[Winstonm]

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please tell me how many positive integers there are.

The technical term is "a shitload".

[Winstonm]

jdonn, on Apr 7 2009, 07:02 PM, said:

Winstonm, on Apr 7 2009, 06:09 PM, said:

And another question for all you geniuses: the universe is supposed to be expanding.  If the universe is defined as all things, including space and time, then into what is the universe expanding?  What is beyond the edge of the universe?

OK, now that I've had one more hit of this stuff, I see that the answer is simple - the universe is exanding into a void due to pressure differences between the something and the nothing. 

You got any more of this *****?

The universe has no boundary. It is not expanding 'into' anything. There is no edge, if you travelled extremely fast you would keep going and potentially even end up back where you started without every changing directions. This is a good explanation.

Thanks.

It took off my mat as it whizzed past above my head. I guess I didn't do enough acid back in the day....

[Lobowolf]

Winstonm, on Apr 7 2009, 07:11 PM, said:

There is no edge, if you travelled extremely fast you would keep going and potentially even end up back where you started without every changing directions.

You know, like playing Asteroids.

[blackshoe]

The universe has no "edge", and it's not expanding "into" anything.

Oops. I see Josh beat be to it.

[Winstonm]

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The expansion or contraction of the universe depends on its content and past history. With enough matter, the expansion will slow or even become a contraction. On the other hand, dark energy drives the universe towards increasing rates of expansion. The current rate of expansion is usually expressed as the Hubble Constant (in units of kilometers per second per Megaparsec, or just per second).

The universe is expanding - but it is expanding within itself? And if it contracts, does it leave behind a void?

[jdonn]

Winstonm, on Apr 7 2009, 09:42 PM, said:

Quote

The expansion or contraction of the universe depends on its content and past history. With enough matter, the expansion will slow or even become a contraction. On the other hand, dark energy drives the universe towards increasing rates of expansion. The current rate of expansion is usually expressed as the Hubble Constant (in units of kilometers per second per Megaparsec, or just per second).

The universe is expanding - but it is expanding within itself? And if it contracts, does it leave behind a void?

A 3 dimensional mind trying to visually imagine a 4 dimensional model is kind of like my dog trying to perform a double squeeze. Good luck with that.

[PassedOut]

luke warm, on Apr 7 2009, 06:10 PM, said:

PassedOut, on Apr 7 2009, 01:42 PM, said:

luke warm, on Apr 7 2009, 12:18 PM, said:

maybe so ken, all we know is what he said, "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." it seems to me that he's saying that an actual infinite is not to be found in reality, only in the mind... do you understand him differently?

Hmm. Didn't you already agree that time can be infinite (using the standard definition of infinite)?

luke warm, on Apr 5 2009, 06:59 PM, said:

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

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

here we go again... i don't know if it's semantics or if there's really a lack of communications, but i don't view "time going on forever" to be an actual infinite *if there were a starting point*, i.e. if there was a bb from which the measurement of time began... that means that time (or space/time if you prefer) and thus the universe isn't eternal - although *something* is... it is the fallacy of composition to assert (and i don't say you do this) that because a set (in this case, of future events) might "go on forever" it is actually infinite - infinity goes not only forward

I know you don't want to engage in word games, so let me point to a couple of things in your response here.

I realize that you don't consider what you agreed to in our exchange as an "actual infinity." You've already made it clear that you consider "actual infinity" to have a special meaning apart from the normal meaning of infinity. We've had quite a few posts about that now.

However, the Hilbert quotes that you provided do not use the phrase "actual infinity" at all. I'm sure Hilbert would have been careful to use the specialized phrase if he'd intended a specialized meaning. If the words "actual infinity" had been in the quotes you gave I would not have asked you the question.

In the same vein, you say that your personal meaning for "infinity" is "eternal," rather than its standard meaning. Given that "eternal" perfectly expresses what you mean and cannot be misunderstood, why not use "eternal" when that's what you mean and let "infinity" retain its normal usage? Wouldn't that ease communication?

Just asking.

[EricK]

jdonn, on Apr 8 2009, 03:11 AM, said:

A 3 dimensional mind trying to visually imagine a 4 dimensional model is kind of like my dog trying to perform a double squeeze. Good luck with that.

