I already responded to this. It's the sequence 1, 1/2, 1/4, 1/8, ..., accompanied by the vocalizations 1, 2, 3, ... Every member of the sequence gets traversed, every natural number gets vocalized. — fishfry

The fact that there is a bijection between the series of time intervals and the series of natural numbers and that the sum of the series of time intervals is 60 does not prove that the following supertask is metaphysically possible:

I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum.

How does one start such a supertask?

Feel free to give a reference, else I can't respond. — fishfry

From Tasks, Super-Tasks, and the Modern Eleatics:

What conclusions are we to draw from this rather heady mixture of genies, machines, lamps, and fair and foul numbers? In particular, has it been shown that super-tasks are really possible – that, in Russell's words, they are at most medically and not logically impossible? Of course not. In a part of his paper that I did not discuss, Thomson does a nice job of destroying the arguments of those who claim to prove that super-tasks are logically possible; had there been time I should have examined them. In the preceding section I tried to do the same for Thomson's own neo-Eleatic arguments. I think it should be clear that, just as Thomson did not establish the impossibility of super-tasks by destroying the arguments of their defenders, I did not establish their possibility by destroying his (supposing that I did destroy them).

Also see my comment here where I try to explain where his arguments fail to "destroy" Thomson's.

This is just a re-iteration of your previous post, which does not address which premise you disagree with. — Bob Ross

I don't disagree with a premise. I simply prove the conclusion false, and therefore prove that one of the premises is false or that the conclusion doesn't follow from the premises. I'll leave it to you to determine where you've gone wrong.

In terms of your “P3”, I responded here. — Bob Ross

"I believe that aliens exist" is a proposition that is made true by my belief that aliens exist. Therefore your conclusion that "a belief cannot make a proposition true or false" is false.

Your argument is:

P1: A stance taken on the truthity of something, is independent of the truthity of that something.
P2: A belief is a (cognitive) stance taken on the truthity of a proposition.
C1: Therefore, a belief cannot make a proposition true or false.

However:

P3: "I believe that aliens exist" is true iff I believe that aliens exist

P3 contradicts C1, therefore at least one of P3 and C1 is false. P3 is true. Therefore, C1 is false. Therefore, either C1 does not follow from P1 and P2 or at least one of P1 and P2 is false.

I don't really care what the answer is; I only care that P3 is true and so that C1 is false.

"I believe one ought not torture babies" is NOT a moral proposition: the moral proposition is that "one ought not torture babies". — Bob Ross

I was addressing this conclusion:

C1: Therefore, a belief cannot make a proposition true or false.

Nowhere in this conclusion is the term "moral" used.

My example of "I believe that aliens exist" being true iff I believe that aliens exist is proof that a belief can make a proposition true or false.

As such you are left with this:

P1: A stance taken on the truthity of something, is independent of the truthity of that something.
P2: A belief is a (cognitive) stance taken on the truthity of a proposition.
C1: Therefore, a belief cannot make a proposition true or false.

P3: Beliefs make moral propositions true or false.
P4: C1 and P3 being true are logically contradictory.
C2: Therefore, moral subjectivism is internally inconsistent.

I'm going to address Benacerraf's Tasks, Super-Tasks, and the Modern Eleatics:

Thomson's first argument, concerning the lamp, is short, imaginative, and compelling. It appears to demonstrate that "completing a super-task" is a self-contradictory concept. Let me reproduce it here:

_{There are certain reading-lamps that have a button in the base. If the lamp is off and you press the button the lamp goes on, and if the lamp is on and you press the button, the lamp goes off. So if the lamp was originally off and you pressed the button an odd number of times, the lamp is on, and if you pressed the button an even number of times the lamp is off. Suppose now that the lamp is off, and I succeed in pressing the button an infinite number of times, perhaps making one jab in one minute, another jab in the next half minute, and so on. ... After I have completed the whole infinite sequence of jabs, i.e. at the end of the two minutes, is the lamp on or off? ... It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off. This is a contradiction.}

Rarely are we presented with an argument so neat and convincing. This one has only one flaw. It is invalid. Let us see why. Consider the following two descriptions:

A. Aladdin starts at t₀ and performs the super-task in question just as Thomson does. Let t₁ be the first instant after he has completed the whole infinite sequence of jabs – the instant about which Thomson asks "Is the lamp on or off?" – and let the lamp be on at t₁.

