Sorry, I overlooked that part.

Can we not count the intervals starting with 1? Would that number not tend towards infinity given time is infinitely divisible or approach a certain value and terminate given a smallest sliver of time exists? — ToothyMaw

"Tending towards infinity" means counting through the natural numbers - the set is infinite. The process has no end.

So they don't want a better government, better economic policies, or better anything. What they want, is to bring down the whole system, because they don't accept the principles on which it was founded. — Wayfarer

What I can't figure out is, what Trump voters think they're voting for. — Wayfarer

My theory is that are motivated to halt, or roll back social changes (e.g.normalization of the LGBTQ communities, perceived special treatment for minorities), and are fed up with the way the goverment works - the negative impacts of government bureaucracy, as well as foreign entanglements. They want a superman who uses his superpowers to solve our most important problems. Trump promises to do the impossible, and thus appeals to their wishes. As example: in 2016, building a (Mexico funded) border wall that was perceived as a solution to many problems (eg unwanted migration, drug trafficking)- an alleged simple solution to a complex set of problems.

It's not that they actively want to do unconstitutional or illegal things, per se, it's that Trump doesn't concern himself with those impediments, so they don't have to even think about those things. They trust Trump will find a way.

I don’t think it entails nihilism and fatalism, but it does set the grounds for them. — NOS4A2

I agree with this much, but disagree with suggesting that the

... implications of determinism...[are] that they have zero responsibility.

I disagree because this sense of responsibility is a part of our mechanism, and contributes to our choices. This is in spite of the fact that all of our decision-making components originated outside ourselves. Indeed, we aren't responsible for our genetic makeup, don't fully control what we learn, are (somewhat) slaves to our conditioned responses, etc, but we still make the choices that we make. As the decision-maker, there is inherent responsibility for those decisions. We hold others accountable, and we ought to hold ourselves accountable. Accountability never means the past can be changed; it is only about the future decisions we (or others) will make. So what if we couldn't have made a different decision in that instant within its circumstances? We can learn from the consequences, and this can result in better decisions in the future.

I actually question the notion that Libertarian Free Will (LFW) entails the Principle of Alternative Possibilities (PAP) - the notion that LFW implies there's true contingency to our choices. Even under LFW, we are guided by our impulses, knowledge, assumptions, etc. Given some series of deliberative thoughts, how could there be a different outcome? We have followed some chain of reasoning, and are subject to the same impulses.

Since x reaches infinity at time 1, all steps are completed at that time, so the task is complete — noAxioms

Infinity is not reached. You're not considering what it means to be infinite in this context: it means continually dividing the remaining time (prior to the 1-minute mark) in half. Because the remaining time corresponds to a real number line, the process proceeds without ending because the remaining time is infinitely divisible. It's limited by the fact that all points of time that are reached by the process are less than 1 minute- so it is logically impossible for this process to reach the point of time of 1 minute.

The source of confusion is that the clock does hit the 1-minute mark. You are incorrectly interpreting this as implying the the process reaches that point. It can't, because it is logically impossible.

The clock reaching the 1-minute mark implies the process ends, but since the process cannot reach the 1-minute mark, the process must be terminated at some point. Laws of nature would clearly provide a limit to how small we divide the time, but even setting that aside - a stopping point is logically necessary. The math doesn't identify any particular stopping point, but it does imply there has to be one.

I also think you are misinterpreting the meaning of limit.
— Relativist

On a forum our words must speak for themselves. But in this instance I can assure you that nothing could possibly be farther from the truth. — fishfry

You explained your interpretation:

You can think of it that way. Or you can think of it "reaching" its limit at a symbolic point at infinity. Just as we augment the real numbers with plus and minus infinity in calculus, to get the extended real numbers — fishfry

So...you're thinking of a limit in a vauge way ("symbolic"), and vaugely asserting the series "reaches" infinity, and then rationalize this with a mathematical system that defines infinity as a number.

Although it's true that there are such mathematical systems, it doesn't apply to the supertask. Time is being divided into increasingly smaller segments approaching, but never reaching, the 1 minute mark. There is a mathematical (and logical) difference between the line segments defined by these two formulae:
A. All x, such that 0<=x < 1
B. All x, such that 0<=x <= 1

Your blurred analysis conflates these, but it is their difference that matters in the analysis. The task maps exactly to formula A, but not to formula B (except in a vague, approximate way). Mathematics is about precise answers.

...like physics is somehow exempt from mathematics (or logic in Relativist's case) or something. — noAxioms

Physics indeed is not exempt from logic. It's logically impossible to reach the 1 minute mark when all steps (even if there are infinitely many of them) fall short of the 1 minute mark.

Calculating the limit does not entail a process that reaches that limit. This is a misinterpretation of the concept of limit.This article describes it this way:
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value...

The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value |an − L| is the distance between an and L.

