specific number. — noAxioms

I agree we can't treat infinity as a number, and haven't suggested you should. But for the supertask to be meaningful, you have to identify where infinity fits in the kinetic task description. I'm saying it entails a never-ending sequence of tasks. Identifying the limit doesn't make this disappear.

I'll add that supertask scenarios actually are NOT coherent- because they entail a contradiction. You seem to be embracing the completeness of the mathematical series, then concluding that there can't be a last step because that would entail a contradiction. So look at it this way:
1) the completeness of the series does not demonstrate an analogous supertask is possible.
2) If there is no last step (or if the process is not consistent with the PSA), then the kinetic process (which is a supertask) is logically impossible.

Very cool! I took only one class in Real Analysis (decades ago), and this is an interesting extension that I was unware of. Gracias!

There are some number systems that define division by zero as a/0=∞. — Michael

I think you're referring to the limit:

Limit (a/n) = ∞
n->0

That's not actually dividing by zero. Here's an article that explains various problems with dividing by zero: https://www.mathsisfun.com/numbers/dividing-by-zero.html

Rather, opinions are propositions that are not truth-functional. — Lionino

Please clarify what you mean. Are opinions not beliefs?

How do we decide what is fact and what is opinion? — Truth Seeker

An opinion is a belief, and let's only consider propositional beliefs. A fact is a true proposition. So your question boils down to: how do we decide what beliefs are true? Here's how: by applying valid epistemological methods, we improve our odds of holding true beliefs. That's as good as it gets.

Things don't pop up for no reason, in fact, that is an assertion that implies a cause(in this case, 'no reason'). Given this, it is wiser to assert that the universe came into existence by some manifestation in, per se, a multiverse, than it is to park randomly on the conjecture it just popped up for no reason — Barkon

The premise that the universe "popped into" existence is incoherent. It implies there existed something, into which the universe popped.

The "universe" is best defined as the entirety of material reality. The universe may very well be finite to the past. If so, this entails an initial state; there can have existed no prior state of its non-existence.

the process of counting steps is not completable
— Relativist
Are you suggesting that supertasks cannot be completed? — keystone

Yes- and that's because the role of infinity in the task. The task entails a sequence of events, so the infinity can only mean an infinite chain of events - one after another without end.

if a physical process ends, there has to be a final step.
— Relativist
This is equivalent to asserting that 'infinity' is the largest integer. — noAxioms

Wrong. The statement (the completion of a consecutive series of physical steps entails a final step) is necessarily true. When we consider this statement in conjunction with a statement about the series being "complete" (in terms of convergence) we introduce a contradiction. This is the point! These statements cannot both be true, but both are entailed by the scenario.

But as Thomson (1954) and Earman and Norton (1996) have pointed out, there is a sense in which this objection equivocates on two different meanings of the word “complete.” On the one hand “complete” can refer to the execution of a final action. This sense of completion does not occur in Zeno’s Dichotomy, since for every step in the task there is another step that happens later. On the other hand, “complete” can refer to carrying out every step in the task, which certainly does occur in Zeno’s Dichotomy."
The definition you appear to be using is the former, which is why Michael's one-digit counter doesn't have a defined output after the minute expires. — noAxioms

The SEP article says:
"Although it has infinitely many terms, this sum is a geometric series that converges to 1 in the standard topology of the real numbers. A discussion of the philosophy underpinning this fact can be found in Salmon (1998), and the mathematics of convergence in any real analysis textbook that deals with infinite series. From this perspective, Achilles actually does complete all of the supertask steps in the limit as the number of steps goes to infinity. One might only doubt whether or not the standard topology of the real numbers provides the appropriate notion of convergence in this supertask. "
Indeed, I'm denying that the topology of the real numbers applies to the execution of the supertask itself - although I agree it applies to the series.

As I noted above, a physical, step-counting process that completes must entail a final step. Your preferred perspective ignores this - or pretends there can't be a final step because that introduces a contradiction. That seems a cop-out. The point of the thought experiment is to highlight the contradiction.