B. Bernard starts at t₀ and performs the super-task in question (on another lamp) just as Aladdin does, and let Bernard's lamp be off at t₁.

I submit that neither description is self-contradictory, or, more cautiously, that Thomson's argument shows neither description to be self-contradictory (although possibly some other argument might).

The fallacy in his reasoning is that it does not acknowledge that for all t_n >= t_1/2 the lamp is on iff the lamp was off and I pressed the button to turn it on and the lamp is off iff the lamp was on and I pressed the button to turn it off.

So the lamp is on at t₁ only if either it was turned and left off at some t_n < t₁ and then turned on at t₁ or it was turned and left on at some t_n < t₁, neither of which are consistent with the supertask.

We have seen that in each case the arguments were invalid, that they required for their validation the addition of a premise connecting the state of the machine or lamp or what have you at the ωth moment with its state at some previous instant or set of instants. The clearest example is that of the lamp, where we can derive a contradiction only by explicitly assuming as an additional premise that a statement describing the state of the lamp (with respect to being on or off ) after all the switchings is a logical consequence of the statements describing its state during the performance of the super-task.

This logical consequence can be shown when the experiment is explained more clearly:

A. At t₀ the lamp is off, at t_1/2 I press the button, at t_3/4 I press the button, at t_7/8 I press the button, and so on ad infinitum

Compare with:

B. At t₀ the lamp is off, at t_1/2 I press the button

The status of the lamp at t₁ must be a logical consequence of the status of the lamp at t₀ and the button-pressing procedure that occurs between t₀ and t₁ because nothing else controls the behaviour of the lamp.

If no consistent conclusion can be deduced about the lamp at t₁ then there’s something wrong with your button-pressing procedure.

So the fact that the status of the lamp at t₁ is "undefined" given A is the very proof that the supertask described in A is impossible.

C1: Therefore, a belief cannot make a proposition true or false. — Bob Ross

"I believe that aliens exist" is true iff I believe that aliens exist

Well, this is a publicly visible forum so nothing stops ChatGPT from visiting the website and copying what shows on the page.

It's possible to stop this by creating a robots.txt file that tells ChatGPT that it's not allowed to visit, but PlushForums doesn't provide such a file.

As for the raw database, the PlushForums FAQ says "we do not sell or share your data with any third parties."

You can't play it in reverse — fishfry

So you're saying that it's possible to have recited the natural numbers in ascending order and possible to have recorded this on audio but impossible to then replay this audio in reverse? That seems like special pleading. Am I metaphysically incapable of pressing the rewind button?

I believe you have agreed with me. — fishfry

I am presenting two versions of your argument; one in which I have recited the natural numbers in ascending order and one in which I have recited the natural numbers in descending order. I am using the second version to illustrate the flaw in the first version.

No, once again you recited the natural numbers in ascending order. — fishfry

No, I'm reciting them in descending order. I'll repeat it again and highlight to make it clear:

I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum – e.g. my recitation ends with me saying "3" at 12:00:07.5 then "2" at 12:00:15 then "1" at 12:00:30 and then "0" at 12:01:00.

What natural number did I not recite? There is no answer. Therefore I have recited the natural numbers in descending order.

Notice that even if the conclusion follows from the premise that the argument fails because the premise is necessarily false. It is impossible, even in principle, for me to have recited the natural numbers in the manner described.

Returning to your version of the argument:

I said "0", 30 seconds after that I said "1", 15 seconds after that I said "2", 7.5 seconds after that I said "3", and so on ad infinitum – e.g. my recitation starts with me saying "0" at 12:00:00 then "1" at 12:00:30 then "2" at 12:00:45 and then "3" at 12:00:52.5.

