Actual Infinity in Reality?

[Angrboda]

It is typically argued that the universe can't be past eternal because there would be no way to traverse an infinity of time.  However, such arguments about being unable to traverse an infinite past up to the present, as implied by such arguments as the one that says you can't count successively to infinity, rely on the A theory of time.  The idea of "traversing" an infinite past is incoherent on the B theory of time.  If the B theory of time is correct, and the universe is infinite in time, such arguments do not apply and you have the case of an actual infinite existing.  This leads to attempts to show that the universe is not past eternal by attempting to directly demonstrate that the universe's past is not infinite because the universe had a beginning.  This is done by invoking things such as the Borde-Guth-Vilenkin theorem and the standard interpretation of the big bang model which supposedly indicates that the universe has a beginning.  However there are theories such as Penrose's Conformal Cyclic Cosmology (see below) in which neither of these objections apply.  Conformal Cyclic Cosmology explains why we would find points "in time" that have the appearance of resulting from a universe that had a beginning in a universe which does not in fact have a beginning.  So, to the best I can tell, the idea that the universe is temporally infinite is consistent with a B theory of time and with some models of cosmology.  So, ultimately, it doesn't appear that the case that you can't have an actual infinite has been made.  (Regarding Hilbert's hotel, supposedly the results are absurd.  This can mean several things.  It can mean that the result is counter-intuitive, or it could mean that the result is logically impossible.  I don't off-hand see that Hilbert's exercise demonstrates anything about logical impossibility so much as it is just showing that such things seem to defy our normal intuitions.  I don't see the latter as any kind of argument that actual infinities don't exist so much as a demonstration that we aren't natively well equipped to think about such things.  That latter fact is of little consequence.  Quantum mechanics presents results that are equally absurd in that sense, that doesn't make quantum mechanics wrong.  If you think Hilbert's hotel demonstrates something more substantial than this, I'd appreciate someone drawing out the relevant connections, because I don't see them.)

I don't think the B Theory of time solves the underlying problem of having a series of cause/effect relationships. It seems to me that even if all points of time are equally real, they are still ordered by a structure we call cause/effect--a tangible series of objects we can use in thought experiments. [NOTE: I say this to start because there are some here who deny even this].

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.

Perhaps a variation of Hilbert's Hotel:

We can conceive of a possible world (much like the one you are proposing) with a beginningless series of discrete successive events of equal duration leading up to the present (real or perceived present). 

[ ...en, ... e5, e4, e3, e2, e1, e0]

We can conceive of another possible world with exactly the same events in the same order, but in between each of those events, another event occurs.

[ ...en, En, ... e5, E5, e4, E4, e3, E3, e2, E2, e1,E1, e0]

In this series, an infinite number of additional events have been added to an already infinite series of events. Are there more events? No. Infinity + infinity = infinity. We can also do the subtraction example from Hilbert, and imagine all the events prior to e3 could have been left out of the chain. 

[e3, e2, e1, e0]

In this series, we have subtracted an infinite number of events from an infinite number of events. Infinity - Infinity = 4.  Alternately, every other event could have been left out:

[ ...en, ... e4, e2, e0]

In this series, we have left out an infinite number of events from an infinite number of events. Infinity - Infinity = Infinity. 

This is not just "counter-intuitive". Actual infinities of real objects leads to absurdities (metaphysical impossibilities). Therefore an actual infinite is not logically possible.

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. 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. 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.

[Example language from a paper from Wes Morrison - http://spot.colorado.edu/~morristo/EndlessFuture.pdf]

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.