Starship Asterisk*

by **charlieo3** » Sat Oct 22, 2011 2:10 pm

"Eternity is very long, especially towards the end." -- Woody Allen (The same could be said for 'infinity.')

by **bystander** » Tue Sep 13, 2011 3:31 pm

mst66186 wrote:I believe I should have written 'cardinal' rather than 'integer' although the difference seems to be slight, cardinals are specifically for talking about the number of elements in a set, as in the discussion above (so the number of Plank volumes in the observable universe would be a cardinal number). I have looked at the definition of cardinal number, and they are transfinite. I have one last question on this topic before I give it a rest and get back to the lectures. I can't find the answer elsewhere on the Internet. I don't know whether you can answer this for me or whether I need to put this on a math forum, but please let me know either way:

is it also true that there is no x, where x is a finite cardinal number, such that the successor of x is Aleph Null?

Unlike ∞, Aleph Null is a member of the set of cardinals, so the above is not true by default, although I believe it is true. That is probably what I should have put originally. Thanks again for your time.

I'm not really sure what your point is here, but to get a definitive answer you will probably need to go to a math forum. IMHO you are not looking for the cardinality of Planck units, but rather a count of them. The set of counting numbers (aka natural numbers) is more appropriate.

by **mst66186** » Tue Sep 13, 2011 11:21 am

bystander wrote:...
∞ is not an integer, so your statement "for all numbers x, where x is a member of the set of integers, there is no x such that the successor of x is ∞" is true by default, because ∞ is not a part of the set. That does not mean there exists a greatest integer.

I believe I should have written 'cardinal' rather than 'integer' although the difference seems to be slight, cardinals are specifically for talking about the number of elements in a set, as in the discussion above (so the number of Plank volumes in the observable universe would be a cardinal number). I have looked at the definition of cardinal number, and they are transfinite. I have one last question on this topic before I give it a rest and get back to the lectures. I can't find the answer elsewhere on the Internet. I don't know whether you can answer this for me or whether I need to put this on a math forum, but please let me know either way:

is it also true that there is no x, where x is a finite cardinal number, such that the successor of x is Aleph Null?

Unlike ∞, Aleph Null is a member of the set of cardinals, so the above is not true by default, although I believe it is true. That is probably what I should have put originally. Thanks again for your time.

by **BMAONE23** » Sun Sep 11, 2011 7:28 am

But what if most of the asterisk posts come from one person with an extremely fractured multiple personality dysfunction? How would you go about proving that Neufer, Owlice, Beyond, Chris, and other posters aren't in fact different personalities of the same person? Perhaps Chris and Harry are different sides of the same coin?

by **Beyond** » Sat Sep 10, 2011 10:24 pm

Vice-versa on that Chris. I would imagine that believing that one exists in the first place, takes a certain amount of faith, based on the experience. And then the faith grows as each experience comes along. Until it becomes 'normal'. Such as chatting with others on the Good Ship Asterisk. Of course if one thinks it's just their imagination, try changing it. If it won't change, then it's not just imagination, i should think.

by **Chris Peterson** » Sat Sep 10, 2011 9:24 pm

Beyond wrote:Existence........I am making this post. Therefore i exist. I am a very small part of something that is either finite, or, infinite. That's about all that is possible to say on the subject for now. Have an infinitely nice, finite day :!:

I don't buy it. I'm reading this post, so I know that I exist, but it certainly doesn't demonstrate to me that you do. You- along with all your posts- could just be a figment of my imagination.

by **Beyond** » Sat Sep 10, 2011 9:13 pm

Existence........I am making this post. Therefore i exist. I am a very small part of something that is either finite, or, infinite. That's about all that is possible to say on the subject for now. Have an infinitely nice, finite day

by **bystander** » Sat Sep 10, 2011 4:07 pm

mst66186 wrote: How about this one:
If entity x 'Exists' then you can possibly find an instance of entity x occurring in the real world.
P = entity x 'Exists'
Q = you can possibly find an instance of entity x occurring in the real world

Now Not P, therefore not Q gives us "entity x does not 'Exist', therefore you can NOT possibly find an instance of entity x occurring in the real world"

Much better! LOL

That, however, does not give you a test for existence.

by **mst66186** » Sat Sep 10, 2011 12:13 pm

bystander wrote:Your Test for Existence is a logical fallacy, denying the antecedent.

