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> > > > What I think we can say for sure is that, although we're not sure about
> > > > the actual curvature of our obserwable universe, and thus we're not sure
> > > > if the space go on and on to infinity, it seems that it's quite sure that
> > > > our universe is accelerating (q<0), and from this point of view we can say
> > > > that if we send out a probe into the space even at the velocity of light,
> > > > it's likely that it will never return regardelss of the the curvature
> > > > of the universe, because it just won't overpass the expantion rate of the
> > > > universe unless there is some nontrivial topology involved. The Big Crunch
> > > > never happens in area where q<0 on Omega_l, Omega_m plane. So from our
> > > > point of view we can say the spacetime is infinite if we're thinking in a
> > >
> > > We can't say "the spacetime is infinite". What you mean is
>
> > why not ?
>
> Because "x is infinite" means that
> "for every y \in {Real numbers}, x > y".
>
> If the Universe has positive curvature with curvature radius R_C (and
> is a perturbed FLRW model, as we think), then there exists a maximal
> length spatial geodesic X_s and a maximal length space-time geodesic
> X_st .

okok but
if we set k=1 the solution of geodesic equation in FLRW metric for
distance is something like arcsine function of comoving distance times
S(t). So it's closed solution and space is finite.
but when additionaly we set for example Omega_l > Omega_cdm then S(t) has
no maximum (as is had in closed model with zero cosmological constant) and
it grows with time to infinity. hmm it sounds simple ;)
however infinity doen't appear anywhere in flormula.
What I said was that I think the data are suffucient to say that
the photon once sent in some direction will never return no matter what
curvature the universe has because of S(t). So from the point of view of
that photon the universe will be infinite

> Then, there exist y \in {Real numbers} greater than these values and
> there are no longer any x in the Universe greater than y. So it's not
> infinite.

>
>
> > event horizon accounts for all evolution of expantion according to assumed
> > model. if it's less than the curent curvature radius then we won't see the
> > probe, if it's bigger we mae but not must see it, for the curvature
> > radius is changing in time.
>
> i'll let you do the calculations here, but they are irrelevant for the
> question of infinity.
>
> > the fundamental nature of the model is that going in space we also move in
> > time.
>
> This is extremely confusing language. What do you mean by "going in
> space", "moving in time" and "we"?  If you mean there's a second time
> variable representing the time variable of a thought experiment, then
> it makes sense.  Or if you're talking about a physical particle, then
> two different time variables are the local (proper) time of the particle
> and the cosmological time.
>
> But it's perfectly possible to imagine comoving space without needing
> any time variable to "go" through it (though having a local psychological
> time variable is convenient).
well, comoving space without remembering of time. how would you describe
how it comoves then ? ;) it wouldn't differ from galileo's static space.

>
> > eg. Imagine that that space is closed, and expands slow enough that
> > a photon emited from your flashlight can round it, but as the time passes
> > expansion rate mae grow up, event horizon falls, (say cosmological
> > constant starts dominate) and the photon won't make it eventually.
>
> IMHO this is a different theme to the universe being infinite or not

why ? this is right the core of this probelm. if we consider only the
spatial part of of the solution of geodesic equation, fine, but it's
multiplied by S(t) which has no bounds because the time dimension in our
universe has no bounds (no upper limit) so the whole distance (to
something) relation has no bounds in spacetime interpretation, if I can
say so.
So here I recall the question why can't we say the spacetime is
infinite, which reduces just to what's the maximal length of your X_st ?


>
> > > If you can think of an alternative model which only models our past
> > > time cone, fine.
> > btw.
> > if the accurate model predict things in the past, I see no reason why
> > it should not predict also things in the future.
>
> You can't predict things in the past. Astronomers doing models often talk about
> "predicting" observations already made, but this is just astropolitics.
>
> The reality is that you can only postdict the past, and that if you make
> predictions they will usually be wrong. Moreover, you're more likely to
> get observing time/grants if you postdict the past (and say that you're
> predicting it) rather than if you make real predictions.
>
> As for accuracy, the model below is extremely accurate, by definition.
>
> > below this this point I don't follow ;)
>
> Another way of describing the model is that the Universe the inside of a
> 6k-light-year sphere and that initial (and continuing) boundary conditions
> on the sphere have been designed in such a way that people trying to
> understand them conclude with a series of simple (but wrong) physical
> laws. Initial conditions throughout the sphere were also set up with
> the same intention (e.g. fossils, distribution of continents, genetic
> mixes of people and other animals, isotopic ratios of uranium etc...)
>
> Maybe another way of describing it is that we live inside a planetarium
> of radius 6kly and that The Designer is pretty good at designing fun
> models with just enough clues that we remain interested and think we
> can understand the model represented in the planetarium.
>
> > > It's a model which perfectly fits all cosmological observations,
> > > including those of WMAP. ;) The Universe in this model is the inside
> >
> > gash, does it says about CMB fluctuations ?
>
> No: but since human beings have interpreted these in terms of a simple
> set of physical laws, the Christian fundamentalist model is satisfied
> - there are simply a new set of boundary conditions designed to make
> physicists interpret them this way.
>