Noah's Ark

[luke warm]

To say that a logical paradox (no matter how cleverly constructed) makes anything impossible in the real world is to claim that the tail wags the dog.

says who? you?

Any logical paradox is an intellectual construction. Unless it is tautalogical (and therefore uninteresting), it cannot by itself (even if devised by a mathematician) make something impossible in the real world. The real world is not bound to follow what anyone, no matter how learned, writes on a piece of paper. You still have to establish that what you write accurately reflects the real world.

i have tried to... you fail to accept it... you simply assert that space/time is beginningless and leave it at that... evidently this is your attempt to reflect the "real world"

The same holds true for the other Hilbert quotes you gave. Suppose I give contrary quotes from other learned people? Those quotes would also prove nothing about the real world.

no, but they might show that you've actually read and/or thought on these things, that they aren't, in your words, "word games"... even such luminaries as hawking have granted that a beginning of time has theological ramifications, which is one reason he has worked on other cosmological theories... as i've said quite often, we all have presuppositions and we all have authorities we can point to who seem to back those presuppositions... if you are arguing that space/time is infinite, you'll need to actually make an argument... i've at least attempted to argue the converse... you have not...

it is nonsensical to say that time is infinite in one direction only, you will have to provide some sort of reasoning for that statement...

Consider a sequence of positive integers beginning with zero. I find it bizarre that anyone would consider that sequence nonsensical. Does it mean that such a sequence accurately reflects time? Not necessarily.

but isn't time what we were speaking of? i find it even more bizarre that one would argue for an actual infinite by positing a beginning point.. why offer an example that doesn't accurately reflect that which you're arguing? btw, a set of positive integers beginning with zero would have a beginning... if we are speaking of space/time starting at zero (the bb?), there is a beginning... you can't say that time is infinite and then point to a beginning as some sort of rationale for that... this is the have-my-cake-and-eat-it-too fallacy...

if time is actually infinite and you can only "look" back, do an infinite or finite number of years separate 9/11/01 from today? if you insist that time is only infinite going forward, how did we get to this day from 9/11/01? weren't an infinite number of seconds, minutes, hours, days, years traversed?

If you recall, seconds, minutes, hours, days, and years have specified measurable durations. The fact that a sequence continues indefinitely does not mean that finite measurements cannot be made within it. We got to this day from 9/11/2001 one measurable second at a time.

'indefinitely' does not translate into 'infinitely'... the whole point is that if time is infinite, there is an infinite period between each second... remember, we are talking about reality here

i never said i accept the bb, i said that present scientific observation seems to support that theory...

Well, do you accept the big bang as the starting point of our universe? It seems that you have argued that way.

i accept a beginning, though not necessarily the bb

However, Jimmy argues (as he must) that numbers do not provide an accurate representation of time. Rather, he claims that time is more like a hotel, which cannot have an unlimited number of rooms. I think that the hotel analogy falls down in many respects, but there you are.

where did i argue that? i expect more from you than straw man arguments

In response to my offering numbers as a good representation of time, you offered the hotel analogy as more apt.

numbers aren't a good representation of time... time is a good representation of time, however... btw, i don't see where you differentiate between actual vs. potential infinites... i think if you would do that you might begin to see the problem

Jimmy,

 

Do you agree that the difference in explaining the universe and its beginning with the utilization of a God Hypothesis versus an Unknown Cause Solution is simply in testability?  In other words, the God Hypothesis can only be tested through logic arguments, while an Unknown Cause Solution not only has to pass the logic test but must also pass the scientific test.

give me an example of a testable 'unknown cause solution'... that sounds suspiciously like

 

something caused the bb

the cause of the bb is unknowable

therefore, we don't know what caused the bb

 

In other words, there is no such thing as "nothing" other than what we can imagine - but the fact that something cannot arise from nothing in no manner implies a Creator was necessary - but it does allow for that possibility, as well as other as yet undiscovered physical explanations.

winston, how many other words can you say the same thing about? we define words so that we can communicate... 'nothing' is defined as, the lack of something, no thing - no movement, no anything

[Winstonm]

give me an example of a testable 'unknown cause solution'

 

I can't and that is the point - the solution is unknown.

 

To be more clear, the difference between saying a Being created the universe and saying I do not know how the universe began is that the second statement accepts testability as a rationale for belief but the first does not.