If it can be found, then it exists. It hasn't been found, therefore it doesn't exist.

Symbolically: If P, then Q. Not P, therefore not Q. This simply doesn't hold.

Post 1 of 2 (it's very late where I am).

I'd like to say thank you to bystander for taking the time to think about what I'd put and taking the time to point out the mistake.

Test for Existence: an entity x 'Exists' if you can find an instance of it occurring in the real world.
If P, then Q.
P = you can find an instance of entity x occurring in the real world
Q = entity x 'Exists'
The error is denying the antecedent.
Not P, therefore not Q.

However, I didn't say in natural language that if you can NOT find an instance of entity x occurring in the real world the entity x doest not 'Exist' so I'd like to try and find a way of expressing the test for existence that can be expressed better formally.

How about this one:
If entity x 'Exists' then you can possibly find an instance of entity x occurring in the real world.
P = entity x 'Exists'
Q = you can possibly find an instance of entity x occurring in the real world

Now Not P, therefore not Q gives us "entity x does not 'Exist', therefore you can NOT possibly find an instance of entity x occurring in the real world"

Much better! LOL

by **Chris Peterson** » Fri Sep 09, 2011 1:54 pm

mst66186 wrote:I do wonder if Chris L Peterson couldn't cite an occurence of infinity inside the observable universe ('it is there... but you can't see it') and you're playing the 'just because I can't find it, that doesn't mean it doesn't exist' card.

How would you propose to ever offer an observational example of spatial infinity? I don't see how that is even possible. That doesn't mean, however, that you must dismiss the idea of an infinite Universe. If the Universe is infinite, that requires a model with certain characteristics which are observable. Thus, specific observations can lead to the logical conclusion that the Universe is of infinite extent, even if we can never make that observation directly. This is, of course, no different from many scientific conclusions that come from indirect observation.

by **mst66186** » Fri Sep 09, 2011 10:35 am

Hmmm... lots to think about there thanks. I'm going to go over my predicate logic before posting a reply to the second part, but there IS a reply. Meanwhile the first part is very interesting too: Cantor had to define integers as infinite because of the axiom of infinity...

I do wonder if Chris L Peterson couldn't cite an occurence of infinity inside the observable universe ('it is there... but you can't see it') and you're playing the 'just because I can't find it, that doesn't mean it doesn't exist' card. You can probably use the same two arguments to cling on to the existence of a supernatural deity! I thought that if it was an open and shut case that infinty Exists, somone could just point to a example of infinity outside pure math and the argument would be settled immediately without having to resort to word play. I didn't think that would be too much to ask of a scientist of any stripe.

by **bystander** » Fri Sep 09, 2011 5:05 am

mst66186 wrote:Well, I have an open mind and I'm prepared to change my point of view. Your task, as an academic, is to make me change my mind! I like to think I'm a rationalist, so if you are too we should be able to reach an agreement, or agree to differ.

I'm not an academic. I'm just a pathetic retiree, an innocent bystander.

I'm not at all convinced your mind is open, and when it comes to mathematics, I must admit mine isn't. Mathematics is a rigorous subject and not subject to whimsy.

Wikipedia wrote:In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. ... The elements of a countable set can be counted one at a time — although the counting may never finish, every element of the set will eventually be associated with a natural number.

What this means, in mathematics, is that it can be demonstrated that there is a mapping function that can map the set to the set of natural numbers (not necessarily that they have been counted). The set of integers has been mapped, as has the set of rational numbers. They are, therefore, countable.

In set theory, an infinite set is a set that is not finite, that is to say, it is a set without bounds. For the set of integers this means there is no greatest integer. For whatever integer you choose, no matter how large, there is always one larger.

∞ is not an integer, so your statement "for all numbers x, where x is a member of the set of integers, there is no x such that the successor of x is ∞" is true by default, because ∞ is not a part of the set. That does not mean there exists a greatest integer.

