beyond infinity

[PrimaVera]

1. Infinity paradoxes. From the book infinity and the mind by Rudy Rucker.

From the book:

1.q. It is sometimes said that if infinitely many planets existed, and then every possible planet would have to exist, including one exactly like earth except with unicorns. Is this necessarily true?

I'm afraid I was "lost" at about this point, because I'm not seeing a paradox. This is, perhaps, a problem when we try to use Mathematical concepts that have very precise meanings within a broader philosophical context. The concepts associated with infinity are certainly seductive, but we must take care not to be equivocal.

In the case above, the "paradox" actually mixes two distinct meanings of "infinity" as used in Mathematics. The first concept is the number signified by a sideways "8". This is a "number" in the sense that it's used in equations (for example in the concept of a limit in Calculus).

The second concept, however, relates to sets, or rather, the cardinality of a set. When we talk about an infinite number of planets, we're really talking about a set, i.e. a collection of distinct objects that share some property that defines membership in the set.

The apparent paradox in this is the failure to realize that the cardinality of two different infinitely countable sets can be the same even though the two sets have different criteria for membership. Consider, for example, sets of numbers. The set of all whole numbers is an infinitely countable set. So is the set of all rational numbers (i.e. numbers that can be expressed as a ratio of two whole numbers). The set of all whole numbers is a subset of the set of all rational numbers, but the two sets have the same cardinality.

On the other hand, the set of all irrational numbers, that is numbers that cannot be expressed as a ratio of two hole numbers (e.g. PI), does not have the same cardinality as the set of all rational numbers. Consider a number line. Pick any two rational numbers. Between those two rational numbers, there is an infinite number of irrational numbers.

The cardinality of sets is denoted using Aleph (of the Hebrew alphabet) notation. Aleph-null is the cardinality of the set of all rational numbers. The set of all irrational numbers has a cardinality of Aleph-1.

Intuitively, it sometimes helps to think of cardinality as relating to the "density" of a set of objects. By that, I mean to say that one can, intuitively, think of the set of irrational numbers as being infinitely more dense than the set of rational numbers.

In terms of Mathematical proof, the difference between the cardianlity of two distinct sets rests on our ability to construct a logical one-to-one correspondence between elements in the two sets. If we can construct a logical one-to-one correspondence between the elements of two sets, then their cardinality is the same. If we can show that it's not possible to construct a one-to-one correspondence between the two sets, then the cardinality of the two sets is not the same.

 

[wil]

I personally like the fact that there are an infinite number of infinities...

 

[rdwillia]

Hello PrimaVera

I'm afraid I was "lost" at about this point, because I'm not seeing a paradox. This is, perhaps, a problem when we try to use Mathematical concepts that have very precise meanings within a broader philosophical context. The concepts associated with infinity are certainly seductive, but we must take care not to be equivocal.

I think that your line of thought is very clear and accurate. You obviously have a much clearer idea of how infinity works than I. But I think that your confusion stems from exactly what you mentioned; precise, mathematical concepts mixed with broader philosophical concepts.

It should clarify things, to state that mathematically, just because an infinite number of planets exist, it doesn't necessarily imply an infinite type of planet. However, I'm still confused, because if there is one thing I've gotten from this topic, it is that an infinite amount (for lack of a better word) of something... is a lot.

With my very limited knowledge of the topic at hand; If there were an infinite amount of planets, wouldn't there have to be an infinite possibility of creation or life on those planets? I'm not being facetious, I'm honestly asking because you seem like you might actually know. I have read before, in more philosophical texts, that the above "must" be true. Or is this exempt from that rule for the same reason that just because a number might be infinite it doesn't mean it's infinite outside of itself... i.e. Just because a number is infinite, doesn't mean it's also a bird.

That's where my part of the paradox fits in. I realize it's not very scientific and it's definitely not the most rational, pretty cooky and much more philosophical but... I'm saying the number is infinite within itself, very true, but then we're limiting it by saying that it can't also be a bird. I was wondering about complete infinity or the void.