 

In other words, I am not arguing against you but simply trying to point out the differences in getting to our respective beliefs.

 

winston, how many other words can you say the same thing about?

 

I thought I was quite plain. I acknowledge the validity of the logic that something cannot arise from nothing, but that only allows for the possibility of a Creator and does not refute other unknown physical explanations that may be uncovered.

 

In other words, saying something cannot arise from nothing may turn out to be a meaningless and irrelevant concept - it may be that there always was "something" but it was not an entity.

 

the whole point is that if time is infinite, there is an infinite period between each second... remember, we are talking about reality here

 

This is point where I wish I had taken that nuclear physicist correspondence course offered by DeVries. :P Still, even with my limited understanding it appears to me a misconception of space-time in your conclusion. If time is an infinite, that could only occur when everything is moving at the same speed as the reference frame - which would have to be the speed of light. But if that were to occur, there would be no passage of time, and time would not exist. So it appears that infinite time and non-time are one and the same.

 

So it seems the conclusion of infinite time would not necessarily mean an infinite period between seconds but infinite time would mean that time does not exist.

[PassedOut]

you simply assert that space/time is beginningless and leave it at that... evidently this is your attempt to reflect the "real world"

This is another false statement, as I believe you know. I've not made any such assertion. In fact, in each post on this topic I have allowed for the possiblity that time might not be infinite in either direction.

 

if you are arguing that space/time is infinite, you'll need to actually make an argument...

As you know, I have not made that argument. I've refuted your statement that infinite time is a logical impossibility. I have thought about and read about these matters, and have determined that providing quotes is useless.

 

it is nonsensical to say that time is infinite in one direction only, you will have to provide some sort of reasoning for that statement...

Consider a sequence of positive integers beginning with zero. I find it bizarre that anyone would consider that sequence nonsensical. Does it mean that such a sequence accurately reflects time? Not necessarily.

but isn't time what we were speaking of? i find it even more bizarre that one would argue for an actual infinite by positing a beginning point.. why offer an example that doesn't accurately reflect that which you're arguing? btw, a set of positive integers beginning with zero would have a beginning... if we are speaking of space/time starting at zero (the bb?), there is a beginning... you can't say that time is infinite and then point to a beginning as some sort of rationale for that... this is the have-my-cake-and-eat-it-too fallacy...

Let's review: You were the one who said, "it is nonsensical to say that time is infinite in one direction only." I simply pointed out that your statement was wrong (regarding "nonsensical") and gave a simple example to illustrate. Whether or not time is really infinite cannot be determined by citing a number sequence or a hotel with an infinite number of rooms.

 

the whole point is that if time is infinite, there is an infinite period between each second...

Simply wrong. Where did you get that idea?

 

btw, i don't see where you differentiate between actual vs. potential infinites... i think if you would do that you might begin to see the problem

I guess, using your terms, that time might be one of your "potential infinities." Whether or not it is an "actual infinity" cannot be determined by contemplation.

[blackshoe]

I think that is completely wrong, but I'm certainly no expert so if you have a link that supports you I'd love to see it.

The "link" is my memory of college courses (I was a physics major) years ago, and reading after that. I imagine you're as capable of searching the internet (or libraries) as I am, so I'll leave that to you.

 

I think it's a misunderstanding or disagreement of what we mean by 'universe'. There are many references to the universe being of finite age, but I believe what is meant by that is 'the universe in its current form which follows the rules of physics and relativity as we know them', not 'any universe of any form that has ever existed, even prior to the big bang'.

 

"Time" ≠ "Universe"

 

In an earlier post, I alluded to this same possibility ("Maybe it depends on the meaning of "universe"").

 

This article may prove useful to the discussion.

[luke warm]

it is nonsensical to say that time is infinite in one direction only, you will have to provide some sort of reasoning for that statement...

Consider a sequence of positive integers beginning with zero. I find it bizarre that anyone would consider that sequence nonsensical. Does it mean that such a sequence accurately reflects time? Not necessarily.

but isn't time what we were speaking of? i find it even more bizarre that one would argue for an actual infinite by positing a beginning point.. why offer an example that doesn't accurately reflect that which you're arguing? btw, a set of positive integers beginning with zero would have a beginning... if we are speaking of space/time starting at zero (the bb?), there is a beginning... you can't say that time is infinite and then point to a beginning as some sort of rationale for that... this is the have-my-cake-and-eat-it-too fallacy...

