On Infinity

I was recently reading The Restaurant at the End of the Universe by Douglas Adams, and in it I found the following passage: Now let's assume for the sake of argument that the first two sentences are correct. Is the conclusion Adams draws from these two statements correct?For simplicity, let's say there is only one uninhabited planet. That follows the premises, right? So all but one of the infinite worlds in the universe are inhabited. My question is: Is the number of inhabited worlds finite or infinite?It seems to me it must be infinite. If it's a finite number, then adding the one uninhabited planet to find the total number of planets would render a finite number, which is impossible given the premise of an infinite number of worlds.So who's right? Me or Douglas Adams?Big Bang and Cosmology? Coffee House? Somewhere else?

Thread moved here from the Proposed New Topics forum.

The conclusion does not follow from the premises. If there are an infinite number of planets, and if not all of them are inhabited, then there may be an infinite number of inhabited planets or just a finite number.There are an infinite number of natural numbers. Not all of them are greater than 10. In fact, only a finite number are less than 10.There are an infinite number of natural numbers. Not all of then are even. In fact, an infinite number of them are odd.Hope that helps.

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Actually, if their god makes better pancakes, I'm totally switching sides. -- Charley the Australopithecine

This seems a little like the old problem of hilberts hotel..
In Hilberts hotel there are an infinite number of rooms,
So, a guest arrives and asks for a room
But he is told there are also an infinite number of guests,
His solution is to move the person in room one to room two, the person in room two to room three and so on, Then the newcomer moves into Room one.
It's kind of a work around for the problem of infinity.There are also level of infinity, Alef-null, alef-one, Epsilon-zero, epsilon-one.... etcThis accounts for a situation where you have infinity within an infinity.
i.e. you break something down into an infinite number of smaller parts. you have infinity.
but the original object is one of an infinite number of parts in a bigger "object" another infinity so you need different levels of infinity.It's early and I'm not doing a very good job at explaining this.there is a very good book called "infinity and the mind" By Rudy Rucker, he also has one called "geometry, relativity and the fourth dimension" which kinda deals with similar topics.

When D. Adams said, “an infinite number of worlds, simply because there is an infinite amount of space for them . ” imagine that he was thinking that it is infinite divisibility (of space) that justified for him the word “simply”. In other words imagine matter that must take up some space in its place. The place of this space is always divisible, so picture a line between two parts of this space as an operation that Adams could always do. Thus no matter how large or small of local area of matter occupancy is there is , as per the assumption, always an infinte number of lines drawn in the “universe” of this ”infinite’ space. Thus the number of material “worlds” is equipollent with the number of divisions argued for. If space is infinitely divisible even starting with an infinite spread of matter in its places there can be an infinite number worlds or an infinite amount of smaller divisions if the initial size is not infinite.So, when Adams, said, “not every one of them is inhabited”, I took him to mean that on one side or the other of at least one division there is some location that is only the place of impenetrable matter and not necessarily having life, a particular kind of organized matter, in it, but as he said “not every one of them” he must have meant that for every division one could not make an orthogonal cut to ALL of the divisions (infinite) and find life in every one. That was what he started with.Therefore, as there are only marginal placements where life may exist, these must be a finite cut-up of the original size no matter what size it is. In this way life only exists at a finite number of locations.I think this reasoning may be more than simply a mind game.I have begun to sketch how Panbiogeography via evolutionary graph theory may indeed justify thoughts about both the infinite (here the dashed lines indicate the infinite divisions of Adams etc) and finite variety. I would simply disagree with Adams that it can not be thought that life is not “in” every one of an infinte number of separations of physics’ space. What is required is a clear notion of a right angle biologically. This seems possible where graph theory may draw a topology that is not on geodesics. I have not proved this yet even for the fusion of the two kinds of graph theory I introduce here:http://axiompanbiog.com/vcarandtclaim.aspxWhat is required as a prerequisite in the thought is to show that Moran processes and HW equilibrium converge for a given track claim but because Panbiogeography do not guarantee equal weights the cause and correlation must be specified “into” the physics somewhat, which we were only assuming in Adams’ case (an infinite amount of space) (this *may* be a linguistic infinity of deep structure instead in reality).You could be correct if the space infinitly overdetermines the places the matter exists in the space but I take it Adams precluded that when he wrote "NOT EVERY"*****ONE*****. I took that to indicate that matter created the oneness but space bounded it, like Newton that Earth may be thought dynamically to a point(one earth- one point).

