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I will try to keep this topic relevant, but GR based theory in answer to the topic requires discussion of spacetime, including what it is. First, to back up a bit... csmyth: Repeating, " I am saying that the obviously curved trajectories of objects effected by gravity does not verify the actual existence of an entity "the spacetime manifold." Also, from a previous but as yet unanswered question, is quantum theory of gravity not an alternative to GR's curved spacetime theory? Since all GR texts and websites assume "curved spacetime" as central to understanding the theory, how is it that the ontology thereof (see recent references) is not relevant to this discussion of the shape and size of the universe? md65536, replying to my, "GR assumes that space is an entity."... I should have said... "that spacetime is an entity," in that "it" is said to be curved by gravity. (Objects' trajectories are obviously curved by gravity, but the theoretical 'malleable medium' begs for ontological inquiry. As for "space itself," mainstream cosmology posits that space is an entity which "expands." ) Yes, not an entity. It is reasonable to assume that space is empty volume, no-thing-ness. If "it" is considered to be some "thing" then ontology asks, "What?" No, GR has "a mountain of evidence" for predicting local gravitational effects. (It's forte', not global cosmology, this thread's focus.) It would not diminish GR to just refer to the curved path of objects, however, without the superfluous insistence on " curved spacetime" with ruts or grooves, or whatever which "guide" objects' trajectories. I rely on reason, that "and end of space" is an absurd concept in the real cosmos/universe, regardless of what kinds of "shape and size" we can conceive of as models or how complicated the many varieties of theoretical "manifolds" can be, whether compact or not, open or closed, based on metaphysical "higher dimensions," etc... etc. Even the three standard varieties of possible "shapes of cosmic space" beg the question, what lies beyond any conceivable shape? The reason an end of space is absurd is that all such boundaries are just concepts, not some kind of "wall out there." And then of course there is the further absurdity of ignoring the question, "What is beyond this wall?" The human mind has a tough time with infinity, but this does not mean that, therefore the universe must be finite, just because are minds and concepts are.

MigL: Presentism says that now is now everywhere, for the sun and earth and the whole cosmos... not about how much time it takes for light/information to travel between locations... so, no "different now" in different places. But of course info about "now" at the sun (present flares, for instance) will take 8+ minutes to reach us. Presentism examines the ontology of time without the usual reification of time as an entity. Check out my "Ontology of time" thread in "Speculations." Why would it require a "formally expressed law" to know that an egg can not be unscrambled and then unlaid, or to know that we can not visit our unborn great grandchildren until they are, in fact, born (as previously expressed in this thread?) I think science fiction has been adopted by a pseudo-science faction in this regard, to the detriment of science in general and reason itself in particular.

DrRocket: I just got Deiks' volumes by Googling "Deiks Spacetime Ontology" and the International Society for the Advanced Study of Spacetime with "Spacetime Society"... both in less than a minute... not that difficult to find... just requires a smidgen of interest in the ontology of spacetime. csmyth3025: I am saying that the obviously curved trajectories of objects effected by gravity does not verify the actual existence of an entity "the spacetime manifold." There is a lively ongoing debate among well credentialed scientists and philosophers of science about what "spacetime" is (if anything.) See key word references above. Also, regarding "curved spacetime" as the established fact for how gravity works, what about the quantum theory of gravity? I don't know the technical/math details, but I think it is a whole different theory of gravity, no? Btw, presentism is another study which inquires beyond local frames of reference and "time environments" and the usual "who sees what and when" of relativity. It can be ignored, but it will not go away. The most simple and obvious illustration of presentism is that now is now for both earth and sun even though it obviously takes over eight minutes for sunlight to travel to earth. Then, this "now" can be extended to "now everywhere," that the universal present is not limited by lightspeed. Calling it wrong doesn't make it wrong. Finally, to this exchange: me: DrR: Do you think Einstein and GR have answered the the two fundamental questions above, the focus of this thread? GR obviously "works well" locally, but this thread is a global/universal focus, which you refuse to address.

DrRocket: Glad you are enjoying your superior perspective. But would you mind answering a few sincere questions?... starting with my last post would be very much appreciated. Also, before you declare victory for the established fact of spacetime, please do the required* research into what the International Society for the Advanced Study of Spacetime (ISASS) has been debating for most of the last decade. *(If you care about what spacetime is) Any search will get you there... if your mind were not already 'made up.'

