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How can something that's already "infinite" expand? ←Baseball Bugs What's up, Doc? carrots→ 21:53, 29 June 2018 (UTC)[reply]
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Yeah, How can something that's already infinite expand? Limited Brain Cells (talk) 22:20, 29 June 2018 (UTC)[reply]
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See Hilbert's paradox of the Grand Hotel. HiLo48 (talk) 23:05, 29 June 2018 (UTC)[reply]
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The flaw is the tendency to think of infinity as a number. It isn't. ←Baseball Bugs What's up, Doc? carrots→ 23:47, 29 June 2018 (UTC)[reply]
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I'm not sure if that is an enlightenment moment or a criticism of Hilbert ;-). To answer your original question: Just think of the real number line and expand each segment by 10 (.e. by mapping it through the function defined by f(X)=10X). The result will have "the same size", indeed, in this case it will actually be the same, and yet the distance between any two original points now is 10 times larger. --Stephan Schulz (talk) 05:04, 30 June 2018 (UTC)[reply]
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See also cardinality. Not all infinite sets are the same size. A simple way of visualizing this is Cantor's diagonal argument (here's a YouTube video walking you through it). --47.146.63.87 (talk) 06:33, 30 June 2018 (UTC)[reply]
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To answer BB's question directly, in the context of the Big Bang, something that's already infinite can "expand" if the objects in it move farther apart.
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In the case of an infinite universe, universal expansion doesn't mean that the universe actually gets larger. It just means that galaxies (or in general, objects that are not gravitationally bound to one another) move farther away from one another.
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On a separate note, yes, there are infinite quantities that can be thought of as "numbers", and they are not all the same size. But that is pretty much unrelated to the Big Bang. --Trovatore (talk) 08:13, 30 June 2018 (UTC)[reply]
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Infinity is not a number. ←Baseball Bugs What's up, Doc? carrots→ 16:39, 30 June 2018 (UTC)[reply]
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That's a common platitude. It's even true, for certain values of "infinity" and "number". But by itself it's not well enough specified to really mean anything.
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There are definitely infinite quantities that are useful to consider "numbers" in some contexts. Whether any of these should be called "infinity" full stop, and whether it or those should be called "numbers", is more an argument about terminology than about anything substantive, as far as I can tell. --Trovatore (talk) 17:10, 30 June 2018 (UTC)[reply]
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My math teachers always said "Infinity is not a number; it is a concept of expansion without bounds." ←Baseball Bugs What's up, Doc? carrots→ 18:14, 30 June 2018 (UTC)[reply]
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Bugs, first of all, you have to remember that math teachers have objectives to meet other than mathematical precision. They have a certain amount of material to get through. They don't want to spend time an awful lot of time answering the question "but Mr Josephson, what happens if you set x=∞?". Especially when they may not understand the concepts related to the mathematical infinite all that deeply themselves.
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That said, there is a longstanding philosophical debate about potential infinity versus actual infinity. Prior to the work of Georg Cantor, it was probably the default position among mathematicians that only potential infinity was worth taking into account. That is no longer so, has not been so for more than a century, but it takes a while for math education to catch up, especially when there is no particularly strong incentive for it to catch up. --Trovatore (talk) 03:59, 1 July 2018 (UTC)[reply]
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Bugs, it depends on what you decide to call a number. You could start with the idea of natural numbers (1, 2, 3), but then add the concept of negative numbers, then fractional numbers, irrational numbers, imaginary numbers, infinitesimals, etc. Infinite numbers are just another extension of the number concept that you can throw into the soup. Trovatore might have been referring to cardinal numbers from set theory, but infinite values occur in other places too. The extended real line is the real numbers from calculus, with +∞ and -∞ added on. Or in complex analysis, the point at infinity corresponds to the north pole of the Riemann sphere. The cool thing about math is you can make up stuff as you go along, and as long as you're consistent about it and interesting stuff comes out, it's "right". 173.228.123.166 (talk) 00:35, 1 July 2018 (UTC)[reply]
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Carl Sagan put it this way: "Think of the largest number you can. That number is no closer to 'infinity' than is the number 1." ←Baseball Bugs What's up, Doc? carrots→ 00:48, 1 July 2018 (UTC)[reply]
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Billions and billions, pah ;-) The largest number I can think of is a large cardinal, which is very very infinite. It's just a matter of what you decide is a "number", which is a human-language term that's mathematically somewhat flexible, so Carl Sagan and your math teachers simply took a narrow view. Transfinite number is an article after all. Philosophers used to lose sleep over completed infinity and maybe still do, but I think in math this is no longer a source of worry (Trovatore would know better than me). Regarding infinity in math, you might like this site: http://cantorsattic.info 173.228.123.166 (talk) 04:16, 1 July 2018 (UTC)[reply]
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On 'If you could magically transport back to the beginning of the universe, by what mechanism would you determine if it was the size of an apple' I would think one could then also magically transport an apple back and compare the two ;-) Even if the magical transport existed how one would actually fit in the space if it was finite is problematic, if it was like the surface of a balloon in 4D then you simply wouldn't fit even though there would be no boundary. Dmcq (talk) 08:39, 30 June 2018 (UTC)[reply]
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If infinity is not a number, then how can the infinite universe expand? Limited Brain Cells (talk) 22:26, 30 June 2018 (UTC)[reply]
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The number line is infinite. Think of moving each number to double its value. You end up with a number line that has expanded, but is still the infinite number line. (As described by Stephan Schulz above.) Dbfirs 00:09, 1 July 2018 (UTC)[reply]
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The number line has a scale? ←Baseball Bugs What's up, Doc? carrots→ 00:27, 1 July 2018 (UTC)[reply]
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Yeah, it does; if we're using the number-line as an analogy for the expanding universe then we might say that its scale is the value of the cosmological constant; or we can rearrange the math and distill it into the cosmological scale factor; or we could use an unusual representation of the Hubble constant; or it might be expressed in some other values in the various equations that physicists write to express fundamental interactions. These important equations, like the Einstein equations of general relativity, are very dense: the positioning and value of each squiggly Greek symbol has a profound meaning that requires intensive study to understand, because each symbol in theae advanced equations of physics is a compact abbreviated representation of a lot of very well-developed earlier work.