However, if you accept that the two (mathematical and philosophical) infinities are separate, it makes perfect sense.

If you could help me clarify the answer to my question (above), it would be greatly appreciated. Thanks for the input!

 

[PrimaVera]

RDWillia,

First, while I appreciate the compliment, I must demure. People like Georg Cantor and Kurt Gödel really got this stuff. I'm barely able to wrap my head around it.

You asked:

With my very limited knowledge of the topic at hand; If there were an infinite amount of planets, wouldn't there have to be an infinite possibility of creation or life on those planets? I'm not being facetious, I'm honestly asking because you seem like you might actually know. I have read before, in more philosophical texts, that the above "must" be true.

I'm afraid I don't know, or perhaps I understand the question. I'm not sure what you mean by "an infinite possibility of creation or life". It seems to me that you're asking whether or not an infinite number of planets implies that there must also be an infinite number of planets on which life can emerge.

If that is what you're asking, I don't quite see how to construct the line of reasoning that produces that conclusion. In order to construct a Cantor-like proof, one would have to come up with a way of enumerating the set of planets on which life can possibly emerge, and I don't know how to do that.

We can, however, say that having an infinite number of planets does not require even one planet that has any particular characteristic. This is analogous to a comparison between the set of whole numbers and the set of rational numbers. The set of whole numbers is a strict subset of rational numbers, meaning that there are elements in the set of rational numbers that are not in the set of whole numbers.

Despite the fact that one set is a strict subset of the other, the cardinality of both sets is aleph-null. In other words, the "size" of the two sets is the same. Georg Cantor proved this to be true using a proof known as the "pigeon-hole" technique.

The key factor in transfinite numbers is some form of recursive (i.e. self-referential) construction of the elements in a set. Cantor's work, for example, employed the technique of building sets out of the cardinality of other sets. Gödel took this same self-referential idea, and started building sets out of theorems, and constructed a theorem whose conclusion is that some theorems are unprovably true.

All I can really say is, when it comes to philosophy, be very careful when you make use of mathematics.

 

[rdwillia]

Hey PrimaVera,

I'm afraid I don't know, or perhaps I understand the question. I'm not sure what you mean by "an infinite possibility of creation or life". It seems to me that you're asking whether or not an infinite number of planets implies that there must also be an infinite number of planets on which life can emerge.

I think I see what your saying, and yes, that is what I was asking... kind of, not specifically but it includes that. I was asking in reference to Z's quote that you referenced in your first post.

1.q. It is sometimes said that if infinitely many planets existed, and then every possible planet would have to exist, including one exactly like earth except with unicorns. Is this necessarily true?

This was, in essense, my question. I gather from your response, that as you understand it, the above wouldn't be true. I'm missing the reasoning behind the leap taken in the above quote. I know there is some scientific theory which explains this. Somehow 'they' (whomever 'they' are) conclude that when you have an infinite number of planets, it would have to mean that every possible planet (terrain, plant life, animal life, etc.) would have to exist. But if I understand this from a purely mathematical view-point, this is not true. If the conditions don't exist (for the above) then they just wouldn't appear. Ours could still be the only planet with life. Unlikely, but possible. The only thing infinite about the planets would be there infinite amount??

Now I'm wondering where the theoretical scientists get this theory from. I can still see that it's possible, but not mathematically necessary. How about you Z? Do you remember how they explain this?

I think our posts got a little surreal for a while, but we were enjoying ourselves. And as I said before, we got into it way deeper in a spiritual/philosophical sense, often neglecting our mathematics. But I maintain my original line of thought, that I think a lot of it makes sense in that aspect as well. Take care,

~Ricky

 

[PrimaVera]

Ricky,

(I have a negative visceral reaction to addressing someone as "Ricky," because I hated being called "Ricky" when I was a child. But, it's your signature, so...)