Good. I'm not just crazy.After discovering this thread's promotion and your and others' responses, I decided to check out what Wikipedia says about the subject. Quoting the Yajur Veda, it says: So, basically, I'm right. As you said, there could be a finite number of planets, but it's not necessarily so. Right. Because -10=, there are an infinite number of natural numbers over ten. Because /2=. I think I've got the hang of this.On a slightly related note, a friend and I were arguing over whether or not .999...=1. He contended that it didn't, while I contended that it did. I used the following proof (which he was previously familiar with): He said the problem with the proof was in the second line. His claim was that 0.999... went on till infinity, but when multiplied by ten it only went to infinity minus one, so the subtraction on the next line would leave one 9 at the end left over. But infinity minus one is still infinity, so there is no 9 left over (nor is there an end).And, if I'm not mistaken, you're a mathematician. I've been dying to hear from a reliable source. (Besides Wikipedia, I've asked two math teachers at my school, who both said .999... doesn't equal one, but I didn't bother with any proofs or challenge their answers.) Does 0.999... equal 1?

Well, they're either very poor math teachers, or they are very good but rather disingenuous to other philosophical persuasions

don't forget, 1 = 0 too. I don't know the proof for that one, though. Would have to ask my friends taking calculus (2?)

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And, if I'm not mistaken, you're a mathematician.

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Heh. A failed mathematician. But better than some of your teachers, it would seem:

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x=0.999...
10x=9.999...
9x=9
x=1

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This is a correct calculation, however the second and third steps do need to be justified. But all this needs to be done using limits; do you remember your calculus? Just nod your head as if you do and we'll proceed.First, we have to figure out what 0.999999... even means. Here is what it means:We have a sequence of numbers:
0.9
0.99
0.999
0.9999
...0.999999... just means the limit of that sequence of numbers. What is the limit? Well, if 1 - 0.999999... = 0, then the limit must be 1. In fact, this is just what it means (the definition) of the limit being equal to 1.So:
1-0.9=0.1
1-0.99=0.01
1-0.999=0.001
1-0.9999=0.0001
...Clearly (meaning that it looks obvious and I don't think anyone wants to see an epsilon-delta proof) the limit really is 0.Hence, 0.999999... = 1.Does that clear anything up?-The whole problem is the idea of infinity. Infinity can get really weird. Look up the Hotel Infinity when you get the chance. (Cool story, even if it doesn't even get to the weirdness of uncountably infinite!)

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Actually, if their god makes better pancakes, I'm totally switching sides. -- Charley the Australopithecine

The funny thing is, as often as these forums (and similar forums) deal with unusual and unintuitive things from physics, I still think that deep down perhaps the Real Numbers are one of the most (the most!) unintuitive things humans ever created/discovered (take your pick).
One of my most well-worn pure mathematics books is Hermann Weyl's "Das Kontinuum" which deals with this fact.
For anybody here who knows this stuff and hasn't read it, he tries to construct the Reals using methods that would satisfy a finitist/intuitionist. In the end he constructs the Reals, but is unable to prove the fundamental theorem of calculus or the intermediate value theorem. Although the proof contains a flaw. It's designed to test your knowledge of calculus. Same for the 2 = 0 "proof".

One can demonstrate that 0.999... = 1.0 by a reductio ad absurdum argument (i. e., by showing that the opposite statement, that they are not equal, leads to a contradiction). If 0.999... is not equal to 1.0, then there must be some number that is in between them in value, that is greater than 0.999... but less than 1.0 (one such number is (0.999... + 1.0)/2, and in fact there would be infinite such numbers). Thus, the decimal expansion of this number must have some digit that differs from the digit in the corresponding place in 0.999..., i. e., a digit other than 9 in some place where 0.999... has a 9. But this would make the 'in-between' number smaller than 0.999... when in fact it must be larger than 0.999...(qed).We have demonstrated that your friend's suggestion is not only absurd, but is vile and disgusting and your friend should be ostracized from society.

In Message 41, Phat asked me:

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If we chopped Infinity into two parts, would each part be finite?

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and I answered:

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Each part would be finite on the cut end.

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And if we cut it in three, we'd have two pieces that were infinite on one end and finite on the other, and one piece that was finite on both ends.

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Was I right or was I joking?

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surely it's either infinite or it's not. to say it's infinite at one end and not at the other suggests It's both infinite and finite at the same time.. logically impossible.If I hold an infinit piece of string at one end, It's still infinite. I.e. it stretches out infinitely from me.

you got to be joking1 is not = to 0, you know thatEdited by fallacycop, : typo

You can make anything equal anything if you use an invalid proof. Usually, this involves division by zero.