Actual Infinity in Reality?

[Angrboda]

I don't see how you get that this is a problem unless you assume that any series must have a first member.  In that case, you would be assuming what you need to prove.  If time is infinite, like the idea of spatial infinity, then the 4-space manifold that is time+space simply has no boundary in either direction, temporally.  I don't see how the idea that there is a cause/effect relationship between every successive part of that manifold undermines the possibility of it being infinite.

That is the question: does a series of events need a first member? Since this thread is on infinity and not something like the PSR, I will continue to limit it to just the infinity question. I brought up cause/effect because some deny there is as a way to avoid the question. I wanted to make sure we were not talking past each other. 

So the cause/effect objection doesn't point to a necessary contradiction in the hypothetical I propose. Then why did you bring it up?

I don't see that Hilbert's hotel demonstrates that an actual infinity is a metaphysical impossibility.  What metaphysical truth is it contradicting?  You seem to be arguing on the surface here, claiming that it is a metaphysical impossibility without showing any actual metaphysics.  It seems to me that you've simply argued in a circle.  Your metaphysics doesn't admit of an actual infinity, so to you an actual infinity is impossible (metaphysically).  In doing so you seem to have simply assumed what you need to demonstrate.  I don't offhand see how Hilbert's hotel advances your argument any.  To me, it's just a distraction.  The hotel produces results that seem absurd.  It's not clear that Hilbert's hotel demonstrates impossibilities.  You need to show the latter, not the former. 

Possible worlds semantics helps us identify metaphysical possibilities/impossibilities.

No, it does not. Possible worlds semantics is just a convention for expressing modal claims. It adds nothing to our understanding. It's just a different language for expressing the same thing as embraced by our modal assumptions. Translating from one set of conventions for expressing things to possible world semantics is akin to translating a sentence from English into French -- it doesn't add any meaning.

I set up four such thought experiments above. We clearly have contradictions that arise when comparing these possible worlds.

You are once again conflating the ability or inability to imagine something as being the same as demonstrating that something is or is not logically possible. It is the strength of imagination that we have the ability to conceive of impossible things. I can imagine that there is a possible world where God does not exist. Have I thus demonstrated that God is not a necessary being? No, I have not. If God is a necessary being and I imagine that God does not exist in a possible world, all I've shown is that my imagination is at odds with my assumptions. Your thought experiments don't add anything to the assumptions and conclusions you had prior to the thought experiment.

As long as we're on the subject though, allow me to make several notes,

1. Infinity, while treated as a number, is not a number in the sense that the counting numbers are. Therefore the equations you are presenting above have to be construed as set theoretic operations. As such, there is nothing contradictory about the set theoretic results. It only appears that way if you are construing the equations as normal numerical operations. Thus presenting the equations adds nothing and seems to serve only to mislead.

2. From what I understand of possible worlds semantics, the idea of comparing one possible world to another, different possible world is not supported. If you think it is, then I'd request that you show which possible world semantics you are referencing. If you can't compare possible worlds meaningfully, then attempting to even formulate Hilbert's hotel's operation in terms of possible world semantics is not possible.

3. Hilbert's hotel applies to sets that are countably infinite. If time is continuous and infinite, it would seem that the set of all possible moments is uncountably infinite. In that event, Hilbert's hotel simply wouldn't apply. As long as we're throwing around burden of proof questions, I think you are obligated to either show that time is not continuous, or that even if it is, that the set of all possible moments is a countable infinity. Otherwise, we can simply dispense with Hilbert's hotel, as it does not cover all the possibilities for a temporally infinite universe that I have raised. An objection which only applies to some of the possibilities but not all cannot possibly demonstrate that all cases are impossible.

But the real problem is that your position is that all of them are true despite the obvious contradictions. You have not shown why we should accept the contradictions other than to wonder if that's just the way it is. It seems to me that you have some burden of proof to shoulder if you are proposing ignoring obvious contradictions. 

Since you haven't actually shown any such alleged contradictions, I have great difficulty making sense of your complaint here. I'm supposed to refute the existence of contradictions you haven't demonstrated? That's ballsy, but ridiculous. I can't refute a case that you haven't made. So, no, I don't assume any burden of proof to show that something you claim exists doesn't exist. You need to first demonstrate the existence of these alleged contradictions. Once you've shouldered your burden of proof, we'll see what obligations I have in return.

If anything, Hilbert's hotel demonstrates that our understanding of the meaning of reference is undermined by an actual infinity, and that seems true. We can't form a sensible relationship between referents and the things they reference under operations involving infinity. Is that a metaphysical problem, I don't think so. 

Why isn't that question begging? The proposition is that an actual infinity does not exist. To dismiss thought experiments on the basis they don't work with actual infinities needs a little more support.

Well, first of all, you're moving the goalpost. The question is whether or not an actual infinity is logically possible. The claim that the proposition is whether an actual infinity actually exists is asking me to demonstrate that a specific actual infinity is in fact actual. Those are different standards. I don't know that I could prove that time is temporally infinite even if I wanted to do so. I never claimed as much. Only that the idea of a temporal infinity is consistent, both logically, and with known models of physics and cosmology. I believe I've done that. Your job as my interlocutor is to show that I've missed a contradiction which exists. In that context, I am suggesting that the so-called absurdity that results in the thought experiment may be a product of an incomplete set of intuitions about reference. It's a possibility. Your task, is to show that the absurdity in Hilbert's hotel is metaphysically real, not just a product of intuitional failure. You so far have not done so, and continue to talk around the problem rather than addressing it.

You need to go further than simply recounting Hilbert's hotel to show that any essential metaphysical assumption has been violated.  When you do, I think you'll find that you've simply assumed your conclusion.

It seems to me you are accepting an actual infinity as a brute fact. Can accepting a brute fact really be considered "logical".

No, as I just pointed out, I'm accepting that the hypothesis that time is temporally infinite is both logically and physically consistent. But that doesn't seem to be your point here. Your question as to whether what I'm doing is "logical" seems to be nothing more than a rhetorical smear. If you're reduced to such smears, I have to question what you hope to achieve with it? Treating something as a brute fact is neither logical nor illogical, so I can only assume that, instead, you are simply trying to suggest that I'm being irrational. I don't see that as a productive path to a convincing argument. It seems little more than an attempt to distract from the point I made, that you had not shown that any metaphysical assumption has been violated, and thereby avoid actually showing such a contradiction. 

Thanks.  I'll look at this in more detail at a later date.  According to Morriston, "It is controversial, of course, whether there is genuine absurdity in either case."  I don't see that you've eliminated the controversy so much as arbitrarily championed one side of it.

The other side seems to be to claim there are no absurdities. Hard to pick that argument apart when you just showed there is.

Um, yeah, whatever, Gumby. When you decide to calm down and actually show such contradictions instead of simply claiming that they exist, I will look forward to it.