Let's review: You were the one who said, "it is nonsensical to say that time is infinite in one direction only." I simply pointed out that your statement was wrong (regarding "nonsensical") and gave a simple example to illustrate. Whether or not time is really infinite cannot be determined by citing a number sequence or a hotel with an infinite number of rooms.

it is nonsensical... you asserted that my statement was wrong and then tried to illustrate this with a set of (supposedly infinite) numbers that have a finite starting point (zero)... how is that anything other than a circular argument? you said

If you recall, seconds, minutes, hours, days, and years have specified measurable durations. The fact that a sequence continues indefinitely does not mean that finite measurements cannot be made within it.

as if "indefinitely" meant the same as "infinitely"... are there an infinite number of measurements that can be made within an hour?

the whole point is that if time is infinite, there is an infinite period between each second...

Simply wrong. Where did you get that idea?

why is it simply wrong? where did you get that idea?

btw, i don't see where you differentiate between actual vs. potential infinites... i think if you would do that you might begin to see the problem

I guess, using your terms, that time might be one of your "potential infinities." Whether or not it is an "actual infinity" cannot be determined by contemplation.

wouldn't it be more correct to say that you cannot determine this by thinking? btw, these aren't my words (but since you've read about these things you already know that)... i know you dislike using the words of others, but this concept can be traced to at least the time of aristotle...

 

in any case, the terms are usually defined:

 

an actual infinite is a set of things that cannot be added to

a potential infinite is a set of things that can be added to

 

if you can add a thing (book, painting, hotel room, historical events) to a set of infinite books, paintings, hotel rooms, or historical events, then that set was not actually infinite...i don't know why this is hard to dontemplate

[PassedOut]

What we have here is a troubling failure to communicate.

 

it is nonsensical to say that time is infinite in one direction only, you will have to provide some sort of reasoning for that statement...

It is clearly not nonsensical for time to be infinite in one direction only. That would indeed be a useful way to describe time if the universe began with the big bang and expands forever, which might well happen. The number sequence I offered simply showed that the idea itself is not nonsensical.

 

Whether or not time is actually infinite going forward is interesting, but not of much immediate consequence. And if some now-hidden matter or force eventually pulls the universe together again into one point, then it might turn out to be useful to describe time as finite. I don't know how this will play out.

 

it is nonsensical... you asserted that my statement was wrong and then tried to illustrate this with a set of (supposedly infinite) numbers that have a finite starting point (zero)... how is that anything other than a circular argument?

What do you mean by "supposedly?" Do you disagree that the sequence of positive integers is infinite? Is that sequence nonsensical to you? Maybe that points to where our misunderstanding of each other starts.

 

the whole point is that if time is infinite, there is an infinite period between each second...

How would the fact that our universe continues to expand forever (if that's what turns out to happen) require "an infinite period between each second" that we experience on earth right now?

[PassedOut]

in any case, the terms are usually defined:

 

an actual infinite is a set of things that cannot be added to

a potential infinite is a set of things that can be added to

 

if you can add a thing (book, painting, hotel room, historical events) to a set of infinite books, paintings, hotel rooms, or historical events, then that set was not actually infinite...i don't know why this is hard to dontemplate

I use "full" and "not full."

[luke warm]

What we have here is a troubling failure to communicate.

 

it is nonsensical to say that time is infinite in one direction only, you will have to provide some sort of reasoning for that statement...

It is clearly not nonsensical for time to be infinite in one direction only. That would indeed be a useful way to describe time if the universe began with the big bang and expands forever, which might well happen. The number sequence I offered simply showed that the idea itself is not nonsensical.

you're right, unless one of us is simply being obstinate there is a failure to communicate... you use words differently from the way i use them, else with meanings that differ...

 

an actual infinite is a set of things that cannot be added to

a sequence of future events is a set of things being added to

therefore a sequence of future events is not an actual infinite

 

that is mine, what is yours?

[blackshoe]

Looked at from "now" a sequence of future events is "being added to" because they haven't happened yet. But suppose you could observe the whole axis of time, past to future, from some outside viewpoint. Then you might see that the sequence is not "being added to", it's just that you see it that way, because you only see events as they happen.

[helene_t]

Is time bounded? Probably. One aspect of "Big Bang" theory is that time did not exist before the Bang, and will presumably cease to exist after what Helene called the "Crunch".

