# A set

So, if we wish to discuss the set of sentient beings S, we know with certainty that it is not empty.

So we have at least

S={h} where h = humans.

If we consider the possibility of degrees of sentience, are there beings with less sentience than humans? Recent work with sign language and primates suggest that they may be closer to sentience than not so it is not entirely unreasonable to and a few species to this set. If we accept degrees of sentience then it can be ordered.

S={S1,S2,....Sh} where S is a specific species and Sh is humans. Humans in this set are 'the most sentient'

If we assume that life exists elswhere in the universe, and that sentience is a probable outcome of evolution then the set can reasonably become

S={S1,S2...Sh...Sn} where Sn is the most sentient being and Sh (humans) exist somewhere in the continuum between least and most.

Any problems with this so far? More to come.

#51Got another one, or at least a continuation to y previous (that there might be an upper limit beyond which you no longer recognise a god).

Has there ever been a recorded case in anthropology where an advanced group met an atheistic group, and the atheistic group hailed them as gods?

Is it even possible?

If it's improbable, then there are an infinite number of atheistic societies of which it will be true, but it may not be true of ours.

Point being: atheism might be that "point" I spoke of above, at which we stop thinking of things as god, and where the Arthur C Clarke argument breaks down.

T="theists who's posts are fun-to-read, truth-seeking and insightful". Your own T will be different, but Tdewi includes { Avecrien, Cory T, crocaduck, JHenson, jread, wavefreak }

#52wavefreak wrote:This is false. Even if you have an infinite set, you can't say that all "possible" things must exist.

Counter-Example: If you have a set of positive integers where each member is described by the function {m[0]=n^2; m[i]=n^2+m[i-1];} (where n is a random positive integer). It is possible that such an infinite set may contain the number 4, however it is also possible that it does NOT contain the number 4. There are several possible ways "4" can come about, however it can be shown inductively that certain sets can NEVER contain 4 once a single member exceeds 4.

In other words, unless you have sufficient information about a particular infinite set, you can't say all things must exist.

Another example: if there is an infinite universe, there is a non-zero probability that there are an infinite number of beings 100% identical to you. This is not gauranteed to be true, as there would also be a non-zero probability that outside a certain finite subset there's nothing but vacuum, black holes, vanilla pudding, etc.

Even if the universe is infinite, given we have a lack of a complete picture of it does not mean that all things that are possible must exist in it.

Why yes, I can believe it's not butter!

#53canofbutter wrote:What can we say about the infinite set of universes that the multi-verse theories (hypotheses?) allow.

1) They contain information

I'm not sure I need to assume anything else. The existence of information is a necessary condition for intelligence. We know that one of these universes evolved an entity capable of processing that information (We exists therefor it is a certainty). Therefore there is a non-zero probability that entities can form in these universes that process information.

So the real question is, what is the probablity that other universes have information processing entities? Well the same processes that spawned our universe from the cosmic foam spawned all the other universes so I can't see any issue with other universes containing such entities. Or am I using some type of circular reasoning?

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#54wavefreak wrote:I agree - IF there are other universes, then there is a non-zero probability that there is intelligence in one or more of those other universes. However, non-zero does not mean 100% - there is also a non-zero probability that any other universes (if they exist at all) do not contain any life at all (let alone intelligent life).

It would also be possible (even given an infinite set of universes) that our universe is completely unique and that all other universes are just really super massive black holes (zero-dimensional, no time, nothing).

My point is that it doesn't make sense to assert anything about them when we don't even know if they exist let alone what properties they may or may not have. We have no concept of the domain of the sets of other universes, so we can not say what the probability is. For all we know, all other universes have properties that make the probability of life 0%...

Other universes are sometimes fun to think about, though, but at this point I find that it's of little to no academic value being their existence and domain is unknown.

Why yes, I can believe it's not butter!

#55LOL good 'ol canofbutter calling people on the same set mistake I made.

Mine was bad wording and I'm sticking to that excuse.

#56Cpt_pineapple wrote:Seems like I spend more time talking about math on this forum than I do atheism, etc...

Mother always told me I should be an actuary...

Seems there are few food names floating around here... the only food I can think of off the top of my head that uses pineapple and butter would be pineapple butter: 8oz crushed pineapple,1 cup softened unsalted butter: Use a food processor to blend pineapple and butter...

Well that was off topic...

Why yes, I can believe it's not butter!

#57canofbutter wrote:I see your point about probablity. I'll work on it.

