[JohnDuffield]

Absolutely. If the universe was flat and the cosmological principle (http://en.wikipedia.org/wiki/Cosmological_principle) is correct (both of which are widely beginning to be accepted as true) then the universe unbounded and not finite, i.e. the space is not bounded and goes on forever which means that the number of hadrons and hence the amount of matter is infinite. I can't imagine Sean disagreeing with me.

I disagree with you. A "principle" is no substitute for scientific evidence. And as for Sean, he thinks there's an evil twin universe where time runs backwards.

[Bill S]

Pete, I have never made any secret of the fact that I lack a background in maths; I have always regarded this as a disadvantage, but perhaps it is not as disadvantageous as the apparent disability to see beyond mathematical definitions when that becomes appropriate or necessary. 

By definition, a quantity is defined as infinite when it increases without bound. It's said not to have a limit. That's what it means to be infinite

I understand that.  I accept that that is the definition of a mathematical infinity. 

You interchange “infinite” and “without bound”.  I would not argue with this as being acceptable in maths, but I think that if you apply this definition to reality it leads to the sort of blinkered thinking that results in the repetition of mathematical rationale as though it were a definitive answer to something it does not address.

Infinity is not a number; to treat it as such is simply to use the term as an approximation.  It is in effect saying “this is so large, or small, it is reasonable to regard it as infinite”.  No problem. 

A limit is the value that a function or sequence "approaches" as the input or index approaches some value. If there is no such number and the sequence increases without bound then

What is the value of infinity?   Perhaps the second part of the quote addresses that question, but it seems a bit inconclusive.

[Bill S]

And as for Sean, he thinks there's an evil twin universe where time runs backwards.

Now - there's a cosmological minefield!   []

[jeffreyH (OP)]

I'm going to do a little more dreaming. Let's consider a cube. We want to extend this cube to infinity so we add 7 more cubes. We still have a bounding surface so we add 19 more cubes. No matter how many extra cubes we add we will never extend this cube to infinity as by definition a cube will always have a boundary. We need a geometry without initial bounds to be able to reach infinity. This then becomes a complex problem. How do we define an object with unbounded geometry? The only 'object' with such a profile is infinity itself which is not a thing made of substance at all. It is a concept of human invention.

[jeffreyH (OP)]

Think of this also, as we have quanta built into the theories of quantum mechanics, which by definition have a boundary, can these then be extended to infinity?

[PmbPhy]

Pete, I have never made any secret of the fact that I lack a background in maths; I have always regarded this as a disadvantage, but perhaps it is not as disadvantageous as the apparent disability to see beyond mathematical definitions when that becomes appropriate or necessary.

We really don't need a solid math background other than simple concepts. We start with the axiom (i.e. law, principle, etc. these terms all mean the exact same thing) cosmological principle. It's a very simply concept and based on solid scientific evidence. The cosmological principle states that the universe is homogeneous and isotropic when viewed on a large enough scale which means that the distribution of matter, and therefore mass, is homogenous and isotropic. Based on the recent Wilkinson Microwave Anisotropy Probe measurements we know that the universe is flat with only a 0.4% margin of error, according to NASA scientists. According to cosmologists the observational data best fits with the conclusion that the shape of the universe is infinite and flat. If the mass density of the universe is uniform and space is infinite then it directly follows that the amount of mass is infinite. It's as simple as that.

You interchange “infinite” and “without bound”.

That's because they're the same thing when used as I posted the definition.

I would not argue with this as being acceptable in maths, but I think that if you apply this definition to reality it leads to the sort of blinkered thinking that results in the repetition of mathematical rationale as though it were a definitive answer to something it does not address.

I'm sorry Bill but you lost me. I have no idea what that means. Please rephrase it so that even I can understand it.

Infinity is not a number; ...

Obviously.

....to treat it as such is simply to use the term as an approximation.

I never said it was a number and if you thought that's what I meant then you misread what I wrote.

It is in effect saying “this is so large, or small, it is reasonable to regard it as infinite”.

I never said or wrote anything that implied such a thing.

What is the value of infinity?

Why are you asking me a question for which you know there's no answer to it, i.e. it's meaningless? We both know that infinite is and we both know what the cosmological principle is and that the universe is most likely flat (or we're assuming that for the sake of argument) then the amount of mass/matter in the universe is infinite. That means that there is no number which equals the amount of mass in the universe. You know what infinite mass is since you know what infinite is and how to arrive at the conclusion.

So what's your problem with that, Bill?

[PmbPhy]

I disagree with you. A "principle" is no substitute for scientific evidence.

That's because you're not a scientist, certainly not a physicist. If you were then you'd understand it. Your problem is that you don't know that the cosmological principle was postulated because scientific evidence shows that the matter density in the observable universe, on a large scale, is homogeneous and isotropic.