 

I am certainly no expert on this subject, but I understood that Einstein in his General Relativity theory showed that time and space are interwoven. Time really has no meaning without a refererence frame - if everything in the universe was moving at the speed of light then nothing would be moving and no time would pass.

 

It may be more correct to conclude that the BB created a reference frame for time.

That's how I understand it, too. Any event that took place "before" the big bang took place on a different timeline, and we might as well say it took place in a parallel universum since we canot compare "their" dates to ours.

 

Or something like that :P Gotta read "the elegant universe".

[PassedOut]

What we have here is a troubling failure to communicate.

 

it is nonsensical to say that time is infinite in one direction only, you will have to provide some sort of reasoning for that statement...

It is clearly not nonsensical for time to be infinite in one direction only. That would indeed be a useful way to describe time if the universe began with the big bang and expands forever, which might well happen. The number sequence I offered simply showed that the idea itself is not nonsensical.

you're right, unless one of us is simply being obstinate there is a failure to communicate... you use words differently from the way i use them, else with meanings that differ...

 

an actual infinite is a set of things that cannot be added to

a sequence of future events is a set of things being added to

therefore a sequence of future events is not an actual infinite

 

that is mine, what is yours?

Good, I was thinking that might be the case. Let me get away from the word "infinite," which I use according to my education in mathematics.

 

In plain English, what I am saying is that time can continue forever, moment after moment (no matter how finely or coarsely one defines "moment") without violating logic. Indeed, this scenario is quite likely, given the evidence we have to date. This is true whether or not you accept the big bang as the starting point for time.

 

Agreed?

[kenberg]

"an actual infinite is a set of things that cannot be added to"

 

I can say with authority that the term "infinite" is not used this way in mathematics. In seventy years of living, I have never before encountered this usage in conversation. I gather some school of philosophers use the term this way? I cannot say that I understand it at all. With the possible exception of the set of all things whatsoever (a phrase of doubtful meaning) it seem any set of things can be added to. Maybe it all hinges on what "added to' means.

 

Anyway, it is possible to avoid the whole issue of competing definitions [Edit; Not for the first tiem a faster typist has made my post redundant'].

 

Instead of saying that time might be infinite into the future, we could say that for each positive integer n there will be at least n more years of time. Same idea, no usage of the troubling work infinite. Alternatively, and equivalently, we could say that for any year the phrase "next year' refers to a time that will exist. This would be in contrast to speaking of ten years before the Big Bang, which at least arguably is not a meaningful time.

 

If Jimmy has some set that nothing can be added to (I would like to see an example) then he could just say that the set cannot be added to. Again, we avoid the disputed term.

 

 

No one needs to give way on their usage of "infinite". Certainly the Passed Out usage is closer to mine as well as to any usage I have ever before encountered, but no matter. The semantic issue can be easily ducked.

 

Except for a brief irrelevancy, I have stayed off this thread until now. Philosophy confuses me.

[Winstonm]

....if we could agree that from nothing nothing comes, it makes it easy to arrive at the conclusion that something has always existed, else there would be nothing now... now all we have to do is try to reason what it is that might have always existed... if you read the above posts re: the eternalness of the universe, you'll see the problem - if the universe, hence time (which is a part of the universe), is actually infinite then it means an infinite regression can be traversed, which is not possible... an infinite temporal series of receding events occurring in nature would be an actual infinite, but this is incoherent... there are many analogies that demonstrate this, such as hilbert's hotel (though this is but one of many)... since ex nihilo nihil fit (assuming one accepts this), and since time isn't an actual infinite, whatever has always existed must be outside of and apart from time... it doesn't matter to me whether this entity, whatever it is, is called "God"... call it (or him) what you will

 

Jimmy,

 

I reread this and appreciate that you lay out in a straightforward manner the reasoning that leads you to a faith in a Creator-Being. At the same time, I am unclear whether what you believe about time and the universe are accurate.

 

You talk about time and the universe as being one and the same thing and that is not accurate in my understanding. Time itself is a relative measurement, but the universe is not measured as a relative. The only bounds on time (that are known) occur due to the speed of light; it is assumed nothing in the universe can move faster than light.

 

Time is also relative to the speed of the observer - in theory space could be moved forward without the traveler moving at all and thus not affecting time at all.