The only thing infinite about the planets would be there infinite amount??

I would say "number" rather than "amount," but that's essentialy correct.

Now I'm wondering where the theoretical scientists get this theory from. I can still see that it's possible, but not mathematically necessary.

Well, you'd have to use something other than set theory to get there, which doesn't mean you can't reach some interesting philosophical conclusions using set theory. Take a look at this.

 

[rdwillia]

Hello Rick,

Yes, I have internally debated dropping the "y" myself. I am fully aware of how 80's child actor-ish it is. And the popularity of Ricky Martin a few years back didn't help matters much. A lot of people have transitioned into dropping the "y" but the majority still call me Ricky. I decided to not put much concern in my name and let people call me whatever they'd like. I'm hesitant to use Richard as that has seemed to lead into other undesireable nick-names... Incidentally, I believe "Z" has signed some posts as Richard. Wonder how that works for him??? So here we are, the three R's...

Back to topic...

Your link was a great read. It covered, generally, just about everything we've discussed here. I particularly enjoyed the part about Descarte's theories being battered by relativism;

More recently, Descartes' formulation has been battered by relativism, to the point where some philosophers now question whether science itself can be objectively valued more than mysticism, intuition, or other "non-rational" belief systems.

I agree with the that we are very limited by our senses. It seems that after you get in to it so deep you'd have to come to emptiness. Set science or theorectical, after a while it gets so jumdled up that nothing make sense any more... Except of course the strict science as we currently understand it, but then that's always changing too, right? How can we possibly know anything... Aaahhh!

Thanks again Rick...

 

[_Z_]

Primavera, hello.

Sorry I didn’t reply sooner, virus problems on my pc.



Hmm, you don’t seam lost at all! I am seeing it from a philosophical perspective not a mathematical one, as you can probably tell, it is the resultant meanings I am after! In the end infinity is enveloped by meaning, and ‘nothing can describe it’. The maths [metaphoric measurement] only comes into the way infinity [perhaps only as a dimension] interacts with the quantum universe.

most interesting!



Rdwillia, hi.

What I would say is that ‘you cannot build up to infinity’ [I like that one], no matter how far we travel we never arrive at the end, and there is continually an infinite ‘amount’ before us, and that indeed we cannot have an infinite amount!

I would presume that if you could have an infinite amount of planets, then you could have an infinite amount of blue planets, an infinite amount of red planets X an infinite amount of blue planets with or without life etc. – but we cannot! [It would be paradoxical all over the place if we did!].

I think we are on the same wavelength, when you describe infinity as whole/complete. Thus it seams to have two distinct natures [another paradox], the ‘beyond infinity’ ‘place’, that is not definable as even infinity, then as soon as we describe it as infinity then ‘it’ tries to take on given natures of the infinite – that is to say- it interacts with ‘finite principle’, and reality is like elastic between the two [etc.]. It is like thought, first there is pure thought of nothing arising, followed by subtle thought [where thought is shaped – and hence similar to reality being shaped by principle! {Belonging to the same universal principles imo}], then fully formed thought that becomes action [in both the actualisation of the thought and synonymous in action- both of which are event manipulating].



Hope that helps chaps!



Thanx for your most interesting replies, oh and thanx for turning my head into spaghetti!



Much respect



Z

 

[Bandit]

Now imagine that situation of you the telescope and the mirrors, then visualise it as a separate entity as if only it exists! At once it is surrounded by the void or infinity, thus is infinitesimal comparatively – no matter how big something is, this always remains so!



I am sure after a couple of drinks down a quiet country pub, we would be talking sh*t until infinity ends.







They will never take me alive! [I wont give in to the scientists and atheists]



Z

i am not giving in to the scientists & atheists either NEVER X beyond infinity.

seriously- if we are going BEYOND infinity, then we wont be at the same place we started & we wont be there right now either. (that is new age stuff) we would be way off the track- a track with a beginning but there would be no end. like a race track where you never run out of gas!
so how is it going Z.

