The contradiction is the result of the fact that there is no criterion set for the final step in your process - i.e., the end state is undefined. — Ludwig V
My point is that I think that the disagreement between you and @fishfry is about different ways to make the same point.That's precisely why supertasks are impossible. — Michael
Quite so. Wittgenstein made much of the endlessness of infinity and asked how it was possible. You may know what his answer is. If you don't, it is easy to look it up. (It would be far too long to try to outline it in this context and you likely know anyway.There's nothing wrong with defining, or performing, a recursive function. There is a problem with claiming that it is possible to have completed a recursive function. — Michael
My point is that I think that the disagreement between you and fishfry is about different ways to make the same point. — Ludwig V
Conclusion: set theory is in violation of the law of identity. I've explained to you why this is the case. Do you agree with me? — Metaphysician Undercover
Surely, the contradiction is the result of the lack of any definition of the terminal state. If the terminal state could be a plate of spaghetti, why couldn't be a lamp that is neither on nor off?
I really cannot see what you two are arguing about. Why does the difference matter? — Ludwig V
The plate of spaghetti is a great dramatic way of making the point that there is no definition. But the series is defined on the basis that its limit is 1. You can't derive 1/2 from a plate of spaghetti. — Ludwig V
My point is that I think that the disagreement between you and fishfry is about different ways to make the same point. — Ludwig V
So why don't you conclude that the use in the context of the law of identity violates the use in the context of set theory? It seems to be an arbitrary choice. — Ludwig V
The meaning of "same" depends on its context. — Ludwig V
Temporal priority is not logical priority.I'd agree except that the law of identity was first, set theory came along after. — Metaphysician Undercover
There are plenty of ways to formulate that law without using the word "same". In any case, "same" in that context just means "same object", so it isn't absolute. moreover, If you drive my car, you don't drive it at the same time.Well, if the law of identity is an obvious self-evident tautology, then it appears like there must be something wrong with set theory if it's in contradiction with what is obvious. — Metaphysician Undercover
There is no unqualified sense of "same".The thing is that everything about it is not the same, only those named qualities are the same, and that's why it's incorrect to say that it is "the same" in that unqualified sense. — Metaphysician Undercover
We agree!Not everything about the two is necessarily the same, only the stipulated required qualities. — Metaphysician Undercover
There is no unqualified sense of "same".So it is incorrect to say that the two sets are the same, in the unqualified sense, — Metaphysician Undercover
I'm sure that Aristotle would not object to my regarding that as not a logical argument.He (sc. Aristotle) claimed that the law of identity was necessary to battle against sophists who could logically demonstrate absurdities. — Metaphysician Undercover
I'm sorry. I was talking about the convergent series. Didn't checkDon't follow. The limit of 0, 1, 0, 1, ... can not be 1. Nor can it be 0. It's a sequence that has no limit. — fishfry
Perhaps not. But if the last term in the series is not defined, contradictions are likely to follow from the attempt to identify it. Equally, if something gives rise to a contradiction, the definition will be faulty. So, if you are right, I need to ask why it matters.I do not think Michael and I are making the same point. — fishfry
Not quite. The code specifies a process which must take time. The function does not.The code here is effectively the same as a recursive function. — Michael
Thank you. I must have got confused.I'm arguing that supertasks are metaphysically impossible. He's arguing that supertasks are metaphysically possible. — Michael
I'm sorry. I was talking about the convergent series. Didn't check — Ludwig V
I do not think Michael and I are making the same point.
— fishfry
Perhaps not. But if the last term in the series is not defined, contradictions are likely to follow from the attempt to identify it. Equally, if something gives rise to a contradiction, the definition will be faulty. So, if you are right, I need to ask why it matters. — Ludwig V
Yes. The exact status of 1 or 0 in these cases is more complicated than I realized.There is no last term in any infinite sequence. There may (or may not) be a limit. Big difference. — fishfry
So can you help me to describe the role of 1 in defining the series 1/2, 1/4, ... when the limit state is 0? (Or indeed when it's the other way round?)Even if you insist that the terminal state must be either 0 or 1, there is no logical way to prefer one over the other. — fishfry
You can define the terminal state to be on, off, or a plate of spaghetti and be consistent with the rules of the game.
— fishfry
No you can't. — Michael
I addressed this in my initial defence of Thomson here, and even more clearly below. — Michael
You're claiming that "a plate of spaghetti" is a coherent answer to the question "is the lamp on or off after two minutes?" — Michael
So I think the confusion is yours. — Michael
He discusses the sequence and its sum, but only to show its irrelevancy, hence the earlier quote. — Michael
From his paper:
What is the sum of the infinite divergent sequence +1, -1, +1, ...? Now mathematicians do say that this sequence has a sum; they say that its sum is 1/2. And this answer does not help us, since we attach no sense here to saying that the lamp is half-on. — Michael
So, the terminal state not being defined does not prevent me defining one arbitrarily?But there's no need for there to be any logical relation between the sequence itself, and the arbitrarily-defined terminal state. — fishfry
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, according to Russell's recipe. 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 seems impossible to answer this question. 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.