 

And as to the Hilbert Hotel example, it seems to prove nothing to me for this reason: the premise makes an assumption that Infiniite number of People (IP) > (IR) Infinite number of rooms, when Infinity (I) must = Infinity (I) at all times. Therefore, if there were an infinite number of rooms they could not all be full and at the same time have more people show up, else IP>IR which cannot be true.

 

I am curious about your beliefs about time - they seem quite straightforward and not in keeping (at least to my) understanding of the General Theory of Relativity.

 

You have said you do not necessarily accept the Big Bang or evolution - do you also remain unconvinced by Einstein's relativity theories concerning space and time?

[luke warm]

What we have here is a troubling failure to communicate.

 

it is nonsensical to say that time is infinite in one direction only, you will have to provide some sort of reasoning for that statement...

It is clearly not nonsensical for time to be infinite in one direction only. That would indeed be a useful way to describe time if the universe began with the big bang and expands forever, which might well happen. The number sequence I offered simply showed that the idea itself is not nonsensical.

you're right, unless one of us is simply being obstinate there is a failure to communicate... you use words differently from the way i use them, else with meanings that differ...

 

an actual infinite is a set of things that cannot be added to

a sequence of future events is a set of things being added to

therefore a sequence of future events is not an actual infinite

 

that is mine, what is yours?

Good, I was thinking that might be the case. Let me get away from the word "infinite," which I use according to my education in mathematics.

 

In plain English, what I am saying is that time can continue forever, moment after moment (no matter how finely or coarsely one defines "moment") without violating logic. Indeed, this scenario is quite likely, given the evidence we have to date. This is true whether or not you accept the big bang as the starting point for time.

 

Agreed?

i agree with this *if* there was a beginning, but not otherwise... i don't care what this "beginning" might be

If Jimmy has some set that nothing can be added to (I would like to see an example) then he could just say that the set cannot be added to. Again, we avoid the disputed term.

i don't have the same background in mathmatics as you or passedout, and sometimes i regret that lack... i find philosophical musings comfortable while some don't, but i still have trouble with this... this concept isn't original, but from a mathmatical view maybe you can tell me this - if there were a museum containing an infinite number of paintings, could you add one more? and, if you could add one more would there have been an infinite number of paintings in the first place?

....if we could agree that from nothing nothing comes, it makes it easy to arrive at the conclusion that something has always existed, else there would be nothing now... now all we have to do is try to reason what it is that might have always existed... if you read the above posts re: the eternalness of the universe, you'll see the problem - if the universe, hence time (which is a part of the universe), is actually infinite then it means an infinite regression can be traversed, which is not possible... an infinite temporal series of receding events occurring in nature would be an actual infinite, but this is incoherent... there are many analogies that demonstrate this, such as hilbert's hotel (though this is but one of many)... since ex nihilo nihil fit (assuming one accepts this), and since time isn't an actual infinite, whatever has always existed must be outside of and apart from time... it doesn't matter to me whether this entity, whatever it is, is called "God"... call it (or him) what you will

You talk about time and the universe as being one and the same thing and that is not accurate in my understanding. Time itself is a relative measurement, but the universe is not measured as a relative. The only bounds on time (that are known) occur due to the speed of light; it is assumed nothing in the universe can move faster than light.

i thought all (or at least most) physicists - including, maybe especially, einstein - view the universe as being the totality of all things... wiki says "The Universe is defined as everything that physically exists: the entirety of space and time, all forms of matter, energy and momentum, and the physical laws and constants that govern them."

You have said you do not necessarily accept the Big Bang or evolution - do you also remain unconvinced by Einstein's relativity theories concerning space and time?

of course not, even if i had the mathmatical acumen to challenge such theories i doubt that i'd have the arrogance to do so... but as i said, i believe even einsten thought of space/time as being part of the universe... do you believe he thought otherwise?

And as to the Hilbert Hotel example, it seems to prove nothing to me for this reason: the premise makes an assumption that Infiniite number of People (IP) > (IR) Infinite number of rooms, when Infinity (I) must = Infinity (I) at all times. Therefore, if there were an infinite number of rooms they could not all be full and at the same time have more people show up, else IP>IR which cannot be true.

which is exactly the point hilbert was making, that an actual infinite could not increase by subsequent addition

[blackshoe]

It may be somewhat counterintuitive, but Cantor (the mathematician, not the actor/singer) showed years ago that the cardinality (the "size" if you will) of different infinite sets is different. For example the cardinality of the set of real numbers (aka "the cardinality of the continuum") is 2^No, where No, called "Aleph null" (I don't have the right symbol for it in this font) is the cardinality of the set of natural numbers. So it is not necessarily the case that all infinite sets are the same size.