What natural number did I not recite? There is no answer. Therefore I have recited the natural numbers in ascending order.

Even if the conclusion follows from the premise I do not accept that the premise can possibly be true. Like with the previous argument, I think that it's impossible, even in principle, for me to have recited the natural numbers in the manner described.

I have attempted at least to explain why this is impossible (e.g. with reference to recording us doing so and then replaying this recording in reverse), but as it stands you haven't yet explained why this is possible. If you're not trying to argue that it's possible – only that I haven't proved that it's impossible – then that's fine, but if you are trying to argue that it's possible then you have yet to actually do so.

Can you prove that it's metaphysically possible for me to halve the time between each subsequent recitation ad infinitum? It's not something that we can just assume unless proven otherwise. Even Benacerraf in his criticism of Thomson accepted this.

It means that is isn't a finite sequence of operations. — noAxioms

No, it doesn't. Saying that it is an infinite sequence of operations means that it isn't a finite sequence of operations.

I'm asking you to make sense of the "every operation is performed" part of "every operation is performed in an infinite sequence of operations”.

By definition, the sequence completes by having every operation occurring before some finite time. — noAxioms

What does it mean for every operation to occur without some final operation occurring?

As it stands your definition is a contradiction.

If you mean that it doesn't complete, it by definition does in a finite time. If you mean that it has no terminal step, then you're making the mistake I identify just above since the definition does not require one. — noAxioms

How can a sequence of operations in which each operation is performed only after the previous operation is performed complete without there being a final operation?

You just seem to hand-wave this away with no explanation.

You also wield the term 'ad infinitum', — noAxioms

Well, yes. That's how to define it as an infinite sequence of operations rather than a finite sequence of operations.

https://en.wikipedia.org/wiki/Subjunctive_possibility#Types_of_subjunctive_possibility

That's all very well. But it also takes us back to the question what this "operation" actually is. — Ludwig V

It could be anything. The problem has nothing to do with the operation being performed and everything to do with continually halving the time between operations.

At 0s A ≔ 1, at 30s A ≔ red, at 45s A ≔ turtle, at 52.5s A ≔ 1, at 56.25s A ≔ red, and so on ad infinitum.

Or:

At 60s A ≔ 1, at 30s A ≔ turtle, at 15s A ≔ red, at 7.5s A ≔ 1, at 3.75s A ≔ turtle, and so on ad infinitum.

That an infinite series with terms that match the described and implied time intervals has a finite sum isn't that it makes sense for either set of tasks to have actually been carried out. This should be self-evident in the second case. You're being deceived by maths if you think the first case is different.

Argument 1
Premise: I said "0", 30 seconds after that I said "1", 15 seconds after that I said "2", 7.5 seconds after that I said "3", and so on ad infinitum.

What natural number did I not recite? There is no answer. Therefore I have recited the natural numbers in ascending order.

Argument 2
Premise: I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum.

What natural number did I not recite? There is no answer. Therefore I have recited the natural numbers in descending order.

---

In both cases for any given natural number I can calculate how long it took me to reach it.

These arguments only show that if I recite the natural numbers as described then I have recited all the natural numbers, but this does nothing to prove that the antecedent is possible, and it is the possibility of the antecedent that is being discussed. As it stands you're begging the question.

Now let's assume that it's metaphysically possible to have recited the natural numbers in ascending order and to have recorded this on video/audio. What happens when we replay this video/audio in reverse? It's the same as having recited the natural numbers in descending order which you admit is metaphysically impossible. Therefore having recited the natural numbers in ascending order must also be metaphysically impossible.

Both Argument 1 and Argument 2 are unsound. The premises are necessarily false. It is impossible in principle for us to recite the natural numbers in the manners described.

After 60 seconds I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum.

What natural number did I not say?

You can't answer, therefore it is metaphysically possible to have recited the natural numbers in descending order.