So, the terminal state not being defined does not prevent me defining one arbitrarily?
Isn't it the case that there is a requirement - that the terminal state not be defined by the function. — Ludwig V
Michael, This post may be of interest to you. — fishfry
I think you'll find that's because it makes no sense to answer the question.It makes no sense to answer this question with "a plate of spaghetti" or "1/2". — Michael
I understand that. What seems important to me is that the convergent series is the result of a calculation which involves 0 and 1, while "0,1, 0, 1, ..." doesn't involve any calculation at all. You could also have a series "a, b, a, b, ..." or "fish, chips, fish, chips, ..." The calculation involves numbers, but "0, 1, 0, 1, ..." only involves numerals.That completion is just as arbitrary as any other. But it has one supreme virtue: 0 happens to be the limit of the sequence. So that's why I call it natural. — fishfry
I think you'll find that's because it makes no sense to answer the question.
In other words, it also makes no sense to answer the question with "on" or "off". — Ludwig V
Exactly. The contradiction follows from the fact that no final state is defined.The lamp is either on or off at t1. The fact that it makes no sense for it to be on and no sense for it to be off if the button has been pushed an infinite number of times before that is proof that it makes no sense for the button to have been pushed an infinite number of times. — Michael
I understand that. What seems important to me is that the convergent series is the result of a calculation which involves 0 and 1, while "0,1, 0, 1, ..." doesn't involve any calculation at all. You could also have a series "a, b, a, b, ..." or "fish, chips, fish, chips, ..." The calculation involves numbers, but "0, 1, 0, 1, ..." only involves numerals. — Ludwig V
Temporal priority is not logical priority. — Ludwig V
There is no unqualified sense of "same". — Ludwig V
It makes no sense to answer this question with "a plate of spaghetti" or "1/2 — Michael
He lingered on the first step, marked "1," for 30 seconds, soaking in the enchanting energy coursing through his veins. Moving to step "2," he paused for 15 seconds, feeling lighter and quicker, like a feather in descent. Driven by an irresistible urge, he continued to step "3," then "4,", and so on, each time halving his rest period. — keystone
The infinite staircase appears to only allow one to traverse it in one direction. It simultaneously exists… — keystone
So there is no "logical" way to connect the sequence, with its arbitrary terminal state, which you can define as on or off. — fishfry
I take the point. I may not have stated it accurately enough, but the crucial thing, it seemed to me, is to realize that the limit is part of the definition from the start - not, as I think you're saying, something that is worked out from the sequence itself.Rather, you are given the sequence; and given the limit; and you can apply a formal definition to see that 0 is indeed the limit of the sequence. It's conceptually sort of the other way 'round from thinking that the limit is the result of some logical process applied to the sequence. — fishfry
If I say that Hesperus is Phosphorus, I am saying that they are the same object.I believe that what is attempted with the law of identity is to express an unqualified sense of "same". You seem to think it fails. Why? — Metaphysician Undercover
You are right, Language is a great trap here. I would like to use "endless" or "endlessly" and even "endlessness" instead. That would make it more difficult to talk about conclusions. But we are lumbered with a world which uses "infinity". Natural language allows this, but has no guard rails to prevent us from talking nonsense.“The infinite” or “infinity” as a noun, is best used for dramatic effect. It’s not a thing, like a noun is best employed. “Infinitely” as an adverb, sets out some activity that, by definition, cannot conclude. Thereby banishing all finitude, which marks conclusion, such as a step, or a series of steps, or a noun. — Fire Ologist
The difficulty here is that it is possible to defined an infinite series in a finite frame, which leads people to think of apply the abstract idea to the physical world. Sometimes that works, as in physics, so we can't just say that such ideas have no place in the physical world.But the infinite finds no home, no place in the physical world, — Fire Ologist
Yes, we do. We don't find them by failing to count them, but through various arguments. The proofs that π or sqrt(2) or that there is no largest natural number are all well established. So is the possibility of a convergent series.But you never find the infinite. There need be no infinitely small fraction. — Fire Ologist
True, if you are thinking of a staircase. But nobody would contest that. But if you think of the distance between my eyes, you can certainly divide that by 1/2 or 1/4 or...There is no such thing as a half step. — Fire Ologist
But if you think of the distance between my eyes, you can certainly divide that by 1/2 or 1/4 — Ludwig V
Unlike Zeno's thought experiments, which deal with examples of ordinary motion — SophistiCat
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