See http://www.gizmag.com/universe-homogeneous-300-million-light-years/24149/

One bit of evidence pointing to a homogeneous Universe is the isotropy of the cosmic microwave background (CMB). The CMB has very nearly the same intensity regardless of which direction you look from Earth - it is isotropic from our vantage point. Moreover, that it still appears isotropic after travelling through the Universe for 13.7 billion years suggests that the early Universe was highly homogeneous, and that it has remained rather homogeneous since then.

You should learn about these things before you try to claim that you're right. You do this way too much.

[Bill S]

Pete, are you, or have you ever been a school teacher?  I ask this because your responses so often seem to follow the pattern: “This is the answer to your question; whatever your question was.”

I will do my best to keep to one question at a time, and if you answer just that question, perhaps we can make progress.

We really don't need a solid math background other than simple concepts. We start with the axiom (i.e. law, principle, etc. these terms all mean the exact same thing) cosmological principle. It's a very simply concept and based on solid scientific evidence. The cosmological principle states that the universe is homogeneous and isotropic when viewed on a large enough scale which means that the distribution of matter, and therefore mass, is homogenous and isotropic. Based on the recent Wilkinson Microwave Anisotropy Probe measurements we know that the universe is flat with only a 0.4% margin of error, according to NASA scientists. According to cosmologists the observational data best fits with the conclusion that the shape of the universe is infinite and flat.  If the mass density of the universe is uniform and space is infinite then it directly follows that the amount of mass is infinite. It's as simple as that.

Does the cosmological principle apply to anything that did not originate in the Big Bang?

[Bill S]

Think of this also, as we have quanta built into the theories of quantum mechanics, which by definition have a boundary, can these then be extended to infinity?

Only if you accept the myth of the infinite series/sequence.   []

Of course there could always have been an infinite number of quanta, but what am I saying?  Infinity is not a number!
 

[JohnDuffield]

...We start with the axiom (i.e. law, principle, etc. these terms all mean the exact same thing) cosmological principle. It's a very simply concept and based on solid scientific evidence. The cosmological principle states that the universe is homogeneous and isotropic when viewed on a large enough scale...

It isn't based on solid scientific evidence. It's a principle. An axiom. A presumption. For all we know, when some guy 46 billion light years looks up at the night sky, he sees half of it all black and empty. 

That's because you're not a scientist, certainly not a physicist. If you were then you'd understand it.

I am a scientist. I have a Computer Science degree. And whilst I don't have a physics degree, my physics knowledge is very good. Let me demonstrate:

Your problem is that you don't know that the cosmological principle was postulated because scientific evidence shows that the matter density in the observable universe, on a large scale, is homogeneous and isotropic.

It's a postulate. We have clear evidence that the observable universe is homogeneous and isotropic, but we have no evidence that the entire universe is like this, and nor do we have any evidence that the universe is infinite.

 

[yor_on]

Not if you want a finite universe John. Because if you do then you already have the answer, it is whatever we can measure. And as we only can measure approximately 13.7 Billion light years, being the 'earliest' light reaching us, then the universe must be homogeneous and isotropic. If you don't trust this but assume that your 'finite universe' continue past this limit you automatically define it as infinite. because you have now passed the earliest light existing, as seen from Earth. Or you will have to find a way to proof that what we call the earliest light isn't the 'earliest' at all, but that it continue, becoming  'earlier' and 'earlier', past that 13.7 billion ly limit.
==

http://www.nasa.gov/audience/foreducators/5-8/features/F_How_Big_is_Our_Universe.html

" It is time, not space, which limits our view. Beyond a certain distance, light hasn't had time to reach us yet."

so what we see far away, defining the 'birth' of our universe, if going there 'instantly', would then change to the exact same view we find from here, homogeneous and isotropic all around us,  but now with our earth non existing as it would be 'placed' at lights limit  (birth) from our new position. And with a 'new' 13.7 billion ly visible in the opposite direction, all the way to lights 'birth'.

[JohnDuffield]

...If you don't trust this but assume that your 'finite universe' continue past this limit you automatically define it as infinite.

No I don't. I just say the size is unknown. There's just no evidence at all for declaring the universe to be infinite.
 

...Or you will have to find a way to proof that what we call the earliest light isn't the 'earliest' at all...

No I don't. I accept that the earliest light was 13.8 billion years ago. What I don't accept is the non-sequitur that the universe must be infinite. It's based on a presumption, not evidence. It's like you've got some pompous guy up a mountain who thinks he knows it all. It's homogeneously and isotropically flat and misty where he is, and he declares it's the same for everybody else in the world, and that the world is therefore infinite. And woe betide anybody who might challenge his assertions. 

[yor_on]

I know John, it's somewhat of a logical conundrum. But the point of it is that the earliest light is around 13.7 ly. And the universe is greater than that. If you define a dimension as being 'bent' you can assume, as a guess, a finite universe, although without a bound. Then you might want to assume that 13.7 is all there is. But if you define dimensions as built through 'degrees of freedom', also defining them as only existing from a inside, infinity does not exist. Infinity is just where our calculations lose their 'finite' meaning.
==

Have a look at http://archive.lib.msu.edu/crcmath/math/math/f/f052.htm and note " The value of the Feigenbaum constant can be computed explicitly using functional group renormalization theory"

Renormalization?

