On Oct 8, 4:57 am, Alfonso <Alfo...@duffadd.com> wrote: > On 07/10/11 15:46, Alen wrote