Still trying to catch up with replies to previous posts, specifically parts of DrRocket's May 11 post. I have a global notion of space as unlimited, i.e., without end. What would an end look like? And beyond that? Also one can reasonably "imagine" that beyond focus on local "time environments" (duration of local events) the present is present and ongoing everywhere, universally. (See intro to Deiks' volumes cited below.) Regarding your matter-of-fact assertion that ... Have you read either of the two volumes of papers compiled by Dennis Deiks on The Ontology of Spacetime as presented at the international conferences on same? It's not as cut and dried (matter of fact) as you present it. But of course, you may not be interested in what "curvature of the spacetime manifold" means in the real world of observable phenomena. Yes, objects effected by gravity often have curved trajectories. But do you care if the phrase "curvature of the spacetime manifold" has a referent in the real world, or is a working abstract concept enough without a referent? (An ontological question?) Of course there must be many things in the cosmos that "go unnoticed." But how would small forms (speaking of real entities, not just concepts) or defined spaces with boundaries existing in real 3-D space make the whole universe or endless space finite? It is a "reasonable" question, and directly to the point of this thread's inquiry. And since three axes describe space as volume (defined with specific size or infinite), to what would a fourth (or fifth or eleventh) spatial axis possibly refer? What empirical facts? What distinguishes the many extra "dimensions" in M-Theory from metaphysical "dimensions", as "spiritual realms" beyond the "merely mundane" world of science? Ps: Still not finished with Weinberg's philosophy bashing paper, but any reply will be posted in the philosophy section. But this post has again addressed the ontology of spacetime, so your reply to the above may be sufficient.

MD65536: GR assumes that space is an entity which is curved by mass as a theory of gravity. Quantum mechanics has an alternative theory, as I understand it. Anyway the status of space as a malleable medium/entity is not a given fact, and ontology asks "What is it, if anything besides empty volume?" This is a very relevant question for science, even if it is ontology, another bastard child of the philosophy of science. Forms made of cosmic "stuff" may be "compact" or closed (even the visible cosmos but the question this thread asks is still about what lies beyond any such form, not to assert as a given that space has form, but rather that stuff in space has form... with no end of space beyond any conceivable form. Btw, DrRocket, I've read about half of the paper, "Against Philosophy" by Weinberg, and I'm working on commentary in reply. A correction to my post 86 (too late to edit it), re: "A curved plane exists as a 3-D form, regardless of shape." I should have said "a curved surface." A plane is flat, not curved. (And "flat space" as describing Euclidean 3-D space is a misnomer, though it looks like we are stuck with it as ubiquitous usage would have it.) I have decided not to post my reply to the Weinberg essay in this thread, as it belongs in the philosophy section. Maybe I'll start a new thread there on the relevance of philosophy to science... "Are epistemology and ontology relevant to science?" (or... "Who cares what it is as long as the math works? )

Thanks. I'll check it out. Meanwhile you've inspired me to dig up my reply earlier in this thread to the proposition that philosophers are "dopey" and that the only reason to tolerate philosophy on a campus is to broaden sophistication in the humanities of the real students and teachers, the people that matter, the scientists. (I saw this, and still do, as the ultimate scholastic bigotry and arrogance... I'll find it.) Iggy, You wrote: I did read your link on how the finite can be "conceived as" unbounded. Here are selected quotes and my commentaries: (My bold.) I am astounded that the development of non-Euclidean geometry and the named references (however well respected) make a fact out of (and legitimized) our ability to “cast doubt” on a previously obvious fact... there is no possible end of space. (See the several absurdity scenarios I have already presented about walls and beyond.) There is still a place for reason in science. We can imagine whatever strikes our fancy. Unicorns are cute too. But the analogy, inter-dimensional (2-D to 3-D) relevance is limited. In the real world, we are 3-D creatures in a 3-D universe... (add time if you like, for motion, which “takes time.”) Recall my earlier posts on the transition beyond Euclidean geometry. (Warning: It involves ontology!) A curved line actually exists on a plane; A curved plane exists as a 3-D form, regardless of shape. A “curved volume” is, so far, just an act of imagination (4-D space.) I’ll dig up the link for the whole ontology piece again if you are interested. ...For no particular reason presented but the whim of the author... but of course it is his thought experiment. What causes this to happen, and how did a counter point develop? “Quite analogous” is not “the same as.” A plane remains 2-D and a sphere remains 3-D They are conceived as analogies to actual unlimited space, infinite space. Yet in reality there is always more space beyond what the finite mind conceives as a “form, a closed space.” In other words (same meaning), no limit to surface travel on a sphere is this analogy’s version of limitless. But any closed space (defined form) exists in the real 3-D universe, and the surface of any sphere is its limit/boundary as measured from its center outward (toward, expanding into, actual universal 3-D space), and beyond that surface must always be more space, as I’ve said a dozen times or more. I'm done saying it now. I think that one must abandon his/her faculty of reason to believe that there is an end of space, that the universe is a "finite form." But of course relativity theory encourages that (abandonment of reason... even for all the good theory it provides for predicting the finer points of gravity... with the magical fabric "spacetime" and the amazing mystery of constant lightspeed proven by SR.) Seriously.