 

[_Z_]

Bandit.

I am fine mate! thanx



Ha yes, except we don’t start – we arrive there at once! But no it’s a bit like taking a leap of faith! We can skirt around the idea of ‘naked reality’ [infinity beyond infinity] only ever enveloping it – at most.

So given an infinite amount of worlds – what would yours be like?! [if you read earlier posts].



I think everyone who posts on this thread should give me a planet!



Z

 

[rdwillia]

Hello!

What I would say is that ‘you cannot build up to infinity’ [I like that one], no matter how far we travel we never arrive at the end, and there is continually an infinite ‘amount’ before us, and that indeed we cannot have an infinite amount!

So true! Kind of along the same lines as "Wherever you go, there you are." But the reverse of that would be we also have an infinite "amount" behind us. Thank goodness! I wonder about this in terms of religion, any religion. Because I believe this to be true, yet in almost any religion there is some sort of 'end'. Perhaps an end that is yet another beginning. Maybe we have a whole other school of thought once we get wherever we're trying to go, that deals with this.

I would presume that if you could have an infinite amount of planets, then you could have an infinite amount of blue planets, an infinite amount of red planets X an infinite amount of blue planets with or without life etc. – but we cannot! [It would be paradoxical all over the place if we did!].

Perhaps I over simplified my example. But it wouldn't be so odd to think that life, infinite planets over, might require that life be carbon based.

As for your ideas on thought... I agree there as well. I believe in some Hindu traditions they refer to Samsara as the endless cycle of thought. Your mind has a cyclic nature and will infinitely revolve around thoughts. The goal is to create grooves in the "wheel" of thought so that you can eventually have a clear mind... Until you get it down into one big groove and then ware that wheel down, etc, etc... In meditation it would go something like this... I'm sitting down and clearing my mind of all thought. I'm not thinking, I'm not thinking.... And finally... Ah, this is nice, no thought. You enjoy it for two seconds and then you think, wow that was really nice. I like having a clear mind. Shoot! That's a thought! Then you spend the next ten minutes trying to clear your mind again... and it goes on and on and on... Viola! Infinite (yet limited) thought.

seriously- if we are going BEYOND infinity, then we wont be at the same place we started & we wont be there right now either. (that is new age stuff) we would be way off the track- a track with a beginning but there would be no end. like a race track where you never run out of gas!

I would have to agree with Z about not travelling, just being there. And hey, it's not our fault the New Agers decided to pick up on a few (emphasis on the 'few') good ideas and run with them. Sorry, that's not nice... I realize none of this really makes sense, it's all theoretical, but I think it's more of a practice to help me expand my mind. I know when my brain hurts... I must be making progress, right? That's the only real purpose of doing all of this thinking, to broaden our "real" world minds.

I think everyone who posts on this thread should give me a planet!

Shoot Z, I'd give you a quarter of all the infinite "number" of planets. There we could have our pub chat.

 

[Bandit]

Hello!


I would have to agree with Z about not travelling, just being there. And hey, it's not our fault the New Agers decided to pick up on a few (emphasis on the 'few') good ideas and run with them. Sorry, that's not nice... I realize none of this really makes sense, it's all theoretical, but I think it's more of a practice to help me expand my mind. I know when my brain hurts... I must be making progress, right? That's the only real purpose of doing all of this thinking, to broaden our "real" world minds.

yah. but for some reason i cant do it by not travelling. unless we are going to beyond infinity & back BUT- i dont plan on coming back if i dont have to .
i think that is why i like the mirror & telescope analogy.
i see it more in distance than numbers or just arriving at where i have always been. that seems to simple for me. i think of the numbers as gallons of gasoline & miles.
actually rdwillia, it does all make sense to me.
or maybe when the brain hurts you are making progress & when it stops hurting you are beyond infinity. or you are beyond infinity when the brain hurts (not sure)

 

[Bandit]

Bandit.

