Originally posted by mikelomThanks. On my way...!
We are simply talking numbers here. Let's go to see about infinite space, as this read will give you lots of relativistic understanding about how infinity can actually have and end, and yet still be infinite. 😉
http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/cosmo.html
yes, html.
Enjoy!
-m.
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Originally posted by vistesdwell the simple countable infinity is what most people mean when they use the term.
I need to think through this (and I saw Agerg's response first) but--
Is this not a matter of defining "infinity"? Some might take "infinity" to be an absolute, not subject to "levels". I am happy with different infinities, but I am not sure that that means anything other than different (competing) definitions of infinity. Can we start with a definit ...[text shortened]... aths and ontology?
[I get the "countable" and "non-countable" distinction, btw.]
If you say space is infinite you mean that you can head off in any given direction (from any
point) and keep going forever without ever reaching an edge or circling back on yourself.
or like the integer number line, you can go on forever counting the next integer without ever
running out of numbers.
So that is definitely infinity.
However the real numbers are also infinite, but bigger.
If you define infinity to be say the countably infinite then you need a new word for the non-countably infinite.
But the problem doesn't go away.
Because there are an infinite number of 'levels' of infinity.... do you want to find a new word for all of them?
So we are stuck with a class of concepts called infinity.
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Originally posted by AgergI would dispute that but not at this hour, and this might not be the place ...
The way you argue that the size of one set (the reals) is bigger than the other (the integers) by mentioning that for any two numbers there are infinitely many reals between them isn't enough unfortunately...look up Cantor's diagonal argument instead ;]
There are infinitely many rationals between any two integers but the size of the rationals is equal to the ...[text shortened]... hough the theist generally doesn't!) then the set of things that can be done is still finite.
[btw disputing the rational numbers countably infinite not Cantors proof]
however I'm linking the wiki pages for this.
http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
http://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory
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Originally posted by googlefudge…a class of concepts called infinity
well the simple countable infinity is what most people mean when they use the term.
If you say space is infinite you mean that you can head off in any given direction (from any
point) and keep going forever without ever reaching an edge or circling back on yourself.
or like the integer number line, you can go on forever counting the next integer find a new word for all of them?
So we are stuck with a class of concepts called infinity.
But that seems to me to be a far better (non-contradictory) definition of “infinity” per se! “By infinity we mean a class of concepts that include … .”
But I still may not be grasping power sets of infinite sets. Let me try this: what does “larger” (or “greater” ) mean in such a context? Surely that, too, must have a specialized definition that is very different from ordinary language. Or are we in the kind of territory that I often find with religionists—where they will use a word like “just” in such a way that it has all the identifiable meaning of “gurdywurt”? Why not say that the relationship of the power set to the original set is “gurdywurt”, and then give an appropriate definition to that word? My problem is not really with the word “larger” (or “greater” ); but it surely needs to be given a contextual definition (“in the mathematics of infinity, it means…” ) that is non-contradictory.
BTW, can one take the inverse (if that’s the correct term) of a power set? That is, in the way that one can take the square root of 2^2? How many?
Originally posted by vistesdThis is a good thread, and thanks for starting it. I'll get back to you on this in a couple of days. Pool match tomorrow after work, and if I do enter the forums in my inebriated or exhausted state afterwards it will be with nothing insightful. 🙂
I seem to recall that the initial controversy over power sets was that you end up with an infinity larger than the infinity of the original set? (Likely I have not said that well; but I am open to correction.) This seems to be a paradox.
With that, I understand what you’re saying. But—can there be a power set of all sets? Or is the number of ...[text shortened]... he power set including the null set { , }. What if there is—ontologically speaking—no null set?
Originally posted by AgergThank you. You know me—I will argue to learn, which often means being willing to take my lumps for arguing from ignorance! 🙂 All arguments that I make here should be so construed.
This is a good thread, and thanks for starting it. I'll get back to you on this in a couple of days. Pool match tomorrow after work, and if I do enter the forums in my inebriated or exhausted state afterwards it will be with nothing insightful. 🙂
[I had a friend of mine who was a mathematician. In his dissertation, he used the word “torsion”. The extra-departmental member of his committee when he was defending was an engineer (that seems to be common in the U.S.—to have a committee member from outside the department; not sure about elsewhere). My friend said that this committee member got all excited about that word “torsion”, and it took my friend some time to explain that—as he was using it—it had nothing to do with whatever it might mean to an engineer…]
Originally posted by vistesdOf course you start two interesting thread on my busiest day of the week. Dammit!
Thank you. You know me—I will argue to learn, which often means being willing to take my lumps for arguing from ignorance! 🙂 All arguments that I make here should be so construed.
[I had a friend of mine who was a mathematician. In his dissertation, he used the word “torsion”. The extra-departmental member of his committee when he was defending wa ...[text shortened]... xplain that—as he was using it—it had nothing to do with whatever it might mean to an engineer…]
Originally posted by vistesdYeah, we say "greater" , but physically speaking it's boulderdash. (Linguistical shortcomings?)
[b]…a class of concepts called infinity
But that seems to me to be a far better (non-contradictory) definition of “infinity” per se! “By infinity we mean a class of concepts that include … .”