The best my dog can manage is a couple of ruffs.

[Wackojack]

Winstonm, on Apr 8 2009, 12:09 AM, said:

And another question for all you geniuses: the universe is supposed to be expanding.  If the universe is defined as all things, including space and time, then into what is the universe expanding?  What is beyond the edge of the universe?

OK, now that I've had one more hit of this stuff, I see that the answer is simple - the universe is expanding into a void due to pressure differences between the something and the nothing. 

You got any more of this *****?

How about looking at it this way:

Space is in the universe rather than the universe being in space.
The big bang happened everywhere, not at one point in space.
The big bang was the explosion of space, not an explosion in space.
There is no centre of the universe.
Space is not nothing.

Measurements taken by astronomers suggest that the universe is 3 dimensionally flat or very nearly flat. Think of a sheet of paper on which a particle enters from the left and exits off the right. Now roll up the paper, gluing the left and right edges. The particle that previously exited off the right would now re-appear on the left. Thus it is possible that light that we take to be far away is in fact wrapping around a number of times thus causing an illusion of distance. More complicated shapes have been suggested.

[kenberg]

I don't know Hilbert's mind but let me take a shot at tis potential and actual infinity. First, trash the hotel, there is too much extra stuff going with it (do the rooms have showers, etc). Take the stars, Does it make sense to ask if there might be infinitely many? And I want to duck modern physics. It wasn't available to Hilbert and it is subject to change.

Here is what I think would be the argument: If there were infinitely many stars the more distant ones would presumably have lesser in\mpact on our environment. Imagine we actually discovered one trillion stars. By measurement, we might infer, much as was done with the planets, the existence of more stars that are as yet undiscovered. By observation, measurement and inference we might be able to say with confidence that there are at least two trillion stars. Beyond that, we are not sure. Perhaps three trillion. That figure of three trillion could perhaps someday be verified. Maybe someday we get to four trillion. But observation and measurement both have limitations (changing limitations but always limitations) and it will always be the case that we will be able to say something like there must be at least N stars but beyond that I cannot say. So any large number of stars might someday be verified, but even when it is it will not be possible to say within the limits of measurement that there are infinitely many stars.

So we might take a position that could be called Strong Rationalism; If it is not possible to imagine a physical experiment that would prove or disprove the existence of infinitely many stars then it is meaningless to assert the existence of infinitely many stars. We could say that there are potentially infinitely many meaning that we do not see that there will come a time when we say that there are 234,635,696,463,578 stars and certainly not even one more, but we also do not see a time when we can be certain that the number is infinite.



I have no idea if Hilbert said or meant anything like this but I can see that it would have some appeal. Whether it leads anywhere is another issue.

[luke warm]

helene_t, on Apr 7 2009, 06:17 PM, said:

If you have read somewhere that set subtraction involving infinite sets is disallowed then you must have misunderstood it.

okay, maybe so... there are many quotes i could find on this, one such is "In connection with transfinite ordinal arithmetic, we can begin by observing that in Cantorian formulations of transfinite arithmetic, the inverse operations in question [subtraction and division] are (conventionally) prohibited or excluded."

maybe that, and others like it, are incorrect

PassedOut, on Apr 7 2009, 11:56 PM, said:

I realize that you don't consider what you agreed to in our exchange as an "actual infinity." You've already made it clear that you consider "actual infinity" to have a special meaning apart from the normal meaning of infinity. We've had quite a few posts about that now.

in all of this i have (or thought i had) been speaking of two separate things - actual infinite and potential infinite... if you view the terms as nonexistent or as meaningless, so be it... that's why i don't view the totality of stars in the universe to be infinite, given the definitions i tried to provide (and they seem to be the most accepted definitions of those terms)... so yeah, it's likely that when you say 'infinite' (of real things, like roses or SUVs) i think 'actual infinite' from within that framework

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However, the Hilbert quotes that you provided do not use the phrase "actual infinity" at all. I'm sure Hilbert would have been careful to use the specialized phrase if he'd intended a specialized meaning. If the words "actual infinity" had been in the quotes you gave I would not have asked you the question.

my mistake, as i said above i had supposed we were speaking of an actual infinite

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In the same vein, you say that your personal meaning for "infinity" is "eternal," rather than its standard meaning. Given that "eternal" perfectly expresses what you mean and cannot be misunderstood, why not use "eternal" when that's what you mean and let "infinity" retain its normal usage? Wouldn't that ease communication?