[mike777]

Put me in the only nothing can come from nothing camp. Example even an all powerful God cannot create God out of nothing. OTOH, logically speaking, perhaps God has always existed.

 

 

OTOH it seems there is considerable debate regarding the arrow of time when it comes to cause and effect; effect may come before cause.

[mike777]

As for Hilbert, he was the Great German mathematician who provided much of the mathematical infrastructure of both the general theory of relativity and quantum theory, he remarked that 'the literature of mathematics is glutted with insanities and absurdities which have had their source in the infinite'.

 

Hilbert proposed the doomed plan to 'establish once and for all the certitude of mathematical methods". See Godel.

 

Hilbert's influential essay 'On the Infinite" ridiculed the finite-number of steps requirement as non substantive but he was mistaken.

 

Godel's incompleteness theorem is a proof that Hilbert's tenth problem cannot be solved. For any set of rules of inference, there are valid proofs not designated as valid by those rules.

[Winstonm]

It may be somewhat counterintuitive, but Cantor (the mathematician, not the actor/singer) showed years ago that the cardinality (the "size" if you will) of different infinite sets is different. For example the cardinality of the set of real numbers (aka "the cardinality of the continuum") is 2^No, where No, called "Aleph null" (I don't have the right symbol for it in this font) is the cardinality of the set of natural numbers. So it is not necessarily the case that all infinite sets are the same size.

Having virtually zero math background I have to rely on whatever basic intellect I have to try to grasp this stuff - but I was of the understanding that Infinity times anything = infinity, so I(X)=I(Y) regardless.

 

but as i said, i believe even einsten thought of space/time as being part of the universe... do you believe he thought otherwise?

 

Jimmy,

 

I only brought up the General Theory as a questionmark about your definition of time and how it functions - all I was attempting to discuss was that my understanding of time did not seem to conform with your definition - and because of that your conclusion seemed faulty to me. To me, the question is more like describing the life-cycle of a frog - one could say the frog began with a big bang,

but that leaves out the time spent as a tadpole. Our universe may have exploded onto the scene with a big bang, but it may well have been a tadpole for eons before it did so.

[kenberg]

Jimmy,

 

In my reply I was consciously avoiding getting into the mathematics of the infinite because you are clearly using the term in a different way than we do. But I'll say maybe a few words now on the subject. Possibly someone will read this but there will not be a quiz.

 

For the most part, mathematicians have long ago learned how to get around without stubbing their toes on infinite sets. Here are a few facts:

 

A set is infinite if it can be put in a one-to-one correspondence with proper subset of itself. I'll explain. Consider all properly married couples. No polygamy or other weird stuff. The match between husband and wife is one-to-one. Each man is matched with exactly one woman, and each woman from this group is a matchee. Contrast this with children and fathers. Several children claim the same father. This is not one-to-one. The standard vowels {a,e,i,o,u} can be matched one-to-one with the numbers {1,2,3,4,5}.

 

Now consider the positive integers {1,2,3,... } and the even integers {2,4,6,...}. Match 1 with 2, match 2 with 4, match 3 with 6 and in general match n with 2n. Amazing (well, not really). The integers {1,2,3...} are matched with the proper subset {2,4,6,...} in a one-to-one way.

 

 

 

This possibility of a one-to-one matching with a proper subset is sometimes taken to be the definition of infinite. At any rate, it characterizes infinite. I am skipping over some technicalities but what I say is basically correct.

 

Think now of two children, each with a bag of marbles, each claiming he has the most. They wisdh to decide, but they do not yet know how to count. So they line them up. Jimmy puts down a marble, Ken puts down a matching marble. They repeat. If Jimmy runs out of marbles before Ken, then Ken has more. But with infinite sets this matching process has surprises. We have seen the first surprise. I may have red marbles numbered 1,2,... and you have blue marbles 1,2,... but if I am tricky I might match my even numbers red marbles with all of your blue marbles so that when the matching is done I still have my odd numbered marbles left over. It might appear I have more marbles than you do. But this is not true, since if I play fair we can match them.

