[david_c]

luke warm, on Mar 29 2006, 03:09 AM, said:

it isn't that i prefer one to the other, it's that one (the infinite one) can't logically exist... i don't know what a 'purely' philosophical reason would be... i'm just saying that the universe, and all it contains, must be finite... and i don't say this on philosophical grounds, but on logical grounds... do a google for 'hilbert's paradox' to get a glimpse of what i'm talking about

"Hilbert's Hotel" is concerned with countable sets and so doesn't seem to be relevant. Jimmy's statement "you can't get here from there" sounds more like Zeno's paradox to me, which is well-known to be fallacious.

Infinity is a difficult subject.

Let S be a collection of subsets of the integers, such that for any A and B in S, either A is contained in B or B is contained in A. Is it possible for S to be uncountable?

[helene_t]

david_c, on Mar 29 2006, 01:17 PM, said:

Let S be a collection of subsets of the integers, such that for any A and B in S, either A is contained in B or B is contained in A. Is it possible for S to be uncountable?

It might or might not be possible. Either assumption is consistent. To get clarrification, you need to accept either the continuum hypothesis, or its negation, as an axiom.

This is called the theorem of Soloway and was proved in 1980, I think.

Most mathematicians prefer the negation of the continuum hypothesis. It is (here it comes again!) a matter of preferences.

[luke warm]

Sigi_BC84, on Mar 29 2006, 05:49 AM, said:

I have just read up on it ([wikipedia] Hilbert's Grand Hotel and [wikipedia] Cosmological argument) and you are stating something for a fact that cannot be simply taken for granted.  Especially Hilbert's Hotel is not necessarily a valid model for the real universe, or a proof of the impossibility that it could be infinite.

i didn't mention the paradox as a model for the univerese, but as an example of why time cannot be infinite

Quote

Now you are shooting yourself into the foot: When mentioning "eternity" you are acknowledging the possibility of infinite time.

no, i don't see it that way... i'm not saying eternity has anything to do with time... as a matter of fact, time itself (if created) must exist apart from eternity... i see eternity as timeless... assume for a moment that God exists, and that heaven exists... assume heaven is located in, or is, eternity... to explain what i mean by timeless: when i die and "step" into heaven (ie, eternity), another who died 100 years prior will also be 'stepping' into heaven... see?

Quote

Reread the passage you have quoted:  The discrete timeline consists of infinitely many, but countable time quanta, and therefore you cannot squeeze additional quanta in ad infinitum.  What you described was an uncountable set.

i think we're missing one another here... i'm saying that the very fact of arriving at point B from A shows there was not an infinite number of points between, not that such points in between are uncountable

Quote

Let's take the amount of time that passed from the moment that I submitted this posting to the very moment that you are reading this.  Let's call this number "Q".  So by tracing Q time quanta I arrive from then to now.  The total number of (countable) time quanta that have passed so far is infinity, so is the total number of time quanta yet to happen.  Where's the problem here?

as i wrote immediately above, this seems to be where we're missing one another... in your mind, mark a point and call it "then"... mark another point and call it "now"... let's assume the 2 points are separated by a unit of measurement called 'seconds'... now then, count... how many increments, if time is infinite, are between the 1st and 2nd second? an infinite number, right? would you *ever* reach second 2 (much less 'now') from second 1, in infinity?

david said:

"Hilbert's Hotel" is concerned with countable sets and so doesn't seem to be relevant. Jimmy's statement "you can't get here from there" sounds more like Zeno's paradox to me, which is well-known to be fallacious.

no, zeno's paradox shows the flaw in the logic of an *actual* infinity existing ... yes, if one attempts to prove that an infinite number of points (meters in the hare and achilles) exist between a beginning and end, the argument is indeed fallacious, merely because a sum is available (one)... however, this is only so because an actual infinity of meters can't exist

i say hilbert's hotel is completely relevant, since it shows what to expect from an *actual* infinite as opposed to an imaginary one

helene said:

Now I'm confused - Hilbert's Hotel can exist in bounded space though not in bounded, discrete space - are you saying that the universe must be bounded and discrete?

i don't have the necessary math skills to speak much on discrete vs. indiscrete sets... i'm only speaking philosophically about a particular problem with an infinite universe... this problem has been acknowledged by others (atheists and christians) and has led to theories and arguments surrounding the origin of the universe... it's why some try to show why the singularity has always existed (which would make it infinite, by definition)... but it also shows why other theories have come into being... now i'm not saying this is the only reason for the string theory, or work in complex math, but i am saying that this logical problem certainly entered the thinking of those who try to prove the existence of a *natural* infinity (ie, one found in nature)

[david_c]

helene_t, on Mar 29 2006, 12:58 PM, said:

david_c, on Mar 29 2006, 01:17 PM, said:

Let S be a collection of subsets of the integers, such that for any A and B in S, either A is contained in B or B is contained in A. Is it possible for S to be uncountable?