I must object to your last statement. Such specualtive thinking, as long as it is grounded in logic, actually drives science. The theories of multi-universes are *required* by certain lines of reasoning that attempt to integrate quantum mechanics and gravity. This is so much fun to think about that CERN is spending large amounts of money to garner more information on the matter.

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#58As I understand it, that claim is like saying "the infinitude of universes is like the set of positive numbers, and we are universe number X. Life can only exist if the universe is X."

Now, you gave 'X' the value 4. So, it sounds plausible. But in an infinite set, we are just as likely to be universe number 3.78*10^498,123,283.

"Life can only happen on universe 3.78*10^498,123,283." doesn't sound so feasible, and we're dealing with an infinite set, so in fact that number is waaaaay too small.

That is, there are an infinite number of sets which include our universe. But only one set includes only our universe. An integer (one) in an infinity has infinitely low chance of occurring.

You could argue that there are an infinite number of sets which include just our universe: but in the infinite infinitude of other sets, the superset of "sets that include only our universe" is aleph-null, and the super-superset of sets that includes all possible combinations of universes would be aleph-one: so the anthropic set is still infinitely unlikely.

My maths is 15 years rusty, but it would seem that the anthropic argument, while "possible", is infinitely improbable in the same way that god, the FSM, and the IPU are.

If I'm wrong, where did I go wrong?

[Edit: actually, by claiming that it's integers and aleph null, I'm making an assumption about the countability of any subset of the infinities: there's no suggestion that this need be the case for infinite universes. Doesn't affect the argument, but felt I should point that out, so that any math geeks reading this can sleep at night.]

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#59DewiMorgan wrote:My use of the number 4 was arbitrary so that it would be easy to see how one would arrive at it in a couple different ways. More to the point was that in a completely random set like this (where we know "some things" about it (allowing for non-zero probabilities), we still can't say things like "has a probability of 1). In the case of 3.78*10^498,123,283 - such a number is equally probable as 4. The specifics don't really matter - the point was more that you can't claim that something MUST exist simply because it CAN exist (as that is a common misconception about infinity and infinite sets that I was trying to clear up).

I used a countable set only because it demonstrated the point. If I had used an uncountable set (such as the power set of the set I used), it would have been more difficult to get the idea.

I'm also not saying that other universes CAN NOT have life, I'm just saying that it isn't 100% guaranteed, even given the existence of other universes (even if it would "feel like" it was really probable, we don't know anything about the domain of other universes, so it's also probable that they don't) - we simply don't have enough information yet.

Regarding the comment about the value of discussion, I was meaning that we can't calculate hard probabilities until there is a known domain for the other universes (until which point we are left simply to speculate without anything testable) The discussions may be useful in helping to determine what we may want to test for, but trying to make claims about things we can't observe or test seems to me like it has little to no scientific value.

Why yes, I can believe it's not butter!

#60Um. I still don't get where you are getting your estimation of probability from. There is a one-infinityeth chance that we are the only universe out there with intelligent life in.

So to claim that there is "not 100% guaranteed" other intelligent life out there is to claim that 99.9999...% is in some way not equal to 100%: and that claim is incorrect. href="http://en.wikipedia.org/wiki/0.9

However, I get the feeling that you know more than I about maths, so I'm not saying "You're wrong", but rather asking: given that's how I see it, where am I going wrong? How is the probability NOT equal to 99.999...%?

T="theists who's posts are fun-to-read, truth-seeking and insightful". Your own T will be different, but Tdewi includes { Avecrien, Cory T, crocaduck, JHenson, jread, wavefreak }

#61DewiMorgan wrote:Sorry for not being clear - I'm not saying any of that. I'm saying that the probability is unknowable right now because we don't know the domain of the other universes. It may be the case that we really are the only universe with intelligent life. It may also be the case that there are a finite number of universes. You are correct that 0.9999... is actually equal to 1 (as it's the sum of 9/(10^n) for n from 1 to infinity, which is convergent at 1).

The complete domain of all universes is currently unknowable, so it is currently impossible to say what the probability is. Also, even with an infinite number of universes, there may be a finite number of universes with the qualities for intelligent life.

You can't make a statement of probability without knowing the domain of the set. That's really all I was trying to say. Sorry if I wasn't clear enough.

Why yes, I can believe it's not butter!

#62We don't know the domain, but we know the... hrm, the "domain of possible domains". So we can establish which domains (those which include only our universe) are likely or not.

Or does the above logic fail, and knowing the "domain of possible domains" doesn't tell you which are likely?