But I still may not be grasping power sets of infinite sets. Let me try this: what does “larger” (or “greater” ) mean in such a context? Surely that, t ...[text shortened]... term) of a power set? That is, in the way that one can take the square root of 2^2? How many?[/b]
good threads dude ..
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Originally posted by googlefudgeIt struck me that my questions about terms such as “greater”, etc., with regard to different classes of infinities is based on a confusion (mine) that is caused by failing to recognize that there are different language games going on—and that there may be no adequate way to translate some concepts from the language of mathematics to another language such as English. And so, even though that is necessary for, say, teaching mathematics, it comes down to something that I think Wittgenstein said in the Philosophical Investigations about such a translation being successful to the extent that the student becomes able to understand and accurately work with the new language (in this case mathematics), without further reference to the old language. In such a case, it is an error (mine) to challenge the meaning of a word like “greater” here, rather than looking at it’s use.
well the simple countable infinity is what most people mean when they use the term.
If you say space is infinite you mean that you can head off in any given direction (from any
point) and keep going forever without ever reaching an edge or circling back on yourself.
or like the integer number line, you can go on forever counting the next integer find a new word for all of them?
So we are stuck with a class of concepts called infinity.
Note: I have spoken with people who are sufficiently fluent in more than one language that they actually think in the alternative languages, without any need for “thought-translation” through, say, their first language. They tell me that they actually think differently about the world, and thus have expanded horizons so to speak (which is what Nietzsche said is the most one can do once one recognizes the inescapable perspectivism we are all subject to).
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Originally posted by vistesdIndeed, which is why language is so important.
Note: I have spoken with people who are sufficiently fluent in more than one language that they actually think in the alternative languages, without any need for “thought-translation” through, say, their first language. They tell me that they actually think differently about the world, and thus have expanded horizons so to speak (which is what Nietzsche sa ...[text shortened]... is the most one can do once one recognizes the inescapable perspectivism we are all subject to).
For example the fact that most major languages distinguish a person by gender as a matter of course,
(ie by referring to him her, his hers, she he ect...) instead of having non-gender specific pronouns as the
norm tends to encourage people to divide people into male and female and often discriminate (often
unconsciously) against one or the other.
[EDIT: and on a side note on this particular subject, the very idea that we are either male or female is itself
damaging. As is pointed out in this article on DSD.
http://www.bbc.co.uk/news/health-14459843 ]
And this can run so deep that recent studies have shown that the very way you perceive the world can be
influenced by how you talk about it.
This was particularly demonstrated by a study of an African tribe who have a smaller set of words for
describing basic colours (and they don't match up with our ones, so one word encompasses what for us
would be a range of greens and blues) who can tell apart easily shades of green that to us appear near identical
and very hard to differentiate but happen to lie under two different words in their language.
And almost fail to discern the difference between a green and blue that to us look totally different, as both
are covered by the same word in their language.
Understanding mathematics makes you think differently, at least when you think about maths problems.
I don't think it impossible in this case to use English to describe the situation adequately for those who already
comprehend the underlying concepts. However whether it is possible to describe the concepts adequately without
mathematics is a different question.
Infinities are very hard to grapple with and I wont pretend to be able to play with them mathematically.
In dealing with problems in physics you generally try to avoid infinities, as they usually mean you got something wrong
or that you can no longer say anything meaningful about what is going on (singularities).
And when infinities do crop up, they are generally of the countable sort, Higher infinities tend to reside purely in the
realm of pure mathematics.
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We believe that the universe is governed by Einstein's theory of general relativity, which among other things addresses such matters as the overall structure of the universe.
In the early 1920s Alexander Friedmann showed that using one assumption (homogeneity - the universe has roughly the same density everywhere), the equations of general relativity can be solved to show that a finite universe must have a larger density of matter and energy inside it than an infinite universe would have.
There is a certain critical density that determines the overall structure of the universe.
If the density of the universe is lower than this value, the universe must be infinite, whereas a greater density would indicate a finite universe.
These two cases are referred to as an open and closed universe respectively.
The critical density is about 10-29 g/cm3, which is equivalent to about five hydrogen atoms per cubic meter.
In comparison the density of water is roughly 1 g/cm3 or about 500 billion billion billion hydrogen atoms per cubic meter.
However, we live in a very dense part of the universe.
Most of the universe is made up of intergalactic space, for which a density as low as the critical density is plausible.
So we should be able to answer the question of the universe being infinite or finite by measuring the density of everything around us and seeing whether it is above or below the critical value.
The problem is that the measured density turns out to be pretty close to the critical density.
Right now the evidence seems to favor an infinite universe, but it is not yet conclusive.
Originally posted by vistesdInfinity is what you get when you add 1+1=2+1=3, and so on, forever.
I wanted to pry this concept loose from some other discussions that seem to be going on. Really, I just want to hear from those who have some maths background that can apply it to philosophical questions (there are a number of you here!).
I read a book called Aleph some years ago, but don’t know if I still have it (will make a search of my booksh ...[text shortened]... e beginning, whether some religionists like it or not—and so I’d like to keep it here if we can.
How freakin' complicated does it have to be? For heaven's sake! 😉
Please don't ask me how long forever is. 😵