Your Test for Existence is a logical fallacy, denying the antecedent.

If it can be found, then it exists. It hasn't been found, therefore it doesn't exist.

Symbolically: If P, then Q. Not P, therefore not Q. This simply doesn't hold.

The most difficult subjects can be explained to the most slow-witted man if he has not formed any idea of them already; but the simplest thing cannot be
made clear to the most intelligent man if he is firmly persuaded that he knows already, without a shadow of doubt, what is laid before him. — Leo Tolstoy

by **Beyond** » Fri Sep 09, 2011 3:46 am

mst66186 wrote:Oh I am SO slow! A line without an end is defined by cDonald's Theorem. http://www.youtube.com/watch?v=Qq4KKnYsWWo

I won't bother to enter the contest. I hardly ever make toast anymore.

by **mst66186** » Fri Sep 09, 2011 3:32 am

babaonet wrote:Can anyone imagine a line which has no end?
I can't imagine anything which has no end. In other words I don't believe absolute infinity.
I am not sure whether the concept of infinity is relative. However, i feel so. Can anyone kindly put their valuable comments on the following two:
1. The universe is infinite.
2. The universe is relatively infinite to us.

Oh I am SO slow! A line without an end is defined by cDonald's Theorem. http://www.youtube.com/watch?v=Qq4KKnYsWWo

by **mst66186** » Fri Sep 09, 2011 3:25 am

bystander wrote:If, as you say, 10⁸⁰ exists, then surely 10²³ exists. By definition, your set of integers are closed under the operation of multiplication.

10⁸⁰ × 10²³ = 10¹⁰³. 10¹⁰³ must exist.

BTW, by definition, your set of integers is countably infinite.

Even with "the business about quantized length/time/energy" and "an integer number of Planck units", since the set of integers are infinite, that doesn't preclude an infinite universe. You use math to argue your point, but the properties inherent in the math you use don't agree with your arguments.

Well, I have an open mind and I'm prepared to change my point of view. Your task, as an academic, is to make me change my mind! I like to think I'm a rationalist, so if you are too we should be able to reach an agreement, or agree to differ.

1. 'Countably infinite' is defined as 'being able to be counted one at a time—although the counting may never finish'. That word 'never' introduces infinity directly into the definition a priori without considering whether it is actually possible to count without finishing; in the Universe if you start counting with 1 at the big bang, you will never get to ∞. This assumes "for all numbers x, where x is a member of the set of integers, there is no x such that the successor of x is ∞" is true. I don't think that's very controversial. Add to that the fact that the universe may also have a definite end, and you have the situation I am banging on about: your system of notation (integers) is a very powerful one and can be extended to represent numbers which will never be seen in the real world.

2. Test for Existence: an entity x 'Exists' if you can find an instance of it occurring in the real world. I don't think that's very controversial either. Which lemma are you going to challenge?

10¹⁸⁵ = the number of Planck volumes in the observable universe and I don't really have any problem with this number. HOWEVER, I'm not convinced that it follows from that that /any/ number you can represent using abstract notation necessarily passes the test for Existence (with a capital 'E').

Lots of abstract concepts don't exist in reality. Your move.

by **rstevenson** » Thu Sep 08, 2011 9:22 pm

"I am incapable of conceiving infinity, and yet I do not accept finity."
-- Simone de Beauvoir

by **Beyond** » Thu Sep 08, 2011 2:11 pm

The problem with thinking about infinity... is that it never ends.

by **Chris Peterson** » Thu Sep 08, 2011 2:00 pm

mst66186 wrote:Wow. I hardly know what to say. When I first started reading what you had put, I thought I was skirting around the issue by only considering the contents of the observable universe. But then you went on to assert that infinity is 'Outside the observable Universe. Just because we can't observe it doesn't mean it isn't there.' (!!) I might accept what you're saying if it was accompanied by a proof - what do you /know/ about the extent of the universe to support that?

