Further elaboration on the above..specifically in reference to the PNC.  This is an assumption of logic.  It's necessary to do logic, but that doesn't mean that the universe must necessarily work that way.  It's only necessary to do logic in that were it not assumed as an absolute semantics would be unworkable..and so the cheif difficulty of propositional logic - natural language- would be insurmountable.

There are, ofc, paradoxes arrived at by the pnc.  These are unresolvable by insistence upon the axiom no matter how well expressed or derived we find it to be in the world around us.  Every system has strengths and weaknesses..sometimes the weaknesses are brought about as a direct consequence of what gives them strength.  

In the case of propositional logic..it's strength is derived from it's ability to describe a universe like ours..but because of the assumptions required it's utterly uninformative with regards to any universe unlike our own. It;s not that no such universe could exist, or the the principles of propositional logic are in some sense absolute...it;s simply that logic as a system is unable to comment upon anything like that.

So there could be some universe where A could be both A and not A?!

The only one thing I can deduce from Steve's mental masturbation is that there are no physical infinities 'cause gawd says it's that way. And, if science ever proves a physical infinity, of coirse, it'll be axactly what gawd ordained

(February 15, 2018 at 3:53 pm)The Gentleman Bastard Wrote: The only one thing I can deduce from Steve's mental masturbation is that there are no physical infinities 'cause gawd says it's that way. And, if science ever proves a physical infinity, of coirse, it'll be axactly what gawd ordained

Shows you are not paying attention. Or that your perception does not match reality.

(February 15, 2018 at 3:35 pm)LadyForCamus Wrote: So there could be some universe where A could be both A and not A?!

We don't know, but if there were, we couldn't cogently describe it or anything about it by reference to a system of axioms derived from this universe where...so far as we know (and that's not exactly a full throated invocation of inviolable truth, now is it, lol)...such is not the case.

Logic, as a system, has all the limitations of this universe..plus whatever additional limitations arise from the axioms of the system itself -and- those who employ it.

Hell, even in -this- universe the system wasn't always formed as such. We didn't discover these "laws" all at once...and I don't know how we could know if we'd found the very last piece yet..today. Propositional logic may still be incomplete just in this world, and whatever else that's left, if there;s anything left, may invalidate some axiomatic x...just as it has in the past. The continued existence of paradox strongly implies that we haven;t gotten it quite right..somehow.

How many virtual photons are emitted and reabsorbed across the universe every Plank interval ??

I suppose it's just a large, very large number, but not infinite . . . .

(February 15, 2018 at 3:58 pm)Khemikal Wrote:

(February 15, 2018 at 3:35 pm)LadyForCamus Wrote: So there could be some universe where A could be both A and not A?!

We don't know, but if there were, we couldn't cogently describe it or anything about it by reference to a system of axioms derived from this universe where...so far as we know (and that's not exactly a full throated invocation of inviolable truth, now is it, lol)...such is not the case.

Logic, as a system, has all the limitations of this universe..plus whatever additional limitations arise from the axioms of the system itself -and- those who employ it.

Hell, even in -this- universe the system wasn't always formed as such.  We didn't discover these "laws" all at once...and I don't know how we could know if we'd found the very last piece yet..today.  Propositional logic may still be incomplete just in this world, and whatever else that's left, if there;s anything left, may invalidate some axiomatic x...just as it has in the past.  The continued existence of paradox strongly implies that we haven;t gotten it quite right..somehow.

I'm always cautious when 'possible' worlds are used to argue anything, they are at best imaginary worlds and don't represent reality (as in what actually happened or is happening)  at all.  We have no way of knowing if even a single atom could be in a different place from what is.

Imaginary worlds are the reserve of a mind that wishes to discuss non existent alternatives to the reality we know, or story makers who want to create some fantasy.

As for logic I agree it is rather limited to the same minds that make stuff up, It's our version of what we think should be given our understanding of what we observe the world around us to be. We seem to have lost the ability to simply say we don't know, logic simply will not help if we lack information on which to base it.

Amusingly, "possible worlds" smuggle in the logic of this world to begin with.  It;s a silent but immensely impactful assumption.  In that regard all possible worlds -are- this world.  This is why "possible world" conclusions are often little more than a mechanical function of whatever assumption we make in this world.

(February 15, 2018 at 12:14 pm)Neo-Scholastic Wrote: As another member noted - eternal and infinite are not synonymous. Eternal is a quality; infinite is a quantity. Eternal is the quality of something for which time has no meaning, like the Principle of Non-Contradiction. The PNC is universally true independent of time or even if there wasn't any time/space at all. Infinite describes the quantity of something, like the amount of patience it takes to read posts by Little Rik.

They're not synonymous, but not for the reason you stated.

Eternal means infinite time, not simply infinite.

Even if it's just one time slice or whatever, eternity can imply an infinite size of that time slice. Or an infinite passage or movement of that time slice.

(February 15, 2018 at 2:39 pm)SteveII Wrote:

(February 15, 2018 at 1:29 pm)Jörmungandr Wrote: 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?

(February 15, 2018 at 2:39 pm)SteveII Wrote:

(February 15, 2018 at 1:29 pm)Jörmungandr Wrote: 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.

(February 15, 2018 at 2:39 pm)SteveII Wrote: 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.

(February 15, 2018 at 2:39 pm)SteveII Wrote: 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.

(February 15, 2018 at 2:39 pm)SteveII Wrote:

(February 15, 2018 at 1:29 pm)Jörmungandr Wrote: 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.

(February 15, 2018 at 2:39 pm)SteveII Wrote:

(February 15, 2018 at 1:29 pm)Jörmungandr Wrote: 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. 

(February 15, 2018 at 2:39 pm)SteveII Wrote:

(February 15, 2018 at 1:29 pm)Jörmungandr Wrote: 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.