Sorry if I bothered you with my earlier comments. I plan to post an upgraded version of the original comment I mentioned above.

I doubt it’s because it’s an old post as I saw others do the same, right?

Anyways, I think you are an interesting guy and I look forward to reading more of your stuff.

Thanks,

AnonDoc

P.S. Is there a limit to the number of links per comment? Are we allowed to post pics, vids or tweets?

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]]>Thanks for the reply. I’m afraid that I was very clumsy when I tried to define V: what I should have said is the following:

V is a set containing ALL of the following elements: the sets that can be obtained by applying the operator P to the empty set a finite number of times.

This was the set that I had in mind. In this case it is easy to see that V contains an infinite number of elements. But it is still the case that V is a set, and that it does not contain itself. This is quite easy to see: suppose that V itself is an element of V. Then that would mean that the set V could be obtained by applying the operator P a finite number of times, say N times. But V also contains the set that is formed by applying P (N+1) times, since (N+1) is also finite, which gives a contradiction. So V is not an element of V.

I think the point is that you can still enlarge this set by a similar operation. In other words, you can consider the set V ∪ {V}. That is, you can apply the operator P to the set V, which gives you a new, bigger set. And then you can repeat the whole process “at a higher level”: you could define the following set:

V_2 is the set containing all of the following elements: the sets that can be obtained by applying the operator P to the set V a finite number of times.

and then you could do that again, to form a set V_3, and so on. You could even do the following: define

V_infty is the union of all the sets V_n, for all finite numbers n.

But then you can still apply the operator P to find a bigger set!

The real definition of V is supposed to include “all” these applications of P, but it’s hard to even say what this means – as soon as you write down some kind of useable definition, you find that you can enlarge this set by another application of P. This construction reminds me very much of Gödel’s incompleteness theorem.

I agree with your thoughts on the infinite past. With regards to the person counting down forever and reaching 0 now: I feel like the “reason” that they reach 0 now is because that is how they are defined in this thought experiment. I think that Craig would like to “define” them only as someone who has been counting down forever at some specified rate – but this doesn’t give a sufficient definition. What number would they reach yesterday? Or how about 100 years ago? If we only define them as “a person who has been counting down forever” then we cannot answer these questions. This is perhaps slightly counterintuitive – if we suppose that they have been counting down from some finite number at a specified rate, starting some finite time in the past, then this is sufficient to answer those kinds of questions. But when we change to the infinite past then we need some extra information to be able to answer these questions – for example, if we know what number they reached 100 years ago, then that would be sufficient to define this counting person.

I’m not sure whether this constitutes a violation of the principle of sufficient reason. If some object in a thought experiment is not fully defined, then we shouldn’t expect to be able to give an explanation for all of their actions. But the moment we do fully define them – for example, by saying that they reach 0 in the count now – it no longer seems mysterious that they reach 0 now.

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]]>I think that there is a very big error in Craig’s argument and I would be grateful if you could critique my argument?

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]]>You say:

///It seems to me that the set you create in this way, by “elaborating this hierarchy forever”, would more accurately be described as follows:

V is a set, each one of whose elements is a set obtained by applying the operator P to the empty set a finite number of times.///

V is a set that has that property. But that isn’t the definition of V. It’s only a necessary condition, but it isn’t sufficient. Consider level ‘1’ (or the second level), which is:

{∅}

It satisfies that definition, because it is ‘a set each of of whose elements is a set obtained by applying the operator P to the empty set a finite number of times’ (namely one time). But, surely, {∅} ≠ V.

Each element of V is obtained by finite iterations of P, but, crucially, V is the result of applying the P operator to the empty set *as many times as possible*. By ‘possible’, we mean mathematically possible, not physically possible. Obviously, I would run out of ink, or grow bored, or die, or whatever eventually if I tried to write it down. But nothing about the mathematics itself ever imposes a limit. And that is the relevant sens of possible here – it is mathematically possible.

Here is how Oystein Linnebo, a prominent philosopher of mathematics, puts it:

“But how far does this hierarchy extend? … We are often told that the hierarchy extends as far as possible. Vague though this may be, it is hard to see how a more definite answer could be provided. For given any attempt to pinpoint the extent of the hierarchy in a more definite way, it seems possible for the hierarchy to extend even further by allowing any objects from the proposed characterization to form a set. And since the hierarchy is supposed to extend as far as possible, this means that the proposed characterization cannot have been correct after all. So there seems to be something inherently potential about the set theoretic hierarchy. Given any attempt at a definite characterization, it turns out to be possible for the hierarchy to extend even further.” (The Potential Hierarchy of Sets, Linnebo, 2013, p 205).

So you say:

///We can also see that this avoids the problem of the “set of all sets”.///

But it only does that because it isn’t a definition of V. It’s a condition that applies to each level of V, but doesn’t characterise V. It’s true that no level contains itself (that’s one of the reasons set theorists originally came up with it). But that observation doesn’t avoid the problem I brought up (which Linnebo also explains above).