I've been using Zeno's definition of complete: That every step has been taken. Given that definition, the supertask can be completed. — noAxioms

I agree with this, but this simply ignores the implication of the physical process of step-counting. For the scenario to be coherent, BOTH view of completeness have to be true. But they aren't - so the scenario is actually incoherent.

As I have been explaining in this thread, you can conceptually adjoin the limit of a sequence to the sequence, as in 1/2, 3/4, 7/8, ..., 1. This is a perfectly valid mathematical idea. This is a representation of the ordinal ω+1

+
1
. In this case, 1 is indeed the "last term," although to be fair, you can no longer call this a sequence, since a sequence by definition is order-isomorphic to the natural numbers. — fishfry

Right! It's not the sequence described in the scenario! There is a background temporal sequence, as the clock ticks, that reaches 1, but we aren't mapping the step counting to the ticks of the clock. The step-counting sequence occurs only at points of time <1. In real analysis, this is called a "right open interval" (i.e.it's open on the right= the endpoint is not included in the interval). 1 is the endpoint, but not included within this interval.

By definition, a limit is not reached, it is approached.
— Relativist

That is sadly a misunderstanding very common among calculus students. So lot of smart people, physicists and engineers and other scientists, have this belief.

In fact a limit IS reached. A limit is exact, it's not merely approached or approximated. It is literally reached.

It's not reached by a single step. Rather, it's reached by the limiting process itself. — fishfry

The limit of the series is "reached" only in the sense that we can reach a mathematical answer. The physical process of sequentially counting steps, doesn't "reach" anything other than increasingly higher natural numbers. Deriving the limit just means we've identified where the sequential process leads. In this case, we've derived that the limit is infinity- but what does infinity correspond to in the scenario? The meaning is entailed by the fact there are infinitely many natural numbers, so it means the process continues without end. It can mean nothing else.

The lesson is that the defined supertask (the fictional, physical process) is logically impossible,
— Relativist

The lamp and staircase scenarios are physically impossible. What law of logic makes them logically impossible? — fishfry

The law of non-contradiction. An infinite series of processes entails never completing, but at points of time that occur after the delinieated interval - the task is necessarily completed.

the process of counting steps is not completable
— Relativist
Are you suggesting that supertasks cannot be completed? — keystone

I'm asserting that an infinite process is necessarily never completed - by definition.

There is no physical process. — fishfry

The scenario describes a fictional, physical process. The lesson is that the defined supertask (the fictional, physical process) is logically impossible, but this isn't apparrent when considering only the mathematical mapping.

Certainly the relationship between time (independent of human control) and physical steps taken over a period of time has ended. — jgill

That's because the physical steps map to an infinite series in an interval with an open boundary. One can't simply declare there's no final step because the mapping implies there isn't. The taking of steps is a repetitive physical process, and if a physical process ends, there has to be a final step.

We of course have the ability to develop our skills of thinking things through, analyzing our opinions and assumptions, and considering other perspectives. But there is a difference between ensuring what you say is correct, and how you conduct yourself in and after saying it. So to say you should “not trust your mind” (yourself)—as I, and Emerson, argue against above—is perhaps different than saying you should not trust the opinions you have or inherited. — Antony Nickles

Consider a devotee of Infowars, who routinely accepts conspiracy theories. Aren't you suggesting they should trust their opinions?

You mention the role of one's conduct, so are you suggesting that the conspiracy theorist just needs to conduct himself in a certain way? Is the right conduct going to lead to him correcting his errors, or are you just suggesting he ought to be polite about his irrational beliefs?

Mathematically, this sequence as a limit of 1.

The sequence never "reaches" 1; nor is there a last step. Neither of these statements is controversial once you understand what a limit is. Sadly, most people have never taken calculus; and most students who take calculus never really learn what a limit is — fishfry

I've taken calculus and I understand what limits are. By definition, a limit is not reached, it is approached. The sequence of steps maps to a mathematical series that approaches, but never reaches 1. The sequence of steps is actually unending (that is how infinity is manifested in this thought experiment)- there is no last term.