It might or might not be possible. Either assumption is consistent. To get clarrification, you need to accept either the continuum hypothesis, or its negation, as an axiom.

I wish that was the correct answer, but it's not, you must have misread it. This is a question we were given in my first year at university, and doesn't require anything deeper than the definiton of countable / uncountable.

[cherdano]

helene_t, on Mar 28 2006, 02:29 PM, said:

Personally, I would prefer an infinite universe, because:
- I like the geometry to be Euclidian
- I like the universe to be symmetric
- A symmetric, Euclidian space in infinite

I don't understand any of this infinity discussion, not even this very concrete statement here... Tori look entirely symmetric and Euclidian to me, but are probably not what you call "infinite".

Or maybe I don't understand what "infinite" is supposed to mean here?

Arend

[helene_t]

david_c, on Mar 29 2006, 02:11 PM, said:

helene_t, on Mar 29 2006, 12:58 PM, said:

david_c, on Mar 29 2006, 01:17 PM, said:

Let S be a collection of subsets of the integers, such that for any A and B in S, either A is contained in B or B is contained in A. Is it possible for S to be uncountable?

It might or might not be possible. Either assumption is consistent. To get clarrification, you need to accept either the continuum hypothesis, or its negation, as an axiom.

I wish that was the correct answer, but it's not, you must have misread it. This is a question we were given in my first year at university, and doesn't require anything deeper than the definiton of countable / uncountable.

You're right, sorry. I thought you were refering to:
R= (the set of all subsets of N)
S=(some subset of R for which there is no surjective mapping S -> R)
Question: can S be mapped injectively -> N?

[helene_t]

cherdano, on Mar 29 2006, 02:18 PM, said:

helene_t, on Mar 28 2006, 02:29 PM, said:

Personally, I would prefer an infinite universe, because:
- I like the geometry to be Euclidian
- I like the universe to be symmetric
- A symmetric, Euclidian space in infinite

I don't understand any of this infinity discussion, not even this very concrete statement here... Tori look entirely symmetric and Euclidian to me, but are probably not what you call "infinite".

Or maybe I don't understand what "infinite" is supposed to mean here?

Arend

If the universe is a torus, Euclidian geometry does not apply. But I thought we were discussing boundedness, I should use that word instead of "infinite".

Let's suppose the vissible universe a 3-dimensional thorus embeded into 4-dimensional euclidian space. Now you could say that the "universe" is the visible universe, hence not euclidian. Or you could say the "universe" is the space in which the thorus is embeded, hence unbounded (or assymetrical, if you prefer).

[helene_t]

Quote

mark a point and call it "then"... mark another point and call it "now"... let's assume the 2 points are separated by a unit of measurement called 'seconds'... now then, count... how many increments, if time is infinite, are between the 1st and 2nd second? an infinite number, right? would you *ever* reach second 2 (much less 'now') from second 1, in infinity?

This is Zeno's paradox and it is not a problem. It's true that we can't observe an infinity of events between "then" and "now". But that doesn't say that there can't be an infinity of time points.

In fact, it is very difficult (to me at least) to immagine space to be discrete. There would have to be some grid of points in space, meaning that you can't travel in certain directions because you wouldn't enounter any points then. Or straight lines must allways be zig-zag lines at the microscopic level (how to explain that light travels with the same speed in all directions, then?) or the grid would have to bend itself to addapt to traveling objects. The math would be quite difficult. So please, give me a continuus space unless you have very strong arguments for something else.

[helene_t]

luke warm, on Mar 29 2006, 02:01 PM, said:

but i am saying that this logical problem certainly entered the thinking of those who try to prove the existence of a *natural* infinity (ie, one found in nature)

I'm not saying that infinity can be "found" in nature - if infinite sets exist we can never observe them. This is more related to our limited perception than to nature.

Much less am I saying that the existence of such sets can be proven.

But theories about non-observable things are everywhere in science. They are even part of human nature - children have been shown spontaneously to develop the concept of "existence" as something independant of "observed".

And many scientific theories require infinite sets for convenient mathematical modeling.

[david_c]

helene_t, on Mar 29 2006, 01:24 PM, said:

If the universe is a torus, Euclidian geometry does not apply.

The natural geometry of the torus is Euclidean: indeed it is a quotient of Euclidean space. Admittedly, you don't "see" this if you try to embed the torus into (n+1)-dimensional Euclidean space, but why would you want to do that?

[helene_t]

david_c, on Mar 29 2006, 02:59 PM, said:

helene_t, on Mar 29 2006, 01:24 PM, said:

If the universe is a torus, Euclidian geometry does not apply.