Proof is a luxury that seldom makes an appearance in science. What we rely on is evidence, and there is plenty of evidence that the Universe extends beyond the observable Universe (which doesn't, of course, imply that the Universe is infinite). Of course, from a purely logical standpoint, it would be as odd to think that the Universe ended at the boundaries defined by local causality as to think the world ended at the horizons. Since the evidence is overwhelming that the Universe is expanding, there must be a point where the expansion rate becomes greater than c, and we can no longer observe material beyond that. So material that was once in the observable Universe no longer is. QED, the Universe exists beyond its locally visible edge.

The question of whether the Universe is finite or infinite, however, is more complex. It depends on subtleties of cosmological theory, and while those are testable, the observations to date are not conclusive. So that question remains unanswered.

I challenge you to tell me where can I go to find an example of infinity. In fact I'll be happy if you can do 10e103 without limiting yourself to pure math.
Surely if something exists in the real world you can show us an instance of it without resorting to hypotheticals that push it outside the observable universe.

The fact that you'd settle for a large number as an alternative to infinity suggests to me a certain lack of understanding. <g>

An example of infinity? I already gave one: the Universe, assuming certain current theories are correct. I don't think you're ever going to get a direct observational example of spatial infinity, because that seems impossible. There are many aspects of the Universe that can only be understood by indirect observations.

by **Ann** » Thu Sep 08, 2011 8:40 am

Are we humans the measure of everything? Must everything conform to our ability to grasp it, let alone make use of it?

I don't think the universe is defined by us humans. We, of course, may use our powers of reason and observation to define it. We may also use our powers of imagination to define it. And we may say, therefore, that what we can't imagine doesn't exist.

Personally I believe that the universe out there couldn't care less. It's interesting to speculate, however, that the Earth and its biosphere, including us humans, might not have existed if the universe hadn't been truly infinite. It's easier to think that we might not have existed if the universe hadn't been unimaginably big, though not necessarily infinite.

Perhaps in the future astronomers will make observations that will make it possible for us to say something about whether or not the universe (or the multiverse?) is truly infinite. For now, all I can say is that I personally think it's utterly impossible for a human being to truly imagine even the distance to the nearest star outside our solar system, Alpha Centauri. Alpha Centauri is four light-years away. It's even more utterly impossible to imagine the distance to the Andromeda galaxy, two million light-years away, even though the Andromeda galaxy is on a collison course with the Milky Way and will likely hit us in a few billion years' time. Even so, it is totally impossible for us humans to truly imagine the distance to the Andromeda galaxy. And if we can't imagine the distance to the Andromeda galaxy, what about imagining the distance to galaxies that are much farther away? What about M101 with its ongoing supernova, SN 2011fe? It may be 20 million light-years away or even more. What about even more distant galaxies? What about the galaxies in the Virgo cluster? What about the galaxies in the Coma Cluster? Are they non-existent because we can't possibly imagine how far away they are?

You have every right to deny the existence of infinity. So far and unless I'm wrong, science hasn't proved, even circumstantially, that true infinity exists. You may also deny the existence of any distance that is too long for you to imagine, but you can't deny these distances on scientific grounds. You may deny them only because you are entitled to your beliefs.

Ann

by **bystander** » Thu Sep 08, 2011 7:50 am

If, as you say, 10⁸⁰ exists, then surely 10²³ exists. By definition, your set of integers are closed under the operation of multiplication.

10⁸⁰ × 10²³ = 10¹⁰³. 10¹⁰³ must exist.

BTW, by definition, your set of integers is countably infinite.

Even with "the business about quantized length/time/energy" and "an integer number of Planck units", since the set of integers are infinite, that doesn't preclude an infinite universe. You use math to argue your point, but the properties inherent in the math you use don't agree with your arguments.

by **mst66186** » Thu Sep 08, 2011 6:34 am

Chris Peterson wrote:
mst66186 wrote:If length is quantized then it follows there's an integer number of Planck lengths from one side of the observable universe to the other. Always have been and always will be. Likewise if time is quantized, which it is, then there have been an integer number of Planck times since the big bang.
If the Universe is of infinite size, or infinite duration, there will not be an integer number of Planck lengths or times defining it. It makes little sense to restrict this question to the observable Universe, which is nothing more than a subset of the entire Universe- which may well be infinite in size, and also in (ultimate) duration.