But you don't get to "give meaning to the factors" if it's all deterministic. Your genetic makeup, experiences, etc., give everything meaning to the group of cells referred to as Relativist. — Patterner

This ignores the fact that your genetic makeup, experiences, etc comprise you. That particular group of cells performs functions, including the cognitive functions of making choices.

when you are in a situation where different directions are taken by different people, the meaning that all those factors have determined you have determine which direction you take.

I agree that the process is, in one sense, programmed, but you are the program. There's also a sense in which you aren't programmed: you aren't the product of design. You weren't built in order to perform the functions you execute.

You do go through a "process" yes. But (while I'm not saying you did so deliberately) note how your wording even implies ultimate passivity. — ENOAH

You're reading passivity into it; that certainly was not what I was trying to convey. The mental processes are under your control, not someone or something else. Determinism implies the process is mechanistic, but even a machine still has to do the work to produce its output - the machine isn't "passive".

The fact that we lack the freedom to refrain from things like breathing seems irrelevant- not everything we contemplate involves a real choice, and that applies even if we have libertarian free will.

Agreed.

You are falling into the trap of thinking a limit "approaches" but does not "reach" its limit. It does reach its limit via the limiting process, in the same sense that 1/2, 3/4, 7/8, ... has the limit 1, and 1 is a perfectly good real number, and we all have had literally billions of experiences of one second of time passing. — fishfry

You're pointing to the limit of a mathematical series. A step-by-step process does not reach anything. There is no step that ends at, or after, the one-minute mark. Calculating the limit does not alter that mathematical fact.

I also think you are misinterpreting the meaning of limit. This article describes it this way:
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value...

The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value |an − L| is the distance between an and L.

You just said to me that one second of time can't pass; and this, I reject. Am I understanding you correctly? — fishfry

No, I didn't. I said the stair-stepping PROCESS doesn't reach the 1 second mark. Are you suggesting it does?

But if the choices are determined, then are they really choices? — Patterner

You do go through a choice-making process, don't you? For important decisions, you may deliberate for a time, weighing the pros and cons of alternatives. Your beliefs and whims will factor in, as will your hopes, desires, risk tolerance - all influenced by your genetic makeup. But all those factors are intrinsic to who you are.

Sure, every one of those factors were caused (by your genetic makeup, upbringing, experiences, education...), but they are bundled together in a unique way to comprise YOU.

If determinism is true, and the person's genetic makeup, upbringing, other past experiences, health at the moment, and all other factors, will allow only one option — Patterner

Of course, whatever choice you make could not have been different at the point you make it, given your life-history. But you will also learn from the consequences of your decision, adding factors that will influence future choices. This is why I have previously argued that moral accountability is at least somewhat reasonable: future behavior is influenced by reward/punishment.

So, although there are causes that necessitate the choices you make, those causes didn't conspire to make the choice. The machinery that is YOU had to do the processing that led to the result (the choice).

And your choices aren't meaningless. You give meaning to the factors, and those meanings influence the choice. The person who chooses to keep the money you drop is doing so because of what money means to him.

If it really is the case that everything that happens couldn’t help but happen and people’s choices aren’t truly free, then those who believe life is meaningless and morality doesn't exist have no choice but to believe that. And nobody has any choice but to live their lives as they do in response to that. — Patterner

Even if determinism is true, we still make choices. It's true that those choices are a product of prior events, but the choices are still made - and we are the agents making them.

As hotly contested as the topic is I can’t recall a philosopher ever using the deterministic nature of the universe as evidence that good and evil don’t exist and the lives of sentient beings have no actual value. — Captain Homicide

First of all, I agree with everything you said. Regarding the above, I don't think determinism (per se) is inconsistent with the existence of objective moral values (OMVs). On the other hand, materialism is inconsistent with OMVs, because OMVs are not material objects.

The task consists of a sequence of actions occurring at intervals of time that decrease by half at each step: 1/2 minute, 1/4, 1/8,.... It is logically impossible for this sequence of actions to reach the 1 minute mark (the point in time at which the descent is considered completed), it just gets increasingly close to it.

That's true, but that just makes it physically impossible. I think it's stronger: logically impossible.

1. A given halfway step cannot reach the goal.
2 There is a specific step that reaches the goal (per PSA)
3 Therefore this final step is not a halfway step (1 & 2)
4 Any given step is halfway (per Zeno)

You don't find this contradictory? — noAxioms

Of course it is, but the the contradiction can be resolved by denying either one of two premises. You chose to deny the PSA, and I responded that the PSA could be true - we'd merely have to reject the other relevant premise - that the goal is reached. You have not made an argument that shows it is more reasonable to deny the PSA than to deny the reaching of the goal. I don't think it make sense to deny that a completed task entails a final step.

Demonstration that immediate contradictions arise from denying either of the premises or presuming your conclusion 3 is also more than just handwaving. — noAxioms

Sure. You have to agree the PSA is true for finite tasks. Is there something different about infinite tasks? It doesn't seem so: consider the process: stepping increasingly closer to temporal point in time 1, but the process never actually reaches it. So the goal is unreachable by the process.

