Ok. Just talking about standard mathematical sequences. It's a common misunderstanding in this thread. The sequence 1/2, 3/4, 7/8, ... has a limit, namely 1, but no last element.

The sequence 1/2, 1/4, 1/8, ... also has a limit, namely 0, and no last element. But if you put the elements of the sequence on the number line, they appear to "come from" 0 via a process that could never have gotten started. This is my interpretation of Michael's example of counting backwards. — fishfry

This is what I mean by reciting backwards:

If I recite the natural numbers <= 10 backwards then I recite 10, then 9, then 8, etc.
If I recite the natural numbers <= 100 backwards then I recite 100, then 99, then 98, etc.

If I recite all the natural numbers backwards then I recite ... ?

It's self-evidently impossible. There's no first (largest) natural number for me to start with.

Given your reluctance to clarify the definition of the verb 'to start', I cannot respond appropriately to this statement. I gave a pair of options, or you can supply your own, so long as it isn't open to equivocation. — noAxioms

Just the ordinary meaning of "start", e.g. "begin".

You ask me, right now, to recite the natural numbers in descending order. How do I begin to perform this?

I think it's self-evident that I cannot begin because there is no first (largest) number for me to begin with.

I've repeatedly challenged you to name the first number not verbalized when we count forward 1, 2, 3, ... at successively halved intervals of time. — fishfry

I accept this:

P1. If we can recite forward 1, 2, 3, ... at successively halved intervals of time then we can recite all natural numbers in finite time

But I reject these:

P2. We can recite forward 1, 2, 3, ... at successively halved intervals of time
C1. We can recite all natural numbers in finite time

If you want to claim that C1 is true then you must prove that P2 is true. You haven't done so.

I think Thomson's lamp and similar examples prove that P2 is false. See here.

I cannot start reciting the natural numbers in descending order because there is no first natural number for me to start with.

That a geometric series has a finite sum is irrelevant to this very simple self-evident fact.

You (as well as Meta above) seem to insist on an additional premise of the necessity of a bound to something explicitly defined to be unbounded. — noAxioms

No, I'm saying that something with no start cannot start and something with no end cannot end.

Your argument is effectively "by definition it has no start therefore it can start without a start." You're trying to take the very thing that makes it impossible as proof that it's possible.

Good! Then it's logically possible for it to. An infinite number of things can complete without blowing up logic. — fdrake

But we're talking about supertasks, not geometric series. That a geometric series is possible isn't that a supertask is possible.

Given that there is no largest natural number it is logically impossible to even start reciting all the natural numbers in descending order.

I don't know why you think the existence of a geometric series proves otherwise.

How does it start? That's easy. When the appropriate time comes, the number to be recited at that time is recited. That wasn't so hard, was it? It works for both scenarios, counting up or down. — noAxioms

There is no first natural number to start with. It is logically impossible to have started reciting the natural numbers in descending order.

So when I hear Michael talking about the impossibility of a geometric series "completing" (so to speak) due to being unable to recite the terms in finite time... — fdrake

I don't think it impossible for a geometric series to complete. I think it impossible to have recited every natural number in descending order.

My issue is with supertasks, not with maths.

Repeating yourself three times, while ignoring my responses, does not help further the conversation. — Bob Ross

Your responses do not address my claim hence why I have to repeat it.

This is false; and does not follow from the former claim you made. — Bob Ross

It does follow. My belief that aliens exist makes the proposition "I believe that aliens exist" true. Therefore, your claim that "a belief cannot make a proposition true or false" is false.

Nah. That's an appeal to metaphysical or physical impossibility. Not logical impossibility! — fdrake

It is logically impossible to have recited every natural number in descending order because it is logically impossible to even start such a task.

So you’re claiming that it’s logically possible to have recited the natural numbers in descending order. That’s evidently absurd.

Which would be odd, seeing as such an object has a model in set theory — fdrake

Does it? Consider these two supertasks:

a. 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

b. 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

The first is reciting every natural number in ascending order and the second is reciting every natural number in descending order.

Does the second have a model in set theory? Is the second logically possible?