Just asking.

i suppose 'eternal' works fine, in the sense that it is a state without beginning or end... i really thought, for some reason, when i said that time (with a beginning) can go on forever it was understood that i was speaking of a potential infinite since, given the definitions, it couldn't be an actual one

[helene_t]

Jimmy, you said you asked me and other mathematicians to correct you if you are wrong. Then I corrected you. The same w.r.t. to the opening post, why the h... does Richard ask about Noah's ark if he already decided the story isn't true?

Btw, you keep confusing the set difference A\B with the difference in cardinality card(A)-card(B). Those are two completely different things. It is correct that subtraction and division is not usually defined in ordinal arithmetics, presumably because it would not be interesting (besides, since ordinal numbers are non-negative, even a mundane difference like 3-4 would probably have to be undefined). But set subtraction is well-defined for all set systems. Whether it is defined in more general theories like class theory I don't know.

[luke warm]

helene_t, on Apr 8 2009, 07:09 AM, said:

It is correct that subtraction and division is not usually defined in ordinal and transfinite arithmetics, presumably because it would not be interesting. But set subtraction is well-defined for all set systems. Whether it is defined in more general theories like class theory I don't know.

in the first instance where i even mentioned this, i said "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." ... i asked for correction as to my understanding of the reason(s) subtraction and division weren't allowed - inconsistencies... the literature on the subject shows that some have recently tried to get around this whole mess by changing the word inconsistent to paradoxical... but the problem remains

[kenberg]

I would try to steer you away from ordinal arithmetic regardless of the issues regarding subtraction. If you are truly interested in the subject I could probably find a reference. I once (c 1960) read some of Naive Set Theory by Paul Halmos, stopping when I had enough (which was exactly what Halmos intended). "Naive" needs to be taken with several grains of salt, and maybe a Margarita, but by comparison with some other treatments it is mild. Halmos' stated intent, as I recall, was to explain enough about the subject so that mathematicians could cope with things such as transfinite induction. I strongly assert that the reason for such a study would be intrinsic interest if by any chance you would find it of interest. The applications to philosophy are non-existent.

A sample: Ordinal addition can be defined inductively (that's transfinite, not ordinary, induction). Once addition is defined then, for ordinals a and b, a-b would be that unique ordinal c such that a=b+c, if there is such a unique ordinal c. If not, either through non-existence or non-uniqueness, then a-b is not defined. Honestly, I forget how it all comes out. I could think it through or look it up, but I don't much care. I suspect there would be trouble with the limit ordinals, a phrase I won't define right now. No paradox, just no unique choice of c in some cases. If there is not a unique c then there isn't. No big deal. The answers will have no bearing on Noah's Ark, the existence of God, or which way to finesse for the Queen. I promise.

[PassedOut]

luke warm, on Apr 8 2009, 06:50 AM, said:

my mistake, as i said above i had supposed we were speaking of an actual infinite

Quote

In the same vein, you say that your personal meaning for "infinity" is "eternal," rather than its standard meaning. Given that "eternal" perfectly expresses what you mean and cannot be misunderstood, why not use "eternal" when that's what you mean and let "infinity" retain its normal usage? Wouldn't that ease communication?

Just asking.

i suppose 'eternal' works fine, in the sense that it is a state without beginning or end... i really thought, for some reason, when i said that time (with a beginning) can go on forever it was understood that i was speaking of a potential infinite since, given the definitions, it couldn't be an actual one

When evaluating limits and functions, it's routine to use a sequence of real numbers or integers going to either positive infinity or negative infinity (or both) as the independent variable.

For a sequence of integers, for example, positive infinity simply means that for any integer n, there is always an n+1. Negative infinity means there is always an n-1. It's very common to start at a particular number and run from there to infinity in either direction, particularly because dividing by zero is not defined. (It's also routine to use real numbers approaching zero as an independent variable.)