---

Obviously the above is fallacious. It is metaphysically impossible to have recited the natural numbers in descending order. The fact that we can sum an infinite series with terms that match the described and implied time intervals is irrelevant. The premise begs the question. And the same is true of your version of the argument.

I've given solid a mathematical argument that your 60 second puzzle guarantees that all the numbers will be spoken. — fishfry

No you haven't. Your premise begs the question and simply asserts that all the natural numbers have been recited within 60 seconds.

7/8 will do just fine. I necessarily had to jump over all but finitely members of the sequence. — fishfry

No, we're reciting the numbers in descending order. It's impossible to do, even in principle. The fact that we can baselessly assert that I recite the first number in N seconds and the second number in N/2 seconds and the third number in N/4 seconds, and so on ad infinitum, and the fact that the sum of this infinite series is 2N, doesn't then entail that the supertask is possible.

That we can sum this infinite series is evidently a red herring.

I go 1 at 60, 2 at 30, etc.

Name the first number that I fail to count

Third time I'm asking you the question.

This is a standard inductive argument. If it's impossible to name the first natural number at which a property fails to hold, the property must hold for all natural numbers.

Please give this argument some thought. — fishfry

It begs the question. Your premise is necessarily false. Such a supertask is impossible, even in principle, to start.

In your opinion. But you have no proof or evidence. On the contrary, the mathematics is clear. — fishfry

You just listed five rational numbers and are claiming that this is proof of you reciting all the natural numbers in descending order? You're talking nonsense.

But counting backward from infinity is always finite! I showed you how that works, counting backward from 1 in the ordered set <1/2, 3/4, 7/8, ..., 1> — fishfry

What number do you recite after 1?

Did I not move you, surprise you, convince you, that if you count 1, 2, 3, ... successively halving the time intervals, that you will indeed count every single natural number in finite time? If not, why not? — fishfry

Because it begs the question.

But counting backward from infinity is always finite! I showed you how that works, counting backward from 1 in the ordered set <1/2, 3/4, 7/8, ..., 1> — fishfry

What number do you recite after 1?

It's easy, I'll do it right here on a public Internet forum.

1, 15/16, 7/8, 3/4, 1/2. Done.

That's because the first step backward from any limit ordinal necessarily jumps over all but finitely members of the sequence whose limit it is. — fishfry

That's not counting down from infinity. That's just reciting five rational numbers.

I don't know what you mean that supertasks are nonterminating by definition. — fishfry

Tasks are performed ad infinitum. I never stop counting. There's always another number to count.

You did lose me when you said that counting 0, 1, 2, ... is "counting down from infinity." I did not understand that example when you gave it earlier. Mathematically, the ordered set <1, 2, 3, ...> exists, all at once. Its counting is completed the moment it's invoked into existence by the axiom of infinity. — fishfry

I'm talking about reciting the numbers. So imagine someone reciting the natural numbers up to infinity. Now imagine that process in reverse. That's what I mean by someone counting down from infinity.

It is a non sequitur to argue that because we can sum an infinite series with terms that match the proposed time intervals that it is possible to have counted down from infinity. It is impossible, even in principle, to start such a count. The maths of an infinite series doesn't change this.

And it is a non sequitur to argue that because we can sum an infinite series with terms that match the proposed time intervals that it is possible to have counted up to infinity. It is impossible, even in principle, to stop such a count. The maths of an infinite series doesn't change this.

Well ok, then why don't I complete a supertask when I walk across the room, first going halfway, etc.? Can you distinguish this case from your definition? — fishfry

If supertasks are impossible and motion is possible then motion isn't a supertask.

* You have not convinced me or even made me understand your reasoning that supertasks are "metaphysically impossible" or that they entail a logical contradiction. — fishfry

By definition supertasks are non-terminating processes, therefore you've gone wrong somewhere if you conclude that they can terminate after 2N seconds.