I am fine mate! thanx



Ha yes, except we don’t start – we arrive there at once! But no it’s a bit like taking a leap of faith! We can skirt around the idea of ‘naked reality’ [infinity beyond infinity] only ever enveloping it – at most.

So given an infinite amount of worlds – what would yours be like?! [if you read earlier posts].



I think everyone who posts on this thread should give me a planet!



Z

i think that is where i am now. infinity beyond infinity because this is not like the local pub.
i will have to get back to you on my infinite amount of worlds & you can have any planet except Jupiter & Saturn.

 

[rdwillia]

Hello Bandit,

yah. but for some reason i cant do it by not travelling. unless we are going to beyond infinity & back BUT- i dont plan on coming back if i dont have to .
i think that is why i like the mirror & telescope analogy.
i see it more in distance than numbers or just arriving at where i have always been. that seems to simple for me. i think of the numbers as gallons of gasoline & miles.
actually rdwillia, it does all make sense to me.
or maybe when the brain hurts you are making progress & when it stops hurting you are beyond infinity. or you are beyond infinity when the brain hurts (not sure)

I understand what you mean about traveling infinity. I kind of visualize myself in my own submarine style space-ship. I usually try to refrain from telling people this.

Apparently we can't quite conceptualize not traveling. Unfortunately with the mirror and telescope analogy everything (for me) keeps getting smaller and smaller and it comes to a point where I can't see it any more. Even with a telescope, I can see more but they still get smaller towards the end. I just picture space and a lot of stars... and I just keep traveling forever. There's another good analogy hidden in that last statement. Perhaps when we stop traveling and just get there, is when we'll really wake up!

But like Z says, "You can't build up to infinity." Traveling would be building up. It just is... in all directions. There I go traveling again... Ciao!

 

[rdwillia]

I should mention that in all reality, I see this 'theoretical' infinity as emptiness or the void in some of my more productive meditations. Complete, unending, bliss where mathematics and making sense of things doesn't matter. There is no paradox, no biases, no differences, just bliss. Where I feel blissful about all of this apparent creation, even it's apparent faults. It's like waking up after a really pleasant nights sleep. It's exists somewhere deep within all of us. I know it's there. I can't always get there but I have glimpsed it for brief periods and plan on a permanent relocation sometime in the near future. That to me, in my very humble opinion, is what it's all about.

I grew up despising the post 70's hippies (as I wasn't around to see the real hippies) and I'm aware that the above sounds silly and hippy-ish but it's so true. I am so ready for all of us to fully and directly experience the spiritual infinity that it hurts.

Peace.

 

[_Z_]

Bandit,

Any planet except Saturn – but I am a Capricorn! .

Infinity beyond infinity because this is not like the local pub

yea, nirvana [beyond infinity - kinda] is nice and peaceful, but I would rather hand out in an infinite pub- with eternity as the beer garden! Not forgetting the nymphs of course!



Rdwillia,



We may have to arrive straight ‘there’, but then of course one can travel to anywhere in eternity! With the infinite eye we can see all things, hmm I wonder if we chose to come here? - if infinity is the home of the original self!

I see this 'theoretical' infinity as emptiness or the void in some of my more productive meditations

As I see it too! Just remember it has no edges – it is always with us and part of us, as we are it. This also applies to an amoeba!



Z

 

[Rouge47]

You have brought up an interesting thread _Z_. I have always wondered if the universe is infinite or if it actually ends.

In the movie "The 13th Floor" the main character comes to the end of the virtual world. The end that has not yet been developed. He gets out of his car and walks to the edge and stares at it. A vast land of nothing but a green and black grided pattern of a virtual computer.

Does the Universe that we know of today ever end? Or does it go on and on with random planets or stars.