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Perhaps most importantly: unlike pseudoscience, cosmology makes quantitative experimentally-verifiable claims - so when we talk about the changing scale/size of the universe, we can draw back to an experimental observation, like measurements of the statistics of observed redshifts in distant astronomical objects. In other words, when we say that the "size" of the universe has changes, this isn't just technobabble: it means something specific that we have measured.
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Nimur (talk) 12:21, 2 July 2018 (UTC)[reply]
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No matter the "scale", and no matter where you are on the number line, you're still just as far away from infinity. ←Baseball Bugs What's up, Doc? carrots→ 14:07, 2 July 2018 (UTC)[reply]
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Yes, that's true. What's your point? --Trovatore (talk) 20:18, 2 July 2018 (UTC)[reply]
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That infinity is not a number. ←Baseball Bugs What's up, Doc? carrots→ 22:22, 2 July 2018 (UTC)[reply]
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That's a meaningless statement. Well, it doesn't have to be meaningless; if you specify what you mean by "infinity" and "number", then it can be meaningful and true. But you haven't specified what you mean by it, or explained what that has to do with the Big Bang in an infinite universe. --Trovatore (talk) 22:24, 2 July 2018 (UTC)[reply]
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If you don't understand what "number" and "infinity" mean, you should file a complaint against your high school math teachers. ←Baseball Bugs What's up, Doc? carrots→ 23:43, 2 July 2018 (UTC)[reply]
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Well, the problem is not that I don't have meanings to assign to the words "number" and "infinity". The problem is that there are too many meanings, and for some of them your statement is true, and for some of them it isn't.
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For example is aleph-naught "infinity", and if so is it a "number"? It's definitely infinite, but that's not necessarily the same as being "infinity". It's not a natural number or a real number or a complex number. But it is a (transfinite) cardinal number, and can be identified with a transfinite ordinal number.
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But more to the point, I don't see how any of this relates to the question of universal expansion in an infinite universe. Can you make your point more explicitly, with attention to that issue? --Trovatore (talk) 23:59, 2 July 2018 (UTC)[reply]
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This is what happens when BB endlessly makes dubious assertions without citations. See [1]128.229.4.2 (talk) 18:06, 3 July 2018 (UTC)[reply]
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Why should I believe you over my math teachers? Or EO, which says a number is a quantity,[3] while infinity is not a quantity.[4] ←Baseball Bugs What's up, Doc? carrots→ 19:15, 3 July 2018 (UTC)[reply]
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So first of all, 128, no need to bring up old stuff. I'm less interested in chiding Bugs for past behavior than in getting him to specify what he sees the putative non-numberhood of infinity as having to do with universal expansion in an infinite universe.
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Bugs, if you're going to get into "argument from authority" using your math teachers, then you might ask yourself whether my PhD in pretty much this exact subject counts for anything. It doesn't, of course, not really — but then neither do your teachers' opinions. As for etymonline, it is not a reliable source as regards mathematics. --Trovatore (talk) 19:22, 3 July 2018 (UTC)[reply]
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Well, you're the one that doesn't seem to understand the terminology. ←Baseball Bugs What's up, Doc? carrots→ 19:31, 3 July 2018 (UTC)[reply]
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Asked and answered, Bugs. There are lots of meanings of both words, "number" and "infinity". Which do you mean, specifically? --Trovatore (talk) 19:34, 3 July 2018 (UTC)[reply]
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The Wikipedia article Infinity defines it as "a concept describing something without any bound or larger than any natural number." Oddly enough, that's exactly what our math teachers told us. And that was long before there was such a thing as Wikipedia. ←Baseball Bugs What's up, Doc? carrots→ 19:36, 3 July 2018 (UTC)[reply]
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Right. So aleph-naught, for example, is larger than any natural number. Do you count it as "infinity"? If not, why not? Do you count it as a "number"? If not, do you consider its classification as a cardinal number to be a misnomer? --Trovatore (talk) 19:38, 3 July 2018 (UTC)[reply]
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Each of those items are referred to as "countably infinite", which means you can make a list of them... or more accurately, you can start to make a list. You can't ever finish the list because... get ready... it's infinite. ←Baseball Bugs What's up, Doc? carrots→ 19:47, 3 July 2018 (UTC)[reply]
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Well, you can't physically make a list, no. But that doesn't distinguish this case from the finite case. You can't physically make a list of, say, a vigintillion items, either.
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From the mathematical realist perspective, such a list exists as a Platonic abstraction, even though you can't physically write it down. Not all mathematicians are realists, but even the formalists still usually find it convenient to talk as though the completed list existed.
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In any case, you still haven't told us what you think all this has to do with universal expansion in an infinite universe. --Trovatore (talk) 19:52, 3 July 2018 (UTC)[reply]
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