Finally, there is a finite amount of matter and energy in the observable universe. So the big question has to be where is this mysterious infinity anyway?
Outside the observable Universe. Just because we can't observe it doesn't mean it isn't there. And it doesn't mean that it isn't important to understand, since material currently outside the observable Universe played a role in defining what we can observe.

...however, when you're dealing with the real world I don't see why you need to visualize anything larger than that - ever! Sorry if I sound like a wet blanket about infinity.
Visualization is a useful tool for most people to use in trying to understand physical concepts. It probably isn't a very good one- at least initially- for trying to grasp infinity. But it's a natural thing for people to try.

Wow. I hardly know what to say. When I first started reading what you had put, I thought I was skirting around the issue by only considering the contents of the observable universe. But then you went on to assert that infinity is 'Outside the observable Universe. Just because we can't observe it doesn't mean it isn't there.' (!!) I might accept what you're saying if it was accompanied by a proof - what do you /know/ about the extent of the universe to support that? I was also interested by the way you were able to throw away the facts about the observable universe that preclude the existence of infinity with the use of 'if' in this one: 'If the Universe is of infinite size, or infinite duration, there will not be an integer number of Planck lengths or times defining it'.
(the above is not supposed to be a flame)

Interestingly, you snipped out the part of the post that I thought was critically important.

OK. I'm a humble scientist of little brain, so I'll say this: I believe that in order to be able to say that something exists, you have to be able to find an instance of it in the real world. You can observe it directly, like an atom, or by looking at the effect it has on other things, like dark matter, or Santa Claus.
Now, my system of mathematics is based on set theory and I have a set-theoretical model of the world around me. When counting, I enumerate.
The number three definitely exists; I can put three beans in a jar and there's an instance of the number 3.
I dare say the number 10e80 exists because there are 10e80 atoms in the observable universe.
Not too sure about 10e102. I agree the system of notation we are using can be extended beyond the point it is practically useful, and arbitrarily far.
I challenge you to tell me where can I go to find an example of infinity. In fact I'll be happy if you can do 10e103 without limiting yourself to pure math.
Surely if something exists in the real world you can show us an instance of it without resorting to hypotheticals that push it outside the observable universe.
It is a symbol used to denote something that is practically too big to measure. The business about quantized length/time/energy above could be taken as a proof that you will never encounter infinity in the real world. -- ever!

by **neufer** » Wed Sep 07, 2011 12:37 pm

Chris Peterson wrote:
mst66186 wrote:
There is a finite amount of matter and energy in the observable universe.
So the big question has to be where is this mysterious infinity anyway?
When you're dealing with the real world I don't see why you need to visualize anything larger than that - ever!
It makes little sense to restrict this question to the observable Universe,
which is nothing more than a subset of the entire Universe
- which may well be infinite in size, and also in (ultimate) duration.

Now who is the Fool

Did you truly thing the aspect of all there is was divided into manageable sections for your convenience

Infinity appears (on right) merged with twin cosmic entity Eternity
in a panel from Infinity War #6 (Nov. 1992).

http://en.wikipedia.org/wiki/Infinity_%28Marvel_Comics%29 wrote:
<<Infinity is a fictional character that appears in comic books published by Marvel Comics.

Debuting in Quasar #24 (July 1991), Infinity featured in an extended storyline that revealed the role of the character in the Marvel Universe. Infinity reappears in Quasar #37 (Aug. 1992) and in the limited series Infinity War #1 - 6 (May - Nov. 1992).

Infinity is a cosmic entity representing the totality of space, while its "brother" Eternity represents time. Together they represent the space-time continuum and the living force of the universe.Infinity’s opposite is another cosmic entity - Oblivion.

The character first appears when the villain Maelstrom, acting on behalf of Oblivion attempts to end the universe. Infinity contacts the astral form of the hero Quasar, and empowers him to act as its avatar. After a series of battles, Quasar defeats Maelstrom, thereby allowing Infinity to prevail. Quasar has another brief encounter with the entity when searching for Eternity.

