Actual Infinity in Reality?

[Edwardo Piet]

An eternal truth is true forever!

He's probably brain farting and he's trying to say that some things are true in all possible worlds. Which is more of a modal logic thing.

Although yeah, absolute reality is ultimately eternal and modal logic refers it. The noumenonal realm and all that jazz.

[The Grand Nudger]

Any true statement is contingent upon the status of it's conditional operants.  An "eternal" truth, then..is not a comment on duration of truth, but on the maintained status of it's conditional operants. The modifier has been misapplied.

[LadyForCamus]

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.

Just some philosophical masturbation here:  Are the laws of logic considered descriptive?  If so, how could they exist outside of space-time?  Without space-time, there couldn’t be an A to be either A or not A.  Right?

Infinite describes the quantity of something, like the amount of patience it takes to read posts by Little Rik.

The Big Bang is what happens to my head when I read one of Little Rik’s posts.  😏

[Edwardo Piet]

Everything is A or not A

Time must have began along with existence. The 'time' that science can study... maybe not.

Time itself, as us lay people speak of it, if it's real at all and non-illusory, must have began along with existence and just be part of it. Why? Because nothing can exist BEFORE time. Why? Because the very concept of 'before' requires time.

And for something to begin at all, you need time. Because something beginning requires something happening. And happening requires temporal movement or IOW: Time.

As for space well, science says it's wrapped in with time... but again they study our experience of it rather than 'time itself' or whatever. As far as I am concerned either 'time itself' is something tied in with existence and logic or it's basically an illusion and we think we experience stuff 'happening' but that's just how our brain translates whatever the hell existence really is.

Space.... absolute space.... as in completely empty space, not empty space teeming with acitvity... but absolutely nothing. Doesn't exist.

So if time and space are truly tied together is that a clue that time isn't real either?

I think it's an illusion in one sense and it's part of logic in another sense. For example, any real sort of time travel is impossible because it violates what time really is if we assume it's real. Despite science saying otherwise.... because science isn't really saying otherwise. Because it's talking about something else. If atoms are by definition the smallest possible objects that are undividible and non-splittable then science never truly split an atom. They split something else slightly bigger and called it an atom. Of course, that version of an atom has become much more well known than the original definition.

In the same way, if time travel to the future but not the past, is at all possible according to science.... it's not talking about time travel in the way we normally think of it.

Look at it this way, the experience of space and time are always interconnected.... Well.... errr.....

To quote the stand-up comedian Ross Noble when heckled by an audience member asking him if he's ever time travelled: "Yeah.... mostly forwards.... and at the same speed as everyone else."

But yeah, the reason why actual real time travel into the future as we normally think of it is NOT logically possible is..... that would be us travelling to a place that doesn't exist yet.... which.... literally makes no sense. If it doesn't exist yet or hasn't happened yet.... then it's unaccessible. And whatever we access or whatever kind of travelling we do and whereever we seem or appear to be going.... it's not to the future.

Maybe if we were able to see the present a lot sooner than most people recognize it.... as there's usually a delay between noticing something.... science would call that time travel

"Hey 3000 AD has already happened but most people take 1000 years to notice it!"

"WUT people don't live for 1000 years!"

"Okay.... 1 nanosecond from now is the true present and most people are focusing on what actually happened a nanosecond ago. I am a time traveller from the future... one nanosecond ahead! Meaning I'm actually in the true present and you are a nanosecond behind in temporal awareness like everyone else! Okay I'm not actually from the future because I'm in the true present and so are you and most people who just haven't noticed it yet, and I'm a fraud and not a time traveller BUT SHUT THE FUCK UP! SCIENCE BITCH! YEAH!"

[Amarok]

The laws of logic are descriptions of how reality is not magic laws commanded from on high . Or the magic qualities of a space wizard .

[SteveII]

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.