This is piecemeal. I hope to get back to more in-depth replies later, after reading some of the recently suggested links. But the following simply addresses MigL's mistaking my "referents" above for "references." Me: MigL: Referents definition (from my computer dictionary): DrRocket is very fluent in math and the "higher dimensional" non-Euclidean geometries, for instance, while my forte' is in the philosophy of science, specifically the ontology of such "dimensions." The two realms of expertise are obviously not in communication in this thread, and DrR seems to believe that ontology is irrelevant to this discussion. (Correct me if I'm wrong.) So when he or anyone speaks of the various theoretical properties/descriptions of different theoretical manifolds and "higher dimensions" or of "spacelike slices of spacetime" as what science means by "space" and I refer to the ongoing investigation of what "spacetime" actually is if anything (see multiple references to the International Society for the Advanced Study of Spacetime)... to the point here... "Is space infinite?"... we talk past each other from different worlds. The "slices" in "spacelike slices of spacetime" are mental concepts in the abstract world of math and non-Euclidean geometry. "They" have no real world referents. Space is not that complicated in the sense queried in this thread i.e., whether or not space is infinite, i.e., does it have an end or boundary or not? If one is proposed, then the ontological questions remain, what is the nature of such a boundary or end, and then, of course, beyond that... what?

Some say up to 26. The 11th "dimension" was first supposedly 'debunked' and then, years later, praised as the way to unite the various (five or six) versions of what a "string" is supposed to be. Please explain each of the string theory dimensions beyond the 3-D axes of volume/space and "time," (if we must call event duration a "dimension.") Or maybe it's still all in the very imaginative minds of M-theory's history of creators (read "theorists.") How does such unobservable speculation become such "popular science?"

I think steevey answered the title question well in post #3: Math and "higher dimensional geometry" by itself does not confer understanding, and all the technical concepts discussed in this thread (mostly without "real world" observable referents) do not answer the simple questions, "How big is the universe... and, Is there an end of space?" Simple answers... "infinite, and no."

Iggy: You don't seem to know what the words finite and infinite mean. You can make up your own definition, of course, but here are the definitions I have already quoted earlier. Wikipedia: Merriam-Webster: Finally, a repeat of Wikipedia on the topic question (from my post yesterday which you apparently didn't read): Logically, what confines the above arbitrary "diameter of the universe" to "d?" (What prevents "d+1" of whatever units of distance?) You seem incapable of thinking about what would lie beyond such a well defined (finite) volume. If "something," what?... if nothing... more empty space, ad infinitum.

I spent much of this morning on the net researching this topic. One thing which particularly struck me is how all the technical lingo of math, geometry and various concepts associated with complex "manifolds" seldom explicitly refers to observable phenomena in "the real world." I'll start with the special use of "compact" gleaned from Wikipedia: In contrast (but still "without boundary") is Mathworld's definition of an open manifold as a " noncompact manifold without boundary." So apparently both an open and a closed manifold can be "without boundary." But an open manifold can be a manifold with boundaries. Wikipedia also says: So "closed" in this context has a special, counter- intuitive meaning. One usually thinks of closed as having an inside and an outside, and in the context of this thread, the space outside of a closed cosmos (I will not say "universe" in this context) has no end or boundary, so is therefore infinite. While I was cruising Wiki's links on the size and shape of the universe, I came across the following: Yes, in the real world, the debate does indeed "go beyond both." And of course, the " 4-dimensional space-time of the Universe" is used without a second thought to the ontology of its referents in the real world, i.e., beyond the obvious, that space/volume has three dimensions/axes and that everything moves "in time." I'll finish my wiki note presentation with it's comments on infinity, global shape, and the observable universe: In other words, in lieu evidence to the contrary, (no wall out there by any stretch of common sense or intelligent thinking) and to the point of this thread, space (the universe) is infinite.