I'm not enough of the mathematician to regurgitate all the axioms and processes involved in the accepted validity of the value of a convergent series. — noAxioms

No need. I understand that the math shows that the series reaches a point of convergence at time 1. However: the kinematic process never actually reaches time 1. That's why the series doesn't adequately account for the kinematic process -and why I've stressed we need to examine the process, not just do the math on the mathematical series.

no impediment to the reaching of the goal has been identified, — noAxioms

On the contrary, there's a logical impediment to reaching the goal through the process: the process does not reach time 1.

You do seem to heavily rely on definitions that come only from finite logic — noAxioms

I'm actually basing my claims on real analysis, which analyzes the characteristics of real numbers - including the associated infinities.

There is a temporal end to it, a final moment if not a final step. — noAxioms

That makes no sense. The process does not have a final moment. because there are infinitely many moments prior to time 1. There is no end to the series of kinematic steps, in spite of the fact that the mathematical series converges.

Relativist: "But this process has a 1:1 correspondence to the supertask -- for every step taken in one scenario, there's a parallel step taken in the other. This suggests that either they both complete, or neither completes."

There is a bijection yes. It does not imply that both or neither completes. — noAxioms

Why not?

Relativist: "The number line in question is an interval that is open on the right: i.e. it includes all points <1, but not including 1. There are infinitely many points in this interval, but the point "1" isn't one of them. So the process cannot reach 1, and 1 is the goal of the process."

The 'process' can go beyond the end of the line despite it ending before the goal. — noAxioms

No it can't - that is logically impossible. The process entails taking steps with increasing shorter durations: 1/2 second, 1/4, 1/8,.... The process can only approach 1, it can never reach it.

. The kinematic process isn't restricted to only points on the number line.

No! Each new step is half the duration of the last step, and this halving process has no end.

Moral law is an invention of mankind for the disenfranchisement of the weak. — Jack Cummins

I disagree with this. IMO, morality is rooted in empathy. It feels wrong to hurt another person, because we empathize with the one who is hurt. The golden rule formalizes this into a "moral law" of sorts. Assessing what is morally good becomes trickier as situations become more complex, and often there's moral ambiguity - partial goods and partial evils. This opens the door for the perceived "disenfranchisement of the weak" in those cases. It's worthwhile to debate those cases, but I disagree that all moral law should be assumed to motivated by such a cynical motive.

She said that as it is a charity supporting children, they will not stock CDs, in case there has been any exploitation of children in the making of the music'. — Jack Cummins

She's not being immoral, she's being cautious - perhaps overly cautious. Why deal in materials that she has suspicions about? Perhaps her suspicions are irrational, but is that relevant?

It made me think of the previous movement of the 'moral right', as represented by Mary Whitehouse, which argued against pornography and art forms which showed forms of violence. It is based on forms of moral absolutism and what is acceptable being enshrined as 'moral law'. — Jack Cummins

This is similar to the charity lady scenario only in that it seems rooted in ignorance and irrationality, but it differs from from the charity scenario in that it represents a movement to generally restrict access to pornography, whereas the charity lady was just choosing not to participate in something she was suspicious about.

"Political correctness" has both positive and negative connotations. On the positive side, it may deter people from offending others. On the negative side, it can be based on false assumptions and become a sword to shame people (sometimes appropriately, sometimes inappropriately).

Or the PSA is correct, and the goal can't be met.
— Relativist
I showed that for a supertask, the PSA is not correct. So no, this cannot be for a supertask. — noAxioms

No, you didn't. You merely asserted: "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." There is no direct violation.

Here's valid logic:
1. A halfway step cannot reach the goal.
2. All steps are halfway
3. Therefore the goal cannot be reached.

You merely asserted the goal is reached (directly contradicting #3) but didn't explain how the sequence of halfway steps somehow reaches the goal. Labeling the process a "supertask" is handwaving, not proof. Show your logic.

If the process continues forever, by definition it isn't a supertask. — noAxioms

Fair enough, I misstated it. The process does not continue forever, however there is no end to the process.

Let's compare the supertask to a scenario in which the time interval between each step is a constant (e.g. 1 second). You'll agree that this process does not complete, right? But this process has a 1:1 correspondence to the supertask -- for every step taken in one scenario, there's a parallel step taken in the other. This suggests that either they both complete, or neither completes.

Points on a number line exist concurrently (in effect).
— Relativist
I don't know what is meant by this. 'Concurrently' means 'at the same time' and there isn't time defined for a number line.
A number line seems to be a set of ordered points represented by a visual line. It can be defined otherwise, but functionally that seems sufficient. It being a visual aid, it seems physical, but a reference to the simultaneity of the positions along the line seems irrelevant to the concept. — noAxioms

My point was that the kinematic stair-stepping process has a temporal element that is not reflected in a number line.

The number line in question is an interval that is open on the right: i.e. it includes all points <1, but not including 1. There are infinitely many points in this interval, but the point "1" isn't one of them. So the process cannot reach 1, and 1 is the goal of the process. The goal is therefore unreachable by the kinematic process.

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?