[PassedOut]

luke warm, on Apr 5 2009, 04:37 PM, said:

PassedOut, on Apr 5 2009, 03:28 PM, said:

What we have here is a troubling failure to communicate.

luke warm, on Apr 5 2009, 09:27 AM, said:

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]

Quote

....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]

PassedOut, on Apr 5 2009, 05:51 PM, said:

luke warm, on Apr 5 2009, 04:37 PM, said:

PassedOut, on Apr 5 2009, 03:28 PM, said:

What we have here is a troubling failure to communicate.

luke warm, on Apr 5 2009, 09:27 AM, said:

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

kenberg, on Apr 5 2009, 06:19 PM, said:

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?

Winstonm, on Apr 5 2009, 06:22 PM, said:

Quote

....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."

Quote

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?

Quote

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]

blackshoe, on Apr 5 2009, 07:07 PM, said:

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.

Quote

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]

Winstonm, on Apr 6 2009, 02:56 AM, said:

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]

kenberg, on Apr 5 2009, 09:20 PM, said:

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.

[luke warm]

kenberg, on Apr 6 2009, 07:35 AM, said:

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.

apparently hilbert spent more time contemplating these things than we might think... the context can only be found, to my knowledge, in hilbert's "on the infinite" which appeared in "philosophy of mathmatics" in 1964... the quote in question read, "The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought... The role that remains for the infinite to play is solely that of an idea."... hilbert in no way denied the importance of cantor's work, even saying (in the same place), "No one shall be able to drive us from the paradise that Cantor has created for us."

this, imo, shows the difference between an actual vs. a potential infinite - the actual infinite is only conceptual, it can only exist in the mind... if, in nature, an actually infinite number of things could exist, there would be all kinds of absurdities.. this is what hilbert's hotel was supposed to show... unlike hilbert's space (whatever that is), i doubt he'd have denied knowledge of his hotel...

it is much like (if i have understood your and helene's explanation) matching {a,b,c} with {1,2,3}, with the difference being we're matching guests to hotel rooms... if an infinite number of rooms were each occupied by one person and if another person showed up asking for a room, he could be accommodated by simply moving each guest from his already occupied room to the next... guest 1 to room b and so forth... subsequent addition of guests to the hotel can be done conceptually only... such a hotel is not possible in reality since, in transfinite arithmetic (correct me if i'm wrong), subtraction is not allowed because of inconsistencies encountered... if this hotel existed in reality, however, a person could conceivably check out if he chose (i.e. the total could be subtracted from)

that was, i'm sure, a poor effort to explain the philosophical difference between actual and potential infinites

[helene_t]

Wackojack, on Apr 6 2009, 01:55 PM, said:

Time is not real. The infinite only exists as a contruct in mathematics.

Time not real? What's that supposed to mean? That timespace has four dimensions and we just by convention chose a particular coordinate system and call one of the coordinates "time"? Or something else?

[TimG]

Wackojack, on Apr 6 2009, 07:55 AM, said:

The infinite only exists as a construct in mathematics.

I think this is what Jimmy means by "actual infinite", the mathematical construct.

Jimmy's "potential infinite" is the real world's "unimaginably large". If you keep adding one grain of sand to your collection, the collection will get very large, but can always be described by a finite number.

Jimmy's view of "always", "forever" and "eternal" are as real world concepts of time and thus are "potential infinite" -- that is they can be a very long time, but no matter how far backwards (or forward) you go, the amount of time will be measurably finite.

(I'm not offering an opinion regarding these concepts, just trying to simplify the definitions in order to make the discussion more understandable.)

How'd I do, Jimmy?

[luke warm]

TimG, on Apr 6 2009, 09:33 AM, said:

Wackojack, on Apr 6 2009, 07:55 AM, said:

The infinite only exists as a construct in mathematics.

I think this is what Jimmy means by "actual infinite", the mathematical construct.

Jimmy's "potential infinite" is the real world's "unimaginably large". If you keep adding one grain of sand to your collection, the collection will get very large, but can always be described by a finite number.

Jimmy's view of "always", "forever" and "eternal" are as real world concepts of time and thus are "potential infinite" -- that is they can be a very long time, but no matter how far backwards (or forward) you go, the amount of time will be measurably finite.

(I'm not offering an opinion regarding these concepts, just trying to simplify the definitions in order to make the discussion more understandable.)

How'd I do, Jimmy?

pretty damn good... this should show why a successive addition of temporal events isn't possible if space/time were an actual infinite... if there were a starting point however (i.e. a beginning), we can so add

[jdonn]

All this talk about the Hilbert quote, I found this on the wikipedia article of Hilbert's hotel:

Quote

Because the Hilbert's paradox is so counterintuitive, it has often been used as an argument against the existence of an actual infinity, for instance an argument for the existence of God posed by the Christian philosopher William Lane Craig is roughly as follows;

Although there is nothing mathematically impossible about the existence of such a hotel (or any other infinite object), intuitively no such object could ever exist, and this intuition is a specific case of the broader intuition that no actual infinite could exist. Since a temporal sequence receding infinitely into the past would constitute such an actual infinite, time must have "started" at some point. Since "time" cannot be started by any temporal thing, and every action must have a cause, this cause must be God.

I guess my question is, how can someone honestly believe intuition proves anything? Or even moreso, how it can disprove anything that is not mathematically impossible?