Also I think the clearest example I gave was that of having counted down from infinity. We can assert (explaining what happened in reverse) that I recited 0 after 60 seconds, recited 1 after 30 seconds, recited 2 after 15 seconds, recited 3 after 7.5 seconds, etc., and we can say that we can sum an infinite series with terms that match the described (and implied) time intervals, but it doesn't then follow that we can have counted down from infinity; we can't even start such a count. The mathematics is evidently a non sequitur, and so it's a non sequitur in the case of having counted up to infinity as well (and so for any proposed supertask).

In the case of Thomson's lamp, nothing ever happens to the lamp except as described by this process: I turn it on after 30 seconds, turn if off after 15 seconds, turn it on after 7.5 seconds, etc. It cannot be on after 60 seconds because I always turn it off after turning it on and it cannot be off after 60 seconds because I always turn it on after turning it off, but it must be either on or off after 60 seconds, and so therefore there is a contradiction.

If you want to say that such a supertask is possible then the burden is on you to explain the state of the lamp after 60 seconds, and your answer must follow from the description of the supertask. If nothing follows then the supertask is impossible.

He does. Most of the paper focuses on rationalizing low probabilities for the first two premises to the point of 3 being likely. — noAxioms

What paper are you reading? From the conclusion to Are You Living in a Computer Simulation?:

A technologically mature “posthuman” civilization would have enormous computing power. Based on this empirical fact, the simulation argument shows that at least one of the following propositions is true: (1) The fraction of human‐level civilizations that reach a posthuman stage is very close to zero; (2) The fraction of posthuman civilizations that are interested in running ancestor‐simulations is very close to zero; (3) The fraction of all people with our kind of experiences that are living in a simulation is very close to one.

If (1) is true, then we will almost certainly go extinct before reaching posthumanity. If (2) is true, then there must be a strong convergence among the courses of advanced civilizations so that virtually none contains any relatively wealthy individuals who desire to run ancestor‐simulations and are free to do so. If (3) is true, then we almost certainly live in a simulation. In the dark forest of our current ignorance, it seems sensible to apportion one’s credence roughly evenly between (1), (2), and (3).

Point is, you are misstating Bostrom's premises. Item 3 doesn't follow from the premises as you word them. — noAxioms

They are not premises. (3) isn't intended to follow from (1) and (2).

Could you give me an example of two incompatible mathematical systems? — Tarskian

There is a universal set in New Foundations but not in ZFC.

The surreal number line, unlike the real number line, includes infinity and infinitesimals.

There are many different, incompatible, mathematical systems. Assuming Platonism, which mathematical system is "correct", and how is this determined?

But I couldn't see why Bostrom thought that one of those three must be true. — Ludwig V

If lots of civilisations are capable of and willing to make simulations then they will, and so simulated persons will greatly outnumber non-simulated persons.

Therefore, if simulated persons do not greatly outnumber non-simulated persons then most civilisations are either incapable of or unwilling to make simulations.

I find both these to be highly unlikely, for the reason stated in this topic and mine. Bostrom of course has motivation to rationalize a higher probability for both of these, but rationalizing is not being rational. — noAxioms

I'm confused by what you're saying.

Bostrom is saying that one of these is almost certainly true:

1. Almost every intelligent civilisation is incapable of creating simulations
2. Almost every intelligent civilisation doesn't want to create simulations
3. Almost every conscious person is living in a simulation

Because if lots of civilisations are capable of and willing to make simulations then they will, and so simulated persons will greatly outnumber non-simulated persons.

Therefore, if simulated persons do not greatly outnumber non-simulated persons then most civilisations are either incapable of or unwilling to make simulations.

He doesn't say which of the three is most likely to be true.

Artificial consciousnesses programmatically fed phenomenal experience, e.g. man-made brains-in-a-vat.

Well between the two of you I have no idea what a supertask is anymore. — fishfry

A supertask is "a countably infinite sequence of operations that occur sequentially within a finite interval of time."