However, the clock does reach 1. At time 1, the stairway descent must have ended, because the descent occurs entirely before time 1. The descent is not a mathematical process (even though it can be mapped to a mathematical series), it is a sequence of movements from one step to the next. No movements are occurring AT time 1. If the descent has ended at this time, how can there NOT have been a final step?

Everyone can be rash, everyone can be stupid, misinformed or otherwise malpracticing adequate reason.

My question is how does one know when that is the case - ie they're chatting sh*t. And to the contrary, when they really do know what they're talking about.

What is the litmus test in the realm of discourse with others which may be either just as misinformed or very much astute and correct? — Benj96

You should NOT trust your mind, but you can gain trust in certain beliefs by applying critical thinking: seek out contrary opinions, test your beliefs through discussion with others (like on this forum), attempt to mitigate confirmation bias by trying to identify objective reasons to support or deny some presumption you may have. Learn at least some basics of epistemology (including the limits of each technique).

I don't even understand what the supposed paradox is. — fishfry

The paradox is this:

1.The bottom of the stairs is reached at the 1 minute mark.
2.Reaching the bottom of the stairs entails taking a final step.
3. Therefore there is a final step
4.The steps are countably infinite (1:1 with the natural numbers)
5. There is no final (largest) natural number.
6.Therefore there is no final step

#3 & #6 are a contradiction.

We can also map the steps to the elapsed time (1 → 0.5, 2 → 0.75, 3 → 0.875, etc.). If we conclude that a full minute has elapsed, doesn't this imply that he has traversed all the steps? — keystone

Indeed, the stipulated elapse of a minute implies all the steps would have been traversed, but that implication is contradicted by the fact that the process of counting steps is not completable. The presence of this contradiction implies there's something wrong with the scenario.

Here's what's wrong: a mapping reflects a logical relationship, not an activity. The activity is a stepwise process: step n+1 is counted AFTER step n; the logical relation is present atemporally - it's an entailment of the way the scenario is defined.

Analogously, a limit entails an abstract operation applying to a mathematical series and shouldn't be conflated with a consecutive process.
— Relativist
Why not?

Same as above: it's a logical relation (atemporal) that does not account for the stepwise process that unfolds in sequence (temporally).

If time is infinitely divisible, the counter would go up to infinity. Not a conclusion that many of us may like, but there doesn't seem to be anything logically absurd with it. — Lionino

I disagree. It's absurd because the counter progresses through natural numbers, and can never reach a final one. Infinity isn't a natural number. In the context of a temporal counting process, infinity = an unending process, not something that is reached (and not a number).

I'm not so sure this will lead to an indictment for Trump. It would be costly to do so, and there's no chance of a trial before the election. If Trump is elected, the case would be put on hold for 4 years - making it all the more questionable as a productive use of resources.

Imagine a universe where not only is everything possible, but that all possibilities must be fulfilled before its natural conclusion.

How might such a universe look? How might you describe it? How would it begin and end? How would it evolve and unfold? What would concepts such as "paradox", "contradiction", "logic", "irrationality", "belief" and "fact" mean in such a universe? How might all these dynamics interact? — Benj96

It seems to me that a universe where everything is possible entails a world with multiple, causally isolated sub-universes. So there wouldn't be a beginning nor end to this universe as a whole, nor would there be a "conclusion" to it. Every possibility is actuallized in one or more sub-universes.

Paradoxes, contractions, logical, irrationality, belief, and facts are epistemological concepts, applying to propositions and reasoning not to ontological reality. The only "dynamics" these apply to are the the processes of reasoning.

What you seem to overlook is that I'm beginning with a premise widely accepted within the mathematical community: the existence of actually infinite objects (like these infinite stairs or the set, N) and the completion of actually infinite operations (such as traversing the stairs or calculating the sum of an infinite series). If you do not accept the concepts of infinite sets or supertasks, then this paradox is not aimed at you. If you claim that an old woman is 2 years old, then you're not basing your argument on any widely accepted concepts of age. — keystone

There's nothing contradictory with the EXISTENCE of an actual infinite, but it's not accepted that an infinity can be traversed in a supertask. In the case of the staircase, there actually is no last step - so it was correct to say the staircase was "endless".That would be analogous to saying the largest natural number can be reached by counting. This same objection has been raised in regard to the Zeno walk (see this SEP article).