 

Another surprise was mentioned by Blackshoe. You might think that given any two infinite sets (infinite used as above) you can find some way of matching them. This is not so. Imagine that someone attempts to math the numbers 1,2,3... with the entire set of numbers that lie between 0 and 1. So there is a number matched with 1, a number matched with 2, a number matched with 3, and so on. I'll now tell you how to find a number that was missed. Expand each of the matched numbers in a decimal expansion. So maybe the number matched with 1 is 0.3527... and the number matched with 2 is 0.8285... . I'll write down the number 0.21... . as follows. For the first digit I use something other than the first digit of the first number. Something other than 3 in the example. For the second digit in the number I am choosing, I se something other than the second digit in the second number. Something other than 2 in the example. In such a way, I write down a number that in some way is different from each of the numbers in the propsed list. There is again a slight technicality: Some numbers have more than one decimal representation, one-half can be written either as .500000... or as .4999999... . This only happens when one representation ends in all zeroes, the other in all nines, and I can easily dodge this by never using 0 or 9 in the number I write down. So the digit at, say, the seventh spot is not 0, not 9, and not whatever digit is in the seventh spot of the seventh number in the list.

 

The result is a number not in the list. Someone gives me a list of numbers between 0 and 1, and then I, using this procedure, give him a number between 0 and 1 which is not in his list.

 

Conclusion: It is not possible to match the infinite set of positive integers with the infinite set of numbers between 0 and 1.

 

Perhaps surprisingly, this all turns out to be important. Cantor actually was looking at some fairly applied stuff when he started developing his theory of sets. Or so I have been told.

 

As you can see, this has nothing to do with sets that cannot be added to. And if there are infinitely many paintings, there is always room for one more.

 

 

It would be incorrect to give the impression that all issues in the foundations of mathematics are settled. Not so. But mostly they are settled sufficiently so that those of us who wish to say "Oh the hell with it, let's just do it" can do so without getting into trouble. Also, folks who are looking to the foundations of mathematics for support for their religious views (for, against, whatever) will, I think, be disappointed.

 

No quiz, I promise

 

Ken

[helene_t]

Having virtually zero math background I have to rely on whatever basic intellect I have to try to grasp this stuff - but I was of the understanding that Infinity times anything = infinity, so I(X)=I(Y) regardless.

Right, but Blackshoe was not refering to Infinity*Infinity.

 

Consider the set of all integers. This is the "smallest" infinite set - some might think that the set of even numbers, which is also infinite, must be smaller, but that is not true. "Smaller" here in the sense of cardinality. That concept is defined on the basis of parings between two sets:

{a,b,c} has the same cardinality as {x,y,z} because you can construct the pairing

(a,x) (b,y) (c,z)

 

By the same token, the set of positive integers can be "paired" with the set of positive even integers, by

(1,2)

(2,4)

(3,6)

more generally:

(x, 2*x)

 

However, some infinite sets have larger cardinality than the integers. Consider for example the set of all subsets of the integers - including all infinite subsets such as for example the set of even integers.

 

Cantor showed that the set of subsets of the integers does not have a pairing with the set of integers. (Ken gives the proof in the above post that the set of numbers in the interval [0,1] has larger cardinality than the integers - this is basically the same).

 

Now a question:

 

Is there a set that has larger cardinality than the integers but at the same time smaller cardinality than the set of integers?

 

The assertion that there is no such set is called the Continuum Hypothesis.

 

Interestingly, the continuum hypothesis can neither be proved nor disproved. When it matters (rarely!), mathematicians conventionally decree by axiom that the continuum hypothesis is false, i.e. there is such an intermediate set. Equally valid, one could decree that it is true, but that would lead to some less intuitive results.

 

Finally, let me give the motivation for why Blackshoe calls the cardinality of the set of subsets of the integers 2^Aleph0, where Aleph0 is the cardinality of the integers.

 

Consider the set {No,Yes}. That set has cardinality 2. Now consider some set, say threefruits = {Apple, Banana, Clementine}. What is the cardinality of the set of subsets of threefruits? Such a subset is specified by, for each fruit, specifying whether it belongs to the subset or not. For example, the set

{Apple,Clementine}

could also be written as the tubble

{Yes,No,Yes}

 

The first element in the tubble can take two values (yes or no), so can the second, so can the third. So the number of subsets becomes 2*2*2. More generally, the number of subsets of Nfruits is 2*2*...*2 (N times), i.e. 2^N. Therefore, if some infinite set has cardinality A, where A is now a transfinite number rather than an ordinary integer, we call the transfinite number of subsets of that set 2^A.