(I could see them using your sig in an ad targetting the Bible Belt. Guy in store does the taste test, says "Why yes, I can believe it's not butter! But then, I'm an atheist!" Sales would skyrocket! )

#63DewiMorgan wrote:All we know is that intelligent life is possible in the domain of universes. Take my example set: we know that "1" is possible in the set, however it also can only occur once. In other words, the probability of "1" occurring is infinitely close to 0 (saying it's equal to zero is actually incorrect; infinity is not a number you can factor, divide, etc. It's just a limit. The limit of the probability is equal to zero, but the probability itself is not. Just like saying the .99999... thing is equal to 1; the limit of the function that describes it is equal to one, but really it's infinitely close to one such that one can prove that the function converges at one (so for all practical and even mathematical purposes, it equals one (the proof is trivial, but it may be best to say that "the function that describes .999... continues to get infinitely close to '1' until it converges on it as n approaches infinity))).

I've made other posts here about the limits of a function not always being "equal to" the value of it (such as when a limit approaches 0 for a hyperbolic function).

Infinity can be a real weird beast... This is generally covered in great detail in the first two semesters of Calculus and Advanced Calculus (Math 165, 166, and Math 431 (respectively) where I went to college). Also number theory (Math 330) is good to have here as well (I would recommend taking Calc. I and II before Number Theory and then taking Advanced Calc.

Even in infinite sets, things that have occurred once are not guaranteed to happen again. We don't know the domain of possible domains - all we know is one possible value for the set (our universe). Like the "1" in my original set function, it's completely possible that we're completely unique (and like the "4", it's possible it only can occur a finite number of times, even if it can come about in different ways).

In other words, yes, even knowing the domain of possible domains, it doesn't tell us which ones are likely (however it could be argued that we don't even know the domain of possible domains, we simply know a single possible value that is in the domain, assuming there are multiple universes in the first place)

DewiMorgan wrote:It's funny, but I

reallycanbelieve it's not butter - my family gives me a hard time about that one, claiming that no one can tell the difference. Just try cooking with the stuff - then you'll believeWhy yes, I can believe it's not butter!

#64canofbutter wrote:Only because that's required by the definition of the set. That's not required by the definition of intelligence or life or universes, so I suspect not relevant.

Except, it's not. http://en.wikipedia.org/wiki/Benfords_law shows us that the number 1 is the most common first digit. More than that, real-world numbers are generally distributed logarithmically so that 1 will be the most common.

That is: we know the "domain of domains", so we can assign probability. Like 4, you chose 1 carefully, knowing that it is common and likely. We do not know this about life.

In any set where 902390898723457 appears, how likely is 1 to appear? Quite likely.

In any set where 1 appears, how likely is 902390898723457 to appear? Not so likely.

That is the agnostic position, yes: that "the tooth fairy" or "easter bunny" are all "infinitely unlikely" and not worth considering, even if that means there is an infinitesimally small chance they may occur.

In my posts above, my thought was that if we pick a random number N within the set 0-infinity, then in an infinite infinity of sets containing N, those subsets that contain only N are infinitely unlikely, and no more worth considering than the easter bunny.

This was because I was picturing the graph of possible set sizes as a single one at 1, an infinite number at 2 (sets of N plus one of infinity-1 other numbers), two infinities at 3 (one each from n, inf-1, inf-2), and so on - increasing by one infinity for each extra item added to the set.

However, I now see an alternative argument: that naturally-occurring numbers appear logarithmically. So in fact, in that infinitude of sets containing N, the ones containing only N should be the most likely, followed by the ones with just one partner, then two, then... any infinitely-large sets containing N should be a little uncommon (infinitely uncommon).

So, fair enough: one data point doesn't give us enough info to tell whether other universes are similar to us, since you could argue either way about the probabilities.

#65I did not chose 1 or 4 because they are common or likely. I chose them because they are small numbers that are easy to conceptualize how they can occur given the function that defines my set domain.

In a true random number generator (i.e. not a pseudo-random number generator based on natural sources (I'm speaking mathematically here (not naturally as Benford's observations are based (also those observations come only from OUR universe - they do not necessarily speak for other universes)))) 1 actually is just as likely as 902390898723457. The probability of 902390898723457 occurring when 1 is present is effectively equal to the probability of 1 occurring when 902390898723457 is present. I say effectively equal only because in this set there may be multiple ways that 902390898723457 can come about, however 1 can only come about in one way (i.e. 1 is the very first random value for n - 1 can never occur if it is not the first random n (I said positive integers, so 0 and negatives are not options; positive integers are all integers from 1 to infinity, meaning that in a truly random number generator you aren't very likely to get 1 at all)).