[yor_on]

Eh, that not meaning that I see the universe as 'finite'
Just the opposite, without infinity having a real meaning, except to us, we're the ones getting it wrong.
=

We built our knowledge out of presumed linear systems, recently we've found non linearity everywhere. Think of decoherence, and the way 'emergences' come. As if linearity have bounds defining it. Another type of 'dimensionality' to me.

[Ethos_]

I just say the size is unknown. There's just no evidence at all for declaring the universe to be infinite.
 

This is true, however, declaring it to be finite is also devoid of positive proof. While our local universe is possibly a finite region, it is very likely that this finite region lies within an infinite bulk. Some will refer to our local region as the whole universe discounting the possibility of the bulk and conclude that it is finite. It all depends on what one means when using the term; "Universe".

Webster's defines "Universe" as; "the totality of all things that exist". Taken in this context, if our local space lies within a larger bulk, the term Universe must also include the infinite Bulk. In my opinion, even though our local region is finite, the extent of all there is includes the infinite bulk making the Universe infinite.

It really all depends upon whether one views the Hypothetical Bulk as proper scientific logic. In my case, I prefer to as many credible scientists also do.

[yor_on]

Well Ethos, you will need 'bent dimensions' in some way to keep the universe relativity defines, if you want it to be finite, although 'infinite', all as I see it. If you on the other hand define it locally, then there are no points differing inside that universe from any other 'point', that is what repeatable experiments tells us, and constants. So where is the bound? It can't be equivalent, even if impossible to observe astronomically. It's not simple to me. the simplest thing is a infinite universe, homogeneous and isotropic without 'bounds' of any kind, more than what causality demands. It's us demanding 'finity', as in a linear definition of physics.
==

It's also so that from a strict locality you can't use 'gravity' as something defining a 'possible universal' curvature. You can use it from a container concept in where you keep one definition of a gravitational field, being the 'inertial' far observers. But locally, free falling, actually inseparable from any other uniform motion ideally defined, there are no 'gravity' for you, inside that 'black box'.

But WMAP, Maxima etc all agree, the universe, at least the 'visible' (13.7 or 13.8 part of it is 'flat' measured at a 'universal scale'. so whatever curvature one need for a 'finite universe' is non existent in the visible part we have. http://curious.astro.cornell.edu/question.php?number=171

(if you read this link you will notice him using an idea of a balloon expanding, as describing inflation. But you can also use a plane, filling in 'new points' everywhere, or some other 'dimensionality'. That we find three room dimensions and one time dimension do not tell us how they come to be for us.)

Thinking of it as a infinite bulk, using my definitions the only bulk existing is what we measure to exist. Without a 'outside' only measurements define what we can observe. So whatever infinity I want to define should already be here, inside our linearity. We so much want a clock work universe, linear. But what I think we will get instead is causality. That's what defines it.

[Bill S]

It all depends on what one means when using the term; "Universe".

I think this might be less confusing if John Gribbin’s usage were followed:

Cosmos = everything that exists, or can exist.
Universe = our (in principle) observable portion of spacetime and its contents.
universe = any other universe that may, or may not, exist.

[jeffreyH (OP)]

We determine that the earliest light was 13.7 billion years ago and came from distance X. That distance existed 13.7 billion years ago dx will determine where that position now resides outside our light cone. As the universe has had 13.7 billion years of intervening time to continue expansion we would need to use Hubble data to see what the extent of the universe is now. That portion may already be outside the Hubble sphere and traveling away at superluminal speed exponentially. If the universe is infinite then it is growing to be larger than infinite simply because an expansion takes up more space. Therefore our view of infinity, which is a human invention, is wrong.

[chiralSPO]

We determine that the earliest light was 13.7 billion years ago and came from distance X. That distance existed 13.7 billion years ago dx will determine where that position now resides outside our light cone. As the universe has had 13.7 billion years of intervening time to continue expansion we would need to use Hubble data to see what the extent of the universe is now. That portion may already be outside the Hubble sphere and traveling away at superluminal speed exponentially. If the universe is infinite then it is growing to be larger than infinite simply because an expansion takes up more space. Therefore our view of infinity, which is a human invention, is wrong.

No, *your* interpretation of infinity is wrong. Infinity is not a number. The usual arithmatic with infinity does not follow the same rules one would expect to find with real numbers:

1 + ∞ = ∞
1000000 x ∞ = ∞
∞ x ∞ = ∞

That does not mean, however that all infinities are the same, but there are an infinite number of infinities that are equivalent.

I know of ℵ null (countably infinite) and ℵ 1 (uncountably infinite), but I don't know if there are other larger (or subtler) infinities out there.

[Bill S]

Infinity is not a number..................there are an infinite number of infinities.

Do you not think those two statements are contradictory?

What is an infinite number, if it is not a number?