Is it like a sphere? Like, whenever you reach the end of it you appear the opposite end of the angle in which you exited or entered?

Or is it like a cube? Bouncing around off the walls whenever the end is reached.
Infinity keeps on going, and going, and going, and going, and going......

.....and going, and going....

Like the energizer bunny with the drum.

What is beyond infinity?
Is there even a beyond?
How can one travel the length of infinite?
How can one reach the end of a universe in which we believe is infinite.
What is the purpose of the vast space that surrounds us?
Why did God (my belief) choose to create a space of darkness (but yet of beauty) that we cannot comprehend?
Why is there even an infinite?
Is infinite chaotic or lenear?

So many questions, infact an infinte number of them.

PJ

 

[_Z_]

Paul James, hi.



Yes I have seen that film. I would say that you cannot have an infinite universe – there are too many paradoxes. Infinity and the universe are imo part of the universal whole, each being a reflection of the others incapability to become the other!

The universe is sphere-like, then imagine everywhere is the centre yet a sphere has only one centre – the surface is then everywhere too!


Infinity is beyond infinity! – When the infinite is used comparatively to anything other than itself!

You cannot travel the length of infinity [or build up to it in any way] – you can only arrive there at once.

There is no beginning or end.

There is a vast space surrounding us, because the universe wants to stretch to all limits finding all its forms – and if you put it all together so there is no space between, then you would arrive at the singularity i.e. the beginning [big bang].

Not sure if I believe in a creator god, as I draw no lines between things thus everything is god [given that there is a god]. And I do believe that we can comprehend it – at least that which is comprehendible.

Infinity is not chaotic or linear, it is incomparative – a beautiful emptiness!


Thanx for reply



Z

 

[Quahom1]

Paul James, hi.



Yes I have seen that film. I would say that you cannot have an infinite universe – there are too many paradoxes. Infinity and the universe are imo part of the universal whole, each being a reflection of the others incapability to become the other!

The universe is sphere-like, then imagine everywhere is the centre yet a sphere has only one centre – the surface is then everywhere too!


Infinity is beyond infinity! – When the infinite is used comparatively to anything other than itself!

You cannot travel the length of infinity [or build up to it in any way] – you can only arrive there at once.

There is no beginning or end.

There is a vast space surrounding us, because the universe wants to stretch to all limits finding all its forms – and if you put it all together so there is no space between, then you would arrive at the singularity i.e. the beginning [big bang].

Not sure if I believe in a creator god, as I draw no lines between things thus everything is god [given that there is a god]. And I do believe that we can comprehend it – at least that which is comprehendible.

Infinity is not chaotic or linear, it is incomparative – a beautiful emptiness!


Thanx for reply



Z

Hmmm, infinity can also be noted between the marks of a ruler. Take one inch (or centimeter if you wish). Half the measurement. Half it again, and again and again...How many times does one need to halve the measurement until there is a zero factor, or nothing left to halve?

It appears that the only reason there is "finiteness", is because we "humans" place borders, either because we can not conceive of the infinite, or we grow tired of trying to calculate it...so we stop at a certain point and say "this is good enough". eh?

v/r

Q

 

[Rouge47]

Hmmm, infinity can also be noted between the marks of a ruler. Take one inch (or centimeter if you wish). Half the measurement. Half it again, and again and again...How many times does one need to halve the measurement until there is a zero factor, or nothing left to halve?

It appears that the only reason there is "finiteness", is because we "humans" place borders, either because we can not conceive of the infinite, or we grow tired of trying to calculate it...so we stop at a certain point and say "this is good enough". eh?

v/r

Q

Interesting. Its like an atom. No matter how much you split it, it is still the exact same thing. You can go on and on and on and on and on and on and on and on.....and on....again....pink bunny....big drum....boom boom....I'm delusional....shouldn't of had that coffee....now I'm going on and on.....


Hehe,
Paul