Here are three different quotes to consider:

In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity [LINKED TO BELOW] is included) of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given, actual, completed objects. This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and each individual result is finite and is achieved in a finite number of steps. https://en.wikipedia.org/wiki/Actual_infinity

In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.[1] https://en.wikipedia.org/wiki/Axiom_of_infinity

Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects  with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.[1][2][3] https://en.wikipedia.org/wiki/Abstraction_(mathematics)

All emphasis added.

Therefore the the abstraction of actual infinity (from the first quote) is based on an axiom that there exists at lease one infinite set. Appropriately, an axiom in mathematics is defined as: a statement or proposition on which an abstractly defined structure is based. In case we are still unclear, the third quote defines an Abstraction. I highlighted the key theme all the way through this. Abstract. 

You have NOT made an argument (in this or the previous thread) where you show how this abstract concept in mathematics applies to the real world. It should be simple to propose some thought experiments or examples for us all to consider. I understand your point that the concept in mathematics exists--now you need to provide some evidence that it applies to real objects. Re-iterating infinite set theory from mathematics will not further this discussion.

[polymath257]

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.

Here are three different quotes to consider:

In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity [LINKED TO BELOW] is included) of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given, actual, completed objects. This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and each individual result is finite and is achieved in a finite number of steps. https://en.wikipedia.org/wiki/Actual_infinity

In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.[1] https://en.wikipedia.org/wiki/Axiom_of_infinity

Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects  with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.[1][2][3] https://en.wikipedia.org/wiki/Abstraction_(mathematics)

All emphasis added.

Therefore the the abstraction of actual infinity (from the first quote) is based on an axiom that there exists at lease one infinite set. Appropriately, an axiom in mathematics is defined as: a statement or proposition on which an abstractly defined structure is based. In case we are still unclear, the third quote defines an Abstraction. I highlighted the key theme all the way through this. Abstract. 

You have NOT made an argument (in this or the previous thread) where you show how this abstract concept in mathematics applies to the real world. It should be simple to propose some thought experiments or examples for us all to consider. I understand your point that the concept in mathematics exists--now you need to provide some evidence that it applies to real objects. Re-iterating infinite set theory from mathematics will not further this discussion.

What this has shown is that an actual infinity is not a self-contradictory thing.

Whether it actually exists in the real world is unknown.

But there is no *logical contradiction* in this concept.

[The Grand Nudger]

@Ham

Extrapolating on silliness and still getting it wrong, "because logic", I see.  I could have sworn we've had this convo before?  Time dilation is not only real, it's observable and has been observed.  So you'll have to take it up with Sergei, who absolutely did arrive at a place that did exist 20milliseconds offset..Earth.  A greater offset would be a simple function of greater distance traveled under greater speed over greater time. It's beyond our practical ability at present, but a part of observable reality all the same.

[polymath257]

All emphasis added.

Therefore the the abstraction of actual infinity (from the first quote) is based on an axiom that there exists at lease one infinite set. Appropriately, an axiom in mathematics is defined as: a statement or proposition on which an abstractly defined structure is based. In case we are still unclear, the third quote defines an Abstraction. I highlighted the key theme all the way through this. Abstract. 

You have NOT made an argument (in this or the previous thread) where you show how this abstract concept in mathematics applies to the real world. It should be simple to propose some thought experiments or examples for us all to consider. I understand your point that the concept in mathematics exists--now you need to provide some evidence that it applies to real objects. Re-iterating infinite set theory from mathematics will not further this discussion.

Yes, I get that. I will happily admit (as I have done repeatedly) that we do not *know* if there is an actual infinity in the real world. None has been given, I agree.

But that is distinct from the question of whether it is *logically contradictory*. The fact that it works in the abstract is enough to show there is no *logical* problem with the concept.

Also, all claimed 'dis-proofs' are based on a confusion between 'size' in terms of subsets, 'size' in terms of cardinality, and attempting to 'subtract' when it isn't well-defined. In other words, the claimed 'dis-proofs' simply don't manage to disprove the concept.

[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.