The natural geometry of the torus is Euclidean: indeed it is a quotient of Euclidean space. Admittedly, you don't "see" this if you try to embed the torus into (n+1)-dimensional Euclidean space, but why would you want to do that?

Right. Arend told me the same. I thought that the fact that straight lines bide themselves in the tail makes a thorus non-euclidian. But now that I think about it I'm not sure if it violates any of Euclides axioms.

So maybe, if Jimmy insists the universe be bounded and I insist it being symmetric and eucidian, we could compromize on a thorus. But in the meantime I've realized that what Jimmy does not believe in is the continous space rather than unbounded space.

Not sure what this has to do with God, though.

[luke warm]

helene_t, on Mar 29 2006, 07:34 AM, said:

Quote

mark a point and call it "then"... mark another point and call it "now"... let's assume the 2 points are separated by a unit of measurement called 'seconds'... now then, count... how many increments, if time is infinite, are between the 1st and 2nd second? an infinite number, right? would you *ever* reach second 2 (much less 'now') from second 1, in infinity?

This is Zeno's paradox and it is not a problem. It's true that we can't observe an infinity of events between "then" and "now". But that doesn't say that there can't be an infinity of time points.

but we can observe the passage of time, correct? that's all i was trying to say..

Quote

I'm not saying that infinity can be "found" in nature - if infinite sets exist we can never observe them. This is more related to our limited perception than to nature.

perhaps my use of the word "natural" was unfortunate... what i meant by it was "actual"... so in case my position hasn't been clear, it's simply this: to say that time is infinite is to say that an *actual* infinity exists... this cannot be shown (proven) to be true by logic, mathmatics, or even simple observance... i come at this from the philosophical side

[Sigi_BC84]

luke warm, on Mar 29 2006, 06:26 PM, said:

perhaps my use of the word "natural" was unfortunate... what i meant by it was "actual"... so in case my position hasn't been clear, it's simply this: to say that time is infinite is to say that an *actual* infinity exists... this cannot be shown (proven) to be true by logic, mathmatics, or even simple observance... i come at this from the philosophical side

OK, I cancel my previous claim that time is infinite (ie. that there is no beginning of time) and that this is logical. I replace it by "I assume that time is infinite and that there is no beginning of time" without claiming that this follows from logic.

However, I also do not accept any claims of the opposite (ie. that time cannot be infinite). Now, if you still want to uphold your claim that time cannot be infinite, then please give accurate reasoning (or provide some links to sources that do so).

Helene has mentioned Occam's Razor already (which in a nutshell says that of two otherwise equally satisfying theories the simpler one is to be preferred). I'm applying this here. If the Big Bang did actually happen, it was replacing something before the singularity, which is beyond our possibilities for observation. Maybe an earlier universe had collapsed into the singularity just to be reborn into the present one, so time never actually ceased to exist -- I do not know.

After all, assuming a creator of the world and a beginning of time is not necessary in my eyes and only makes the theory more complicated without explaining more.

--Sigi

[luke warm]

Sigi_BC84, on Mar 30 2006, 02:50 AM, said:

If the Big Bang did actually happen, it was replacing something before the singularity, which is beyond our possibilities for observation.  Maybe an earlier universe had collapsed into the singularity just to be reborn into the present one, so time never actually ceased to exist -- I do not know.

After all, assuming a creator of the world and a beginning of time is not necessary in my eyes and only makes the theory more complicated without explaining more.

--Sigi

but sigi, this is merely an assertion on your part stated as a fact... as i've stated, an actual inifinity needs imaginary numbers used in complex math... as for stating sources, we're each free to draw our own conclusions from debates held by others... as for me, when hawking says (in 'a brief history in time')

"The idea that space and time may form a closed surface without boundary also has profound implications for the role of God in the affairs of the universe. With the success of scientific theories in describing events, most people have come to believe that God allows the universe to evolve according to a set of laws and does not intervene in the universe to break these laws. However, the laws do not tell us what the universe should have looked like when it started – it would still be up to God to wind up the clockwork and choose how to start it off. So long as the universe had a beginning, we could suppose it had a creator. But if the universe is really completely self-contained, having no boundary or edge, it would have neither beginning nor end: it would simply be. What place, then, for a creator?"

it makes it obvious (to me at least) why it was imperative to prove the existence of an infinite universe... and if the only way to do that was to use imaginary numbers, well that's better than having to use God, right? notice that even hawkings admits that if there was a beginning, God (or at least a creator) exists... to me hawkings is saying that the universe is either infinite or created... since it can't be created (his mindset), ergo ...

so while you contend that assuming a creator makes the matter more complicated, i contend the opposite - what could be simpler?