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 says nothing about the possibility of (b) and so says nothing about the possibility of (a) either.

I think you're missing Bob Ross's point.

A belief that "aliens exist" is not the same as a belief about the proposition "I believe that aliens exist" — ChrisH

I haven't claimed otherwise.

I have only claimed this:

"I believe that aliens exist" is true iff I believe that aliens exist. Therefore his conclusion that "a belief cannot make a proposition true or false" is false.

Ok. I asked for a reference. Now I have no idea what I'm supposed to conclude from this. — fishfry

I said this:

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. — Michael

You responded with this:

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

I gave you this reference:

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.

So I ask again: can you prove that it's metaphysically possible for me to halve the time between each subsequent recitation ad infinitum?

Maybe I'm not being clear, so I'll try one more time.

Here are two proposed supertasks:

a. 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

b. 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

Here is our premise:

P1. In both (a) and (b) there is a bijection between the series of time intervals and the series of natural numbers and the sum of the series of time intervals is 60.

However, the second supertask is metaphysically impossible. It cannot start because there is no largest natural number to start with. Therefore, P1 being true does not entail that the second supertask is metaphysically possible.

Therefore, P1 being true does not entail that the first supertask is metaphysically possible.

If you want to argue that the first supertask can end despite there being no largest natural number to end with, and so is metaphysically possible, then you need something other than P1 to prove it.

you are thinking that "aliens exist" is true or false relative to a belief — Bob Ross

No I'm not.

I'm only saying that "I believe that aliens exist" is true iff I believe that aliens exist. Therefore your conclusion that "a belief cannot make a proposition true or false" is false.

Yes that is a proposition, and whether or not it is true or false is independent of any belief about it — Bob Ross

As per Tarski’s T-schema, “P” is true iff P. As such, “I believe that aliens exist” is true iff I believe that aliens exist.

Might show it's logically possible tho. — fdrake

You think that the super task I described might be logically possible? How would you start it?

By providing a standard mathematical object which is infinite, has no final element, tends to an end state, and has an infinite number of occurrences ("steps"), but occurs in finite time. — fdrake

I’ll repeat something I said above:

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.

A clock ticks 1 time per second.
You start with a cake.
Every second the clock ticks, cut the cake in half.
Make the clock variable, it ticks n times a second.
The limit clock as n tends to infinity applies an infinity of divisions to the cake in 1 second. There is no final operation.

There's nothing logically inconsistent in this, it's just not "physical". — fdrake

The lamp starts off. Every time the clock ticks a lamp turns from off to on or from on to off as applicable. Thomson's lamp shows that this leads to a logical inconsistency.

I don't see that. At best he showed that one example is undefined. To prove something impossible it must be shown that there is not a single valid one. To prove them physically possible, one must show only a single case (the proverbial black swan). Nobody has done either of those (not even Zeno), so we are allowed our opinions. — noAxioms

Take my explanation of Thomson's lamp above:

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 metaphysically impossible. — Michael

The important part is in bold. If there is a problem with the button-pressing procedure, which there is in the case of A, then this problem remains even if the button is broken and doesn't actually turn the lamp on – it turning the lamp on and off isn't the reason that the supertask is impossible but simply demonstrates that the supertask is impossible.

And this problem remains even if rather than press a broken button we recite the natural numbers or even recite a single digit on repeat.

The reason that the supertask in Thomson's lamp is impossible isn't because of what operations are performed but because of how the operations are performed: halving the time between each subsequent operation ad infinitum. This is proven impossible, and as such Thomson's lamp proves that all such supertasks are impossible.

and all you have done is taking a claim that I am obviously going to deny — Bob Ross

So you deny that “I believe that aliens exist” is a proposition?

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 button was pushed when the lamp was off to turn it on and the lamp is off iff the button was pushed when the lamp was on to turn it off. The lamp "arbitrarily" being on or off at t₁ makes no sense.

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

B. 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

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

With (A) we can deduce that the lamp is on at t₁. But what of (B)? If no consistent conclusion can be deduced then its button-pressing procedure is proven metaphysically 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.