Bostrom's Simulation Argument is that one of these is almost certainly true:

1. The fraction of human-level civilizations that reach a posthuman stage (that is, one capable of running high-fidelity ancestor simulations) is very close to zero, or
2. The fraction of posthuman civilizations that are interested in running simulations of their evolutionary history, or variations thereof, is very close to zero, or
3. The fraction of all people with our kind of experiences that are living in a simulation is very close to one.

He then argues that if (3) is true then we are almost certainly living in a simulation.

I'm only asking how far 1,1 is from 1,2 in a discrete space system. As far as I can tell, it's 0 units, right? — Hanover

I don't think the question makes sense, but you'll have to ask a physicist who knows more about quantum gravity to explain it. I can only point out to you that there are physical theories that take spacetime to be discrete.

The problem is adjacency. If object A is adjacent to object B on a finite grid, what is the distance from A to B? If it's 0 units, then A and B occupy the same space and A = B. — Hanover

You seem to be imagining a model of discrete space overlaying some model of continuous space and then pointing out that in continuous space there is always more space between two discrete points.

That seems to be begging the question.

Best I can do is point you to something like quantum spacetime and quantum gravity.

There are physical theories that treat spacetime as discrete. They are not supported to the extent that General Relativity is, but given that quantum mechanics and General Relativity are known to be incompatible, it would seem that at least one of them is false, and my money is on General Relativity being false.

Given the logical paradoxes that continuous space and time entail, I think that discrete spacetime is not just a physical fact but a necessity.

However, the thing measured is the passage of time which occurs. — Metaphysician Undercover

And the passage of time that we would measure as being 60 seconds occurs even when we don't measure it.

If it's at L-1 at T-1 and L-2 at T-2, how long did it take to get from L-1 to L-2? — Hanover

The question makes no sense. You're asking for some second "level" of time to define the time between T1 and T2. There's no such thing. The only time is T1, T2, T3, etc.

Assuming at the most microscopic level, the object is on an 8x8 chessboard. The pawn moves from e2 to e3. There is no e2.1 or other smaller increments in this finite world. At T1 it's at e2 and T30 it's at e3. The assumption is that at some point in time, it was no where while transitioning (moving?) from e2 to e3. — Hanover

We can't examine this at the macroscopic scale. At whatever the smallest scale is: at Time1 it's at Location1 and at Time2 it's at Location2. There's no intermediate Time1.5 where it doesn't exist or Location1.5 that it moves through.

And you're just making the same mistake again and falsely claiming that indirect realists believe that our eyes respond to light reflected by sense data. They don't. If you're going to continue to argue against this strawman then I'm out.

Acquaintance primarily concerns knowledge. — Luke

Yes, hence the epistemological problem of perception.

The direct/indirect realism dispute primarily concerns sensory perception — Luke

It concerns whether or not sensory perception provides us with direct knowledge of distal objects.

Naive realists claim that sensory perception does provide us with direct knowledge of distal objects because distal objects are literal constituents of conscious experience, and so we are acquainted with distal objects.

Indirect realists claim that sensory perception does not provide us with direct knowledge of distal objects because distal objects are not literal constituents of conscious experience, and so we are only acquainted with mental phenomena.

My usage is consistent. Indirect realists equivocate over the meaning of "perception", using it to mean both the sensory perception of external objects and the Russellian acquaintance of mental representations. — Luke

It's only equivocation if they start from the premise that we are acquainted with mental phenomena and then conclude that our eyes respond to light reflected by mental phenomena, but they never make this conclusion. This is the strawman conclusion that you are fabricating.

Except your explanation of what indirect realists believe is that our perceptions of material objects are not mediated by the perception of some other entity, which is therefore not indirect realism. — Luke

What indirect realists mean by "perception of some other entity" isn't what you mean by "perception of some other entity". You're equivocating.

Indirect realists do not and never have believed or claimed that our eyes respond to light reflected by sense data.

That our perceptions of material objects are mediated by the perception of some other entity, such as sense-data. — Luke

Except by this you mean "our eyes respond to light reflected by sense data" which isn't what indirect realists believe.