Not much time tonight, just a quick reply. I wrote: ... and md65536 replied: Posit any form with boundaries/edges any size and shape you want. Beyond that, there is no limit, no "brick wall", and if there were (duh!), ... beyond that, either more stuff scattered about in space or infinite empty space. No, I have not concluded "that there has to be something beyond." Nothing beyond is still empty space, whether or not there is more "stuff" out there or just endless (what end!?) no-thing-ness, empty space.

I'll start with your last post and then go to previous unanswered posts. You could benefit from a little study of ontology. When cosmologists claim that the Bang commenced a very rapid expansion of space, it is essential to understand what is said to be expanding. Likewise when every relativity text and website begins by "explaining" that "spacetime is curved" by the gravitational influence of mass, the question "what is curved?" is essential to understanding the theory. That is the focus of the several conferences over the years on the ontology of spacetime. . So you assign "properties" to "nothing." Still, if there is nothing out there beyond what we can see, how can there be an end to space-as-emptiness, no-thing-ness? There can not. You posit "things with no volume." I guess this is like "nothing" with "properties." You make no sense to me. Now to previous posts. Post 55, zapatos: What does empty space mean if not absence of things, no -thing. Then you introduce a future house with a not yet existing basement as an argument against the infinity (endlessness) of space? It just does not fly... confused thinking, in my opinion. Post 56 by md65536: "The universe" means all there is (known and unknown) as contrasted with the visible cosmos. The question still remains, even with a supposed "finite universe," what lies beyond an "enclosed, finite universe?" See comment above. An "edge of space" (as emptiness) makes no sense, again as above. Right. No-thing-ness, emptiness, space... is not an entity, and there can be no end or edge, boundary, or wall-out-there to empty space. As I’ve repeated many times, if one posits such a boundary, what is beyond the boundary but more space, infinite space? zapatos, post 60, replying to my, Finite space is easy to imagine. The space within any geometric form will do, or any form at all of finite size and the boundaries which de-fine it. Infinite means without boundary, edge, or end. My point was that just because you can not imagine endless space does not mean that there must be such an end. Your "logic" is backwards.

Me too. Thanks for the links you shared. I found the spherical shell theorem most interesting.

"No thing" means emptiness... empty space. Beyond any cosmic "structure," shape, or form you can imagine is more space, whether more stuff exists in it or or not. The finite mind has always had trouble "grasping" or comprehending infinity... endless space, but that doesn't make the universe finite just because our reach of mind may be finite and "create" a finite universe. md65536: I have always argued that space (and time and spacetime) are not entities. Things are entities. Some argue that relationships between things are entities too... as in the substantive vs relational spacetime debate, which I am not going into here. If you posit that something is being curved or stretched (space in this case), it is up to you to explain what that something is in a coherent ontological argument. That is why the spacetime ontology argument is still ongoing with no consensus in sight. Distance usually means the linear measure of space between two points or objects. But does this mean that there is no space beyond things between which we can measure? No. As Chris astutely pointed out above: MigL: My "point" was that a geometric point is just a locus without dimension. Specifically a point has no volume, which is required to "contain" anything... like all the matter in the cosmos squeezed into such a point of zero volume, as per Hawking's absurd statement quoted above (re: the ultimate singularity.)

Spyman: I have no idea. What has happened to material from a long, long ago Bang way beyond our sphere of visibility is clearly speculative, with no information upon which to base such numerical/geometric guesswork. Dust and gas could be "blown around" by local solar winds, for instance or not travel as far out as more dense supermassive cumps. I just don't think we can assume spherical symmetry for stuff that has interacted gravitationally (and drifted around) for ages, or having been already turned around and on the way back. I could be wrong. Yes. And of course if many Bangs happen all over the universe, not from the same epicenter, all bets are off as to overall shape of "all there is." I think your last three paragraphs are imposing math upon a model without sufficient information. I'm willing to leave it there, unless you want to pursue it further. This is, of course, all "off the deep end."

Iggy: Space-time is now an established entity with intrinsic curvature? Inform the International Society for the Advanced Study of Spacetime! Write a paper on it and maybe you can deliver it at their next conference. ..."and there is no 4th spatial dimension." Didn't I just say that... more than once? Define "closed." (No, don't. Just explain what lies beyond the "closure" or "form" on whatever scale.) Again, I do not advocate (and never have) a 4th dimension. Still batting zero in communication here, as always with you. Why don't you just leave it alone if you can't understand it?