The entity also appears during the Infinity War storyline, and in a final battle appears with fellow Eternity as a composite being, incapacitating the villain the Magus.

As a virtually omnipotent abstract entity, Infinity has no physical body but exists everywhere simultaneously. Like Eternity, the character can manipulate the universe to achieve virtually any effect. When needing to address lesser beings, Infinity can create a humanoid form that can be perceived by others; one that typically defaults to a female form, whereas its counterpart (Eternity) usually defaults to male.>>

by **Chris Peterson** » Wed Sep 07, 2011 4:27 am

mst66186 wrote:If length is quantized then it follows there's an integer number of Planck lengths from one side of the observable universe to the other. Always have been and always will be. Likewise if time is quantized, which it is, then there have been an integer number of Planck times since the big bang.

If the Universe is of infinite size, or infinite duration, there will not be an integer number of Planck lengths or times defining it. It makes little sense to restrict this question to the observable Universe, which is nothing more than a subset of the entire Universe- which may well be infinite in size, and also in (ultimate) duration.

Finally, there is a finite ammount of matter and energy in the observable universe. So the big question has to be where is this mysterious infinity anyway?

Outside the observable Universe. Just because we can't observe it doesn't mean it isn't there. And it doesn't mean that it isn't important to understand, since material currently outside the observable Universe played a role in defining what we can observe.

...however, when you're dealing with the real world I don't see why you need to visualize anything larger than that - ever! Sorry if I sound like a wet blanket about infinity.

Visualization is a useful tool for most people to use in trying to understand physical concepts. It probably isn't a very good one- at least initially- for trying to grasp infinity. But it's a natural thing for people to try.

by **mst66186** » Wed Sep 07, 2011 4:06 am

babaonet wrote:Can anyone imagine a line which has no end?
I can't imagine anything which has no end. In other words I don't believe absolute infinity.
I am not sure whether the concept of infinity is relative. However, i feel so. Can anyone kindly put their valuable comments on the following two:
1. The universe is infinite.
2. The universe is relatively infinite to us.

Why would you ever want to visualize it anyway?

If length is quantized then it follows there's an integer number of Planck lengths from one side of the observable universe to the other. Always have been and always will be. Likewise if time is quantized, which it is, then there have been an integer number of Planck times since the big bang. Finally, there is a finite ammount of matter and energy in the observable universe. So the big question has to be where is this mysterious infinity anyway? It's just a mathematical construct - a trick - that translates to 'it goes off the edge of the graph' in plain English. The only place I can think of infinity genuinely applying in the 'real' world is in QED (Quantum Electro Dynamics) in counting the number of possible ways a two particles can couple in order to pedict the probability of an outcome after renormalization. To be frank, I suspect the infinity there is an artefact of the model there too.

We seem, to me at any rate, to live in a world where any number could in principle exist (in the sense that if there are three beans in this jar then the number 3 exists), and it's easy to write down representations for numbers greater than the number of particles in the observable universe(10e80), or even the entropy of the universe (10e102); however, when you're dealing with the real world I don't see why you need to visualize anything larger than that - ever! Sorry if I sound like a wet blanket about infinity.

by **StormyKnight** » Wed Aug 17, 2011 2:25 pm

Imagining infinity takes practice, and meditation. That is something most of us don't take enough time for. We limit ourselves too much. Even the idea of a line on a sphere or cylinder has limits. If you place yourself on that line and follow it, yes it seems infinite. Drop a marker of some kind and you will eventually come back to it. Perspectively back away from the sphere/cylinder and imagine the object transparent, the line becomes a circle. It has limits. It is not infinite. Now imagine a line that passes through the earth and the moon, (or any other celestial object). Where are the ends? By definition, a line does not have ends. For me, thinking about spacial geometry on such a large scale helps to imagine or wrap ones mind around concepts such as infinity.

Starship Asterisk*

I can't imagine infinity

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Eternity and Infinity together ~?!

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