 

As you noted, Aleph0+Aleph0 = Aleph0 and also Aleph0*Aleph0 = Aleph0. However, 2^Aleph0 is different!

 

Lol, just noticed the overlap between this and Ken's post.

[luke warm]

No quiz, I promise

 

Ken

thank God... i appreciate your time and effort... was hilbert wrong when he said that an actual (as opposed to a potential) infinite cannot exist in "the real world of rocks and trees"?... speaking of cantor, didn't he (in correspondence with the Pope) suggest that the existential impossibility of the actual infinite could be used in a mathematical-metaphysical proof for the existence of God? i understand, or think i do, that cantor himself was a religious man and wanted the church to see the correctness and logic of his work in set theory

 

in any event, i know it must be difficult to pretend that a neophyte can understand that which you are conveying, so i do appreciate the effort

[PassedOut]

Ken and Helene, thanks much for these last two posts. Wow!

[Winstonm]

Thanks Ken and Helene - when did you ever find time to learn bridge?

[kenberg]

Even in death, Hilbert is probably a smarter guy than I am so I won't be explaining why he is wrong. But if you have a context for the quote I could try thinking about it. Maybe something got lost in the translation from the German. I suppose there are only finitely many rocks on the face of the earth, and maybe only finitely many in the universe, but I doubt that Hilbert spent much time reflecting on such a matter. I don't know what he said about the actual infinite or the potential infinite. I am sure that I have never before heard either of these phrases.

 

The story goes that Hilbert was attending a mathematics lecture and the speaker began discussing Hilbert Space. Hilbert turned to the guy next to him and asked "What's a Hilbert Space?". Words sometimes have a local meaning that outsiders have no idea of. "Hilbert Space" is now universal. "Potential infinity" is new to me.

[Wackojack]

Jimmy's agenda (at least latterly on this thread) I believe is to reach the conclusions as asserted by St Thomas Aquinas in the 13th century NB:

 

Nothing is caused by itself. Every effect has a prior cause. This has to be terminated by a first cause, which we call God.

 

or

 

There must have been a time when no physical things existed. Since physical things exist now, there must have been something non-physical to bring them into existence. That something we call God.

 

Objections: If we agree that time is part of the physical universe and inseperable from space and matter. Then "God" the cosmic designer must lie outside time and be the creator of time. Otherwise God would have to come into being at the beginning of time and cease to exist at the end of time. Then what does it mean to design something "timelessly"? In human experience a designer is a being who thinks of certain choices in advance and then selects a judicious one. But "thinking" and "in advance" are temporal descriptions.

 

Even if the notion of "timeless design" is accepted, then could the designer have chosen a different universe, or to have chosen not to make a universe at all? Christians, tradititionally believe something quite different. They believe that God created this universe as a free act. That is: God was free not to make this universe. Then it must be very uncomfortable for Christians to answer the question. Why did God create a universe with so much suffering?

 

Hence the discussions in the thread about what we mean by "time", "nothing", and "infinite". FWIIW I feel that these words have no independent existence, just like the words "good" and "evil". They are merely human constructs. Yes the universe exists. Space exists. Nothing does not exist. Time is not real. The infinite only exists as a contruct in mathematics. It make me laugh when someone asks me to imagine a hypothetical hotel with an infinite number of rooms.

 

Story:

Part of the way through a lecture by American philosopher William James about the nature of the universe a woman stands up and denounces the lecturer, claiming she knows how the universe is put together. "The earth rests on the back of a giant elephant which stands on the back of a giant turtle." The bewildered lecturer responds by asking what the turtle was standing on. "You may be very clever young man" the woman shoots back. "But you can't fool me. Its turtles all the way down"

 

Amusing..... However, what if instead of turtles all the way down, there was at the bottem, a levitating "super turtle"? It seems to me that all religeous and scientific explanations need as a pre-requisite a levitating "super turtle" All camps denounce the other's "super turtle" in equally derisory measure.

 

Getting back to the origin of this thread which was questioning the belief in Noah's Ark. This story along with many other even nastier stories in the Old Testament tell of a vengeful God prepared to destroy the innocent and favour sychophantic cronies. Not the God I would like to believe in.