[Sigi_BC84]

luke warm, on Mar 30 2006, 11:39 AM, said:

so while you contend that assuming a creator makes the matter more complicated, i contend the opposite - what could be simpler?

Having to explain the creator makes it more complicated. If you don't suppose the existence of a creator (while choosing a model for the universe that doesn't need one) you don't have to struggle with proofs for the existence of the creator as well.

Also, if you assume that a creator exists, you expose yourself to questions like "where is this entity now?" which lead to very unsatisfactory answers at most.

--Sigi

[david_c]

luke warm, on Mar 30 2006, 11:39 AM, said:

hawking says (in 'a brief history in time')

"The idea that space and time may form a closed surface without boundary also has profound implications for the role of God in the affairs of the universe. With the success of scientific theories in describing events, most people have come to believe that God allows the universe to evolve according to a set of laws and does not intervene in the universe to break these laws. However, the laws do not tell us what the universe should have looked like when it started – it would still be up to God to wind up the clockwork and choose how to start it off. So long as the universe had a beginning, we could suppose it had a creator. But if the universe is really completely self-contained, having no boundary or edge, it would have neither beginning nor end: it would simply be. What place, then, for a creator?"

it makes it obvious (to me at least) why it was imperative to prove the existence of an infinite universe... and if the only way to do that was to use imaginary numbers, well that's better than having to use God, right? notice that even hawkings admits that if there was a beginning, God (or at least a creator) exists... to me hawkings is saying that the universe is either infinite or created... since it can't be created (his mindset), ergo ...

Hawking [no 's' please] does not say "if there was a beginning, a creator exists". He is actually arguing for the converse statement, that is, "if there was no beginning, then there was not a creator". Note that even if you accept that this second statement is true (and Hawking is careful to avoid making such a concrete statement in this passage), it does not imply the first.

An atheist would say that there are two possibilities - either there was a beginning, or there wasn't - but that neither requires a god in order to work.

Hawking also seems to have a rather unusual idea of what God is - "God allows the universe to evolve according to a set of laws and does not intervene in the universe to break these laws". This premise seems completely incompatible with any religion that I know of.

[helene_t]

david_c, on Mar 30 2006, 02:15 PM, said:

Hawking also seems to have a rather unusual idea of what God is - "God allows the universe to evolve according to a set of laws and does not intervene in the universe to break these laws". This premise seems completely incompatible with any religion that I know of.

Hawking writes in "A brief history of time" that he was invited by Pope John Paul as a member of a scientific advisori board that should help the pope formulate the official catholic view on .... well, "life, the universe and everything" or some such. The pope asked the scientists not to come up with any atheistic vision on the creation of the universe since that was the domain of God. For the rest, the scientist were free to say whatever they wanted.

I'm not sure how the story ended but is sounds as if the pope was willing to accept the view on God that Jimmy quoted.

Also, it occurs to me that polytheistic religions do not necesarily attribute an active role for the creator after the creation. But I could easily be wrong.

Anyway, I don't think Hawking (or any other non-theologist) should be seen as an authority on theology. Surely, when the ancient religions were founded there was an overlap between theology and everything else, including cosmology (many astrononomers functioned as astrologists until a few centuries ago) but academic disciplines drift farther and farther away as int becomes increasingly imposible for an individual academic to span multiple diciplines.

[david_c]

helene_t, on Mar 30 2006, 01:36 PM, said:

I'm not sure how the story ended but is sounds as if the pope was willing to accept the view on God that Jimmy quoted.

Well, if you accept that God does not intervene in the universe, then that pretty much rules out Him having a son.

[helene_t]

david_c, on Mar 30 2006, 02:47 PM, said:

helene_t, on Mar 30 2006, 01:36 PM, said:

I'm not sure how the story ended but is sounds as if the pope was willing to accept the view on God that Jimmy quoted.

Well, if you accept that God does not intervene in the universe, then that pretty much rules out Him having a son.

Modern theologist tend to see the biblical accounts of miracles as allegoric. I'm not sure about the particular case of the holy virgin, but since Johannes Paul accepted Darwinism he probably also had some view on this that is compatible with biology.

[luke warm]

i simply quoted a passage attributed to hawking... i don't know his theology, or even if he has one... i do know that a God who doesn't intervene in his creation is counter to my beliefs, even if such a God is acceptable to a pope ... i think if you want views on beliefs, and reasons for those views, a new thread is needed

Sigi said:

If you don't suppose the existence of a creator (while choosing a model for the universe that doesn't need one) you don't have to struggle with proofs for the existence of the creator as well.

Also, if you assume that a creator exists, you expose yourself to questions like "where is this entity now?" which lead to very unsatisfactory answers at most.

i don't struglle with proofs... as for exposing myself to questions, that has never bothered me...