Actual Infinity in Reality?

[Whateverist]

This is all to argue that the past can't be infinite, therefore God made it on His cosmic watchmaker's bench.

Not really. Thomas Aquinas, at least, felt it was indeterminate and did not matter with respect to the 2W once you account for the distinction between accidentally and essentially ordered sequences. Why do you believe it impossible to discuss have a purely metaphysical question without delving into the theological implications? Seems to me like yer itchin' for a fight.

Fight?  No, an interminable argument initiated by our local theists regarding the possibility of an eternal past that is motivated by their need to think it was God who turned the light switch on all at once.  But you'd have to go back to that thread to understand what motivated Steve to reshape the issue in a form he thought more salvable.  Of course that hasn't worked so well thus far.

[Neo-Scholastic]

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.

[The Grand Nudger]

A distinction that makes no difference.  Infinite time is eternal time.  There is an eternity of moments in every minute just as there are infinite points in every inch.  The definition of eternal is, literally..infinite time.......lol?

[pocaracas]

Also remember that, as long as everything happens at the speed of light in vacuum, time doesn't pass.

[SteveII]

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].

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. 

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

[Angrboda]

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.

I'm not sure I agree with you here. Regardless, using eternal in the sense of time having no meaning for something is common in theological discussion of God, I'm just not sure that one can come up with a sensible metaphysics in which to ground such a notion. Failing that, the notion of eternal that you are promoting here seems little more than empty words.

eternal,

A. adj.
1.

a. Infinite in past and future duration; without beginning or end; that always has existed and always will exist: esp. of the Divine Being.

b. By those who hold that time, i.e. the relation of succession, pertains merely to things as viewed by finite intelligence, and not to absolute reality, the word as used of God or His actions is interpreted in the sense: Not conditioned by time; not subject to time relations.
    Phrases in which the word has properly this sense are, however, often used in religious language without any definite recognition of the metaphysical theory which they imply, being taken as figurative expressions of the divine omniscience.

[polymath257]

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].

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. 

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

When you ask if there are 'more', there are, again, two senses for this.

There is the sense of subsets. When you remove elements, the result is a subset of the original. When you add elements, the original is a subset of the result. We usually say that subsets have 'fewer' elements than the supersets.

In all your cases, the subset relation correctly describes the notion of 'more' that you are seeking.

When when talking about cardinality (infinite in your case, is the countably infinite cardinality), subtraction is not well defined. That means that different situations can lead to different answers. There is a similarity with division by 0: 0*3=0, so 0/0=3. But 0*5=0, so 0/0=5. That isn't a contradiction. it is simply that you used division inappropriately. In the case of infinite sets above, you used subtraction inappropriately.

So, yes, this *is* just counter-intuitive: subtraction is not well-defined. That's all.

---

I'd also point out that there is a notion of *set* subtraction (A-B is the set of things in A and not in B) that is more appropriate these situations.

The 'size' of A-B is not just determined by the sizes of A and B separately, but also how they 'fit' into each other. This is why it works better for the Hilbert Hotel.

[Neo-Scholastic]

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.

I'm not sure I agree with you here.  Regardless, using eternal in the sense of time having no meaning for something is common in theological discussion of God, I'm just not sure that one can come up with a sensible metaphysics in which to ground such a notion.  Failing that, the notion of eternal that you are promoting here seems little more than empty words.

eternal,

A. adj.
1.

a. Infinite in past and future duration; without beginning or end; that always has existed and always will exist: esp. of the Divine Being.

b. By those who hold that time, i.e. the relation of succession, pertains merely to things as viewed by finite intelligence, and not to absolute reality, the word as used of God or His actions is interpreted in the sense: Not conditioned by time; not subject to time relations.  
    Phrases in which the word has properly this sense are, however, often used in religious language without any definite recognition of the metaphysical theory which they imply, being taken as figurative expressions of the divine omniscience.

Yes, I am using connotation (b). I see (a) as common usage and confusing for a philosophical discussion. I do not see it as "empty words" to say, for example, that there are eternal truths.

[Edwardo Piet]

Fight?  No, an interminable argument initiated by our local theists regarding the possibility of an eternal past that is motivated by their need to think it was God who turned the light switch on all at once.  But you'd have to go back to that thread to understand what motivated Steve to reshape the issue in a form he thought more salvable.  Of course that hasn't worked so well thus far.

The whole "There had to be a beginning... but not for God because... he makes stuff and stuff." thing just makes me wanna laugh my man nipples right off.

[The Grand Nudger]

 I do not see it as "empty words" to say, for example, that there are eternal truths.

What's the difference between an eternal truth and truth?