It is an agnostic position. With current means, the multiverse, the properties of its component universes, and their overall domains are unknowable.

Taking any other stance of belief or making any statements of probability about it is currently completely unfounded as it is trivial to conjecture about any number of possibilities (e.g. only one universe with intelligent life, only one universe at all, a finite number of universes with intelligent life, or an infinite number of universes with intelligent life). They are within the realm of possibility, but there is absolutely no way to assign a probability to them (let alone say "100%". Summary: assigning probability requires that the domain of the set be known; we don't even know the range of the possible domains - so probability can not be assigned - it's all just complete guessing based on little to no supporting evidence at all. The multiverse is only more useful than the "easter bunny" or whatever because it does explain some things and some people think there may be a positive way to test for it (though I never really intended to make comments on this thread about that - I really just intended to clear up the misconception about how the "infinity" concept works - I don't know much about physics; I do know quite a bit about calculus, set theory, and math in general).

I take a fairly straight-forward approach to issues like this: I will say "I don't know" until there is sufficient information to support a theory that allows making predictions.

Also, there is no such thing as a "number of infinities" - you can have countable infinities (such as the set I previous described) and uncountable infinities (such as the power set of an infinite set). Infinity is a limit concept, not a number - "infinity-1" is the same thing as "infinity". So for your N, its occurrence in a set such as the one I described is still infinitely close to 0.

I'm hoping my original post's intent is now clear.

Why yes, I can believe it's not butter!

#66Infinity is actually classified as NAN (not a number) in computer science. But mathematics includes ways to handle infinity and infinite sets. Integrals are a perfect example.

But back to my "proof".

The problem with the way I've applied probablity is that I didn't assign a likelyhood to any of the events. If we are flipping a fair coin then the possible events are heads, tails and each are of equal probability. So what is the probability in a set of n events that heads will never come up? Easy. 1/(2^n). If n is an infinite set then you have to take the limit and the probablity is zero.

So the question becomes what is the probability that any particular configuration of universe will arise in a multi-verse? Well, technically we don't know but there are two basic possibilities. One is that any universe is equally as probable as another. The other is that there is some dynamic in play that makes certain configurations more likely.

Side note: The second possibility is actually more interesting because this dynamic would imply some structure in the cosmic foam that influences the spawning of universes. This in turn implies the type of energy gradients required for information and also then allows for intelligent entities to exist in the cosmic foam. Cool - but irrelevant.

So basically we can have an even distribution of probablities or an un-even distribution. Either way you get some type of distribution function F(e) that expresses the probabiliy of any event e. An it then follows that the probability of any particular event not happening in any set of random events is (1-(1/(F(e))^n). The limit of this at infinity is also zero.

Having said all this there is still one major problem. This assumes there are finite possible events. In other words, the cosmic foam can only generate a finite number of universe configurations. I don't know how to deal with infinite possible events. I'm going to have to look it up.

But I am also wondering if this is how some of the theories arrive at the possibility of an infinite number of copies of every possible universe. If there is only a finite set of possible universes then in an infinte multi-verse each one will be represented infinitely many times. Is there something in quantum mechanics that requires a finite, though vastly large number of possible universe states?

Hmmmm ...

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#67It really depends on which infinitude of universes you are referring to.

First here's the "trouserlegs of time" universes, when there are countless similar universes, different by just a single event and its outcomes if any. In an infinite number of other universes, an atom in a distant star didn't decay at the precise instant that it did in our universe, and decayed at slightly different times. A vast, vast number of alternative possibilities, but it's debated whether they are made physical. Under this set of universes, there are clearly plenty of universes in which there is intelligent life, and even plenty of universes where I am typing this.

I've never liked the trouserlegs of time, and felt any hypothesis which ended up requiring it was a bit suspect, just like any hypothesis that invokes the anthropic principle.

Then there's Stephen Hawking's "naked singularities" universes (where, if you leapt through a singularity and somehow survived, you'd arrive in a new universe, and if you leapt back through it, you'd arrive in yet another universe and so on infinitely, never arriving back at your original universe), then the only thing implied about them is that they are capable of containing at least one singularity. But Hawking has been wrong before: maybe if you leap back through it, you come to your original universe.

Then there's the branes, where a universe is created whenever two branes knock together.

And there are various other hypotheses about it. None of them have supporting evidence yet, so far as I know.

#68DewiMorgan wrote:Quantum mechanics has been astonishingly successful as a scientific explanation of physical reality. But it as certainly opened a can of worms. The geniuses of the world (Hawking for one) have to invoke these astonishing ideas because they are consistent with their description of things they *can* measure.

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