How about a little basic Euclidean geometry for openers into what space is, what "shape" it has (or not) and whether or not it is infinite. And before anyone blasts me with an anti-Euclidean, "we are past all that" lecture, please review the Ross paper on the Ontology and Cosmology of Non-Euclidean Geometry. at http://www.friesian.com/curved-1.htm... and my selected quotes from post 47. Before a "fourth spacial dimension" was invented, still with no ontological referent in the real world beyond math and abstract geometrical concepts, there were three spacial dimensions and one time "dimension" (which is simply event duration)... 1: the line (length with no area), 2: the plane (flat area) and 3: volume or space, whether defined with boundaries (all possible bounded volumes with whatever shapes) or endless, infinite space (without any boundary.) So, right off the bat, "flat space" makes no sense unless it is confined to a 2-D plane. Likewise "curved space" makes no sense in basic Euclidean unless describing a 3-D object like a curved surface, and the surface is curved, not "space." Review, Ross: A 4th spacial dimension still makes no sense. Space, whether confined (bounded as defined geometric shapes) or infinite, is fully described by three axes. A fourth is superfluous... a product of imagination. I'll go to specific replies now. To first comment: Absolutely! Second: Cute. A solid brick universe, or infinitely thick "wall!" No boundary (unbounded) means infinite, unless specially defined and ontologically ungrounded... like "4-d space." There is no limit to the space outside any defined sphere. What "curves" if space is just volume, regardless of what it contains or how empty it is? Stuff has shape, curved trajectories, etc., not empty space. See above simple Euclidean geometry. (Non-Euclidean is still math/model/conceptual, as per the Ross ontology.) A sphere is 3-D geometry. 4-D space is a pseudo-science myth. No agreement. I know "flat space" is in common usage, but it invents a new meaning for "flat," as I explained above. No. "It all" exists in 3-D space. No 4th required. As above, three axes describes all volume. You can add time, but to be clear, I would not call it a "dimension" but just the movement factor... that "it takes time" for all stuff to move around, and it is a dynamic, not a static snapshot universe. Back later for further replies. PS: Btw, needless to say, a point has no dimension, but is the 'starting point' for geometry. (A point can not "contain" anything... including Hawking's primordial 'singularity', which he said is "infinite mass density in a point of zero volume." Totally absurd of course, even if Hawking said it.)

I thought the my quoted phrase above was my answer... earlier bang(s) morphed from spherical shell/membrane shape, after eons of gravitational coalescence into non-uniform clumps beyond our visible horizon. No shell beyond our visible horizon but randomly distributed clumps. (I set aside the large scale balloon membrane containing our micro-scale visible cosmos deep in its membrane quite awhile back because the super-large scale cosmology will forever probably remain unknown.) But if we acknowledge the possibility that similar "bangs" occur throughout the universe, from different loci/centers, that cosmology would supply the "interactions from the outside." I keep an open mind to all the possibilities and remain open to falsification of each... like the shell theorem invalidating my previous model with multiple concentric shells. Here is a relevant piece from a Kelley Ross paper I've been studying in depth for the dialogue in the "Is Space Infinite" thread: Ross quotes Scientific American, "Is Space Finite?" [Jean-Pierre Luminet, Glenn D. Starkman, & Jeffrey R. Weeks, April 1999, pp. 90-97]:

This is the conclusion of Ross' paper, not a reply to Iggy, which I may or may not get around to. It does address the basic issue of the relevance of philosophy and the extrinsic vs intrinsic point of departure from Euclidean to non-Euclidean ontology and cosmology. BTW, I have made notes and commentaries throughout the paper if anyone wants to get into specifics... including the departure from Euclid's fifth postulate. Here are quite a few quotes up front... inviting comments:

I think my comments here belong in the philosophy section rather than in the relativity section of Physics. Everyone by now knows that I have an ontological beef with "spacetime" as an assumed entity, and with the concepts central to non-Euclidean geometry as per, for instance, intrinsic vs extrinsic curvature as pertaining to what conceptual manifold (Euclidean space vs the Lorentzian 4-D manifold.) I have studied the different manifold "dimensions." (See the Wikipedia intro: http://en.wikipedia.org/wiki/Manifold ) But I am also well aware of the ontological study of these "dimensions," and the assumptions inherent in the transition from the Euclidean paradigm to non-Euclidean. I have before cited an in-depth paper on this by Kelley Ross on the above. Here are two links to his papers, the second being very clear and specific on the varieties of manifolds and their assumed "dimensions"... for anyone not yet convinced that non-Euclidean is the absolute truth on the subject: http://philpapers.org/rec/ROSTOA http://www.friesian.com/curved-1.htm I think it is a mistake to dismiss all philosophy as irrelevant to science. How can it be irrelevant to know exactly how we know what we think we know, i.e., epistemology? Once one abandons simple Euclidean geometry, what do these higher dimensional manifolds actually refer to in the "real world?" (See the Ross links above.) How is it that a curved surface is two dimensional since it is not a flat, 2-D plane? What happened to the line, plane, and volume geometry of the real world, being one, two, and three dimensional respectively? (Please read Ross before lecturing me yet again on higher dimensional manifolds.) Anyway, I jumped into this thread to agree that space is infinite on the logical grounds that there can be no "end of space," therefore it must be infinite by definition.* In a nutshell, as I said, "A 'wall out there' as an end of space is of course, absurd." *Here is the relevant Merriam-Webster online definition of infinite: I'll go back to the philosopher's corner now unless someone here wants to give me a reasonable account of an end of space or discuss Ross's Ontology and Cosmology of Non-Euclidean Geometry. This is bigotry and arrogance at its worst, unworthy of science, which must ask not only what we know but how we know it, the field of epistemology... a branch of philosophy direcly relevant to science. The ontologists I've studied who are specialists in spacetime, for instance, are as far from "dopey" as it gets, brilliant and well credentialed philosophers of science and scientists workinf respectfully together to answer the basic "What is it?" questions.

Yes, but I thought that after cosmic eons a previous bang or bangs, originally in spherical distribution as from any symmetrical explosion, would have had time to gravitationally interact and coalesce into super-SMBH's in non-uniform clumps distributed more randomly than the symmetrical sphere configuration. I've wondered for a long time what shape cosmic "stuff" would take by the time (if ever) it reached the "gravitational net" required for reversal in the oscillating model. If coalesced into mega-masses all around our cosmic horizon (but no longer uniformly distributed), they would eventually pull parts of what we see at different rates of outward acceleration and begin a chaotic phase of gravitational interaction between returning matter and outgoing matter. Everything would eventually come back to the common Bang center/locus but not all at the same time... hence the "cosmic juggling act" of multiple bangs and crunches. But you have given me much to think about, re-considering this model, and none of us yet understand the mystery of the present acceleration of expansion. Maybe after the complex gravitational interactions above, a new overall center of mass would be established... a new locus for the next round of crunches and bangs... a new center for each bang/crunch cycle. But if we must assume that symmetry is maintained throughout the bang/crunch cycle(s), then the latter cosmology would be invalidated.

I don't know. Looks like my model is in trouble. I was thinking that eventually* we would see uneven rates of expansion as previous bang material in super- SMBH clumps reverses, implodes and approaches our visible cosmos. (*Possibly way beyond the life of our sun.) Understood. But astronomers keep finding more matter (not even counting "dark matter") with improved instruments and techniques, which makes reversal and oscillation more likely. The jury is still out on that, and we still don't know what is causing the accelerated expansion. We don't know what kind of "supermassive" objects might lie beyond our cosmic horizon or how they may be distributed after eons of gravitational interaction, so I don't think universal cosmology is all that well understood, and maybe never will be. Interesting links. Still looking for time for a deeper study. How does the definition of a "closed system" apply to my 'universal cosmology' comments above, given the unknown nature of what is "out of sight." This may kill my model, but I'm not yet totally convinced. I'll need to study in more depth. No can do. Back to the drawing board, and more study of extrapolations beyond our present limits. Maybe for far future civilizations to see, if the "far out stuff" ever turns around. Thanks again.

The topic is Infinite Space, not "Is the surface of the earth infinite?" The metaphor does not address the topic. Of course one will not run into an insurmountable wall while circumnavigating the globe. But beyond any sphere is space, ultimately infinite space. A "wall out there" as an end of space is of course, absurd. I know science has a special meaning for the phrase "finite and unbounded space" using a spherical surface as a metaphor. But unbounded means no boundary, and that means infinite in my vocabulary, the metaphor notwithstanding. For example, DrRocket wrote: ...Meaning if you stay on the surface, you will not run into a surface boundary. In the first place, a sphere occupies 3-D space,* not 2-D. (A plane is 2-D, and a line is 1-D... basic geometry.) * Any curved surface is a 3-D object. And, again, beyond the surface of any sphere is more space. The challenge is addressed to those who believe that space is finite.