I agree with you to a large extent about what is happening when people talk about why the past cannot be infinite. You can’t ‘start’ counting backwards from ∞. The usual example given (which comes from Wittgenstein I think) is about finding someone who has been counting forever, who is just now saying “-3, -2, -1, 0. There, I finished!” Now, this is obviously a really weird idea. But if we say that he has been counting forever (in the sense that for any amount of time in to the past we pick, he was already counting at that point) it’s hard to see any contradiction. The best Craig brings up is that there would be a flouting of the principle of sufficient reason. Why had he not finished 5 mins earlier, or 2 years later? True, that seems to be impossible to answer. But picture someone else who finishes their countdown 5 mins earlier. We can say that this guy was always ahead of the first guy, by 5 mins. That is why he finished earlier. Again, no contradiction as such. If we picture one of Morriston’s angels beginning his count up the integers now, he would continue forever, and if his buddy waited five mins and then began his count, this would explain why his count is always behind. But why did they start at these points and not some others? That doesn’t seem mysterious. That is just when it started. So why is the reverse so mysterious? I’m not sure. It’s a hard area.

Not sure if that helps or not! Let me know.

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]]>I am a little bit confused about your example involving sets. To me, it seems like the construction you outline is actually parallel to the construction of the natural numbers.

Let me define an operator P which acts on sets in the following way: for any set S,

P(S) = S ∪ {S}

where ∪ means “union”. Then in your text, you basically begin with the empty set, and repeatedly apply the operator P. It seems to me that the set you create in this way, by “elaborating this hierarchy forever”, would more accurately be described as follows:

V is a set, each one of whose elements is a set obtained by applying the operator P to the empty set a finite number of times.

By saying that we are “elaborating this hierarchy forever”, we are basically saying that V contains all the sets that you can make by repeatedly applying the operator P. But you can only ever apply the operator P a finite number of times in each case. So in some ways this still seems like a “potential infinite”, but on the other hand V clearly contains an infinite number of elements. Thus V incorporates both an “actual” and a “potential” infinity, in some sense.

We can also see that this avoids the problem of the “set of all sets”. Consider the question: does V (as I defined it above) contain itself? Well, if it contained itself, this would mean that we could construct the set V by some finite number of applications, say N applications, of P to the empty set. But clearly this is not true: V also contains the set obtained by applying P (N+1) times. In this way, this construction using sets really mirrors the construction of the natural numbers, and the set V is similar to the set of all natural numbers, each one of which can be reached by counting upwards some finite number of steps.

You might wonder what happens if we consider the set V ∪ {V}. This would be like taking the set of natural numbers and then adding an additional element ∞. There is nothing wrong with this on the level of sets, but we cannot expect this additional element to follow the same rules as the other elements of the set – for example, it is not clear how to “count up or down” from ∞, whereas this is built into the construction of the natural numbers, which are the other elements of this set.

My own thoughts on this actual/potential infinity question are the following: as a mathematician, I was rather surprised to discover this concept, which I have never heard discussed by mathematicians. But the more I look into it, the more I feel like every time someone tries to back up a claim that there is something wrong with an “actual infinity”, their example of an “actual infinity” is really just an incoherent statement which tries to treat infinity as though it were a number.

Take the following argument, variations of which I have heard or read a few times. “The past cannot be infinite, because if it were, we could never have arrived at the present.” What does this mean? Suppose the past were infinite. To me, this means something like the following: for every natural number N, the world existed N days ago. Now, the argument that we would never reach the present goes something like this: start with N = ∞ and count backwards, and you will never reach 0, that is, you will never reach the present. But N cannot equal ∞, N is supposed to be a natural number! In other words, we are only supposing that the universe existed a finite number of days ago, but where this finite number can be anything we like – that is what it means to exist forever in the past! It certainly does not mean that there is some natural number ∞, and that the universe existed ∞ days ago, which is just incoherent.

I think that the reason for the injunction against “actual infinities” arises from this kind of confusion. People realise that there is something wrong with treating ∞ as a number, because it can lead to all sorts of contradictions. But people mistakenly think that these contradictions arise from the “physicality” or “actuality” of the situation they are imagining – for example, an infinite past would be an “actual” infinity because the past has some kind of ontological status which differs from, for example, the natural numbers. On the other hand, if you pretend that ∞ is a number and try to use it mathematically, you will run into contradictions just as fast. In the latter case, everyone can agree on what went wrong: you illegitimately used ∞ as if it were a member of a set (say the natural numbers) of which it is not a member.

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]]>Is there any way to discover/derive moral oughts? You refer to an intuition that you ought not to kidnap someone and tie them up in your basement. Do we just take it as given that we ought to do what our intuitions tell us we ought to do?

Cheers,

Nick

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