If you think back, perhaps to high school, I think you'll recall both plus-infinity and minus-infinity notations when limits were introduced to you. I grant that many folks find professions where they don't do many calculations and forget about those notations. However, many professions make use of them all the time. In my case, I've been in business for many years and have found many valuable applications for the math I learned as a young man.

The way I use infinity, positive or negative, the real-world applications are many and the theological implications are nil. Therefore, it's a bit strange to read a comment like this:

luke warm, on Apr 7 2009, 06:10 PM, said:

it is the fallacy of composition to assert (and i don't say you do this) that because a set (in this case, of future events) might "go on forever" it is actually infinite - infinity goes not only forward

Because most people (at least those people who use math at all) think of infinity pretty much the way I do, it seems to me that your posts would be a lot clearer if you used "eternal" whenever that is what you really mean.

[helene_t]

Just learned a couple of things about ordinal numbers. It appears that they do not form a set: http://en.wikipedia....i-Forti_paradox

Also, this must be Ken Rexford's favorite mathematical theory: http://en.wikipedia..../Surreal_number

It turns out just to be another name for what we called "extended real numbers" back in my college days.

[helene_t]

jdonn, on Apr 8 2009, 04:11 AM, said:

Winstonm, on Apr 7 2009, 09:42 PM, said:

  The universe is expanding - but it is expanding within itself?  And if it contracts, does it leave behind a void?

A 3 dimensional mind trying to visually imagine a 4 dimensional model is kind of like my dog trying to perform a double squeeze. Good luck with that.

The number of dimensions is not the issue AFAIU.

The thing is, space is curved. A "straight" line is in fact a circle with a finite circumsphere, some 70 billion lightyears or whatever. This means that although space is of finite size there is no point that is more "central" than other points, just like a circle is completely symmetric in that none of its points are more central than others (unlike a piece of a straight line, which has endpoints and a center).

Exampansion of the universe means the circumpherence gets larger.

Something like that. Disclaimer: I am no expert in cosmology.

[mikeh]

helene_t, on Apr 8 2009, 12:43 PM, said:

jdonn, on Apr 8 2009, 04:11 AM, said:

Winstonm, on Apr 7 2009, 09:42 PM, said:

  The universe is expanding - but it is expanding within itself?   And if it contracts, does it leave behind a void?

A 3 dimensional mind trying to visually imagine a 4 dimensional model is kind of like my dog trying to perform a double squeeze. Good luck with that.

The number of dimensions is not the issue AFAIU.

The thing is, space is curved. A "straight" line is in fact a circle with a finite circumsphere, some 70 billion lightyears or whatever. This means that although space is of finite size there is no point that is more "central" than other points, just like a circle is completely symmetric in that none of its points are more central than others (unlike a piece of a straight line, which has endpoints and a center).

Exampansion of the universe means the circumpherence gets larger.

Something like that. Disclaimer: I am no expert in cosmology.

The analogy that has always stuck with me is to imagine that we are on the surface of a balloon. The balloon is being inflated, so all points on the surface of the ballon are moving away from each other, and none of them can claim to be the 'centre'.


Moving from that easy to grasp situation to expansion in 3 or more dimensions makes my neurons shrivel.

[blackshoe]

General Relativity posits 4 dimensions, 3 of space, and one of time. So the "straight line" of which Helene speaks is a line in a 4 dimensional space. It looks straight to us because we perceive it in 3 dimensions.

That space is curved (and it is) means only that a "straight line" must be curved in a higher-dimensional space of which the space in which the line exists is a sub-set . Example: Consider the surface of a sphere. This is a two dimensional space. A straight line in this space (the equator, for example) is not curved in that space. It is, however, curved in the 3 dimensional space in which the sphere is an object, a sub-set or "sub-space" if you will.

Flatland and its successor Sphereland give a pretty good exposition of the dilemma involved in trying to understand more dimensions than we perceive.

[jdonn]

I haven't read Flatland in probably a decade. It was awesome. I never caught the sequel but thanks for reminding me it exists, I will one day.