We can consider the steps to be implicitly numbered - they map to the natural numbers. Traversal is one step at a time, moving from step n to step n+1. Every such n is a member of the set of natural numbers, but the supertask obviously never runs out of these. The contradiction is introduced by the stipulation that the end (of something endless) is reached by this process.

One reason the thought experiment can be misleading is that we're accustomed to treating infinite sets as mathematical objects. So we can consider the set of natural numbers and discuss it's cardinality (aleph-0). The set of supertask steps (step 0 to step 1, step 1 to step 2...) is also an infinite set with cardinality aleph-0 so it maps 1:1 to the set of natural numbers. The mapping is "complete" because it's defined for each member of the sets, but a supertask is a consecutive PROCESS, not a formulaic mapping identifying the correspondence. So a complete (i.e. well-defined) mapping shouldn't be conflated with a completed PROCESS.

Analogously, a limit entails an abstract operation applying to a mathematical series and shouldn't be conflated with a consecutive process.

I question the meaningfulness of blaming/crediting "Christianity". It's people who are engaging in good/bad behavior. Sometimes they point to scripture to rationalize their behavior (eg slavery), but that's generally post hoc.

Despite the staircase being endless, he reached the bottom of it in just a minute. — keystone

There is a contradiction in the stated scenario: there's an END to the ENDLESS staircase. Better to ask where he is after a minute.

Assess progress after each step he takes by noting the number of steps yet to be taken: there are always infinitely more to take. So at no point does he actually make progress - even after traversing infinitely many steps because that relation holds at all points along the way.

I expect that title to appear after Stormy gives her testimony.

Here's a guy that I could attack both ways: https://www.foxnews.com/video/6329053321112

Instead of attacking the perpetrators of this anti-Trump information (and risking committing a genetic fallacy), why don't you point out some disinformation they've put forth? TBH, I've seen some of their material, and although it's certainly slanted and conveys some wishful thinking when predicting trial outcomes, I haven't noticed factual falsehoods, like we see from Trump-friendly sources. I invite you to disabuse me.

Are there things in the physical universe that we can never find out? — Vera Mont

Absolutely. For example: what is the ontological bedrock of physical reality? No matter how deeply we explore, we can't know we've reached rock bottom.

the truth is that they have been planning for such a war since at least 2009. — Tzeentch

I watched the video, and read the Brookings report. The person in the video grossly misrepresents the report. Brookings does not state a plan, it lists options - and identifies potential negative and positive consequences of each. The author's premise is that there is some secret plan to go to war with Iran, and he interprets points in the Brookings document to in light of this premise. The fact that certain events have unfolded with some of the anticipated consequences is a testament to Brookings' analysis, not an implication that one particularly nefarious path has been chosen by the US, among all the permutations of paths outlined by Brookings.

I agree, but chip availability is critical to national security.

I agree we need to address the trade imbalance with China, but a sudden, across the board 60% tarriff would hurt consumers and be inflationary. The results of the current tarriffs have been mixed (see this). Going forward, we need to carefully target them to minimize consumer price impacts and avoid supply chain distuptions.

Biden has generally retained Trump's tarriffs on China, even increasing them in some areas. So in practice, there may be little difference between them on tarriffs, although Biden's subsidies for building chip factories is a positive in his corner. The "60%" threat may be campaign talk to create a contrast that's not real. If real, it seems dumb.

Yeah, you're right about it being about perception.

Regarding taxes, it appears that Trump has only promised to extend the current tax schedule that's soon to expire - not lower current taxes. By contrast, Biden says he'll keep current tax rates for everyone making under $400K. Trump has also said he's thinking about lowering corporate tax rates. So I don't see a real benefits for most folks.

Trump has also talked about imposing more tarriffs, including 60% on imports from China, and 10% on all others. This will increase the costs of many things, and likely lead to a trade war.

The culture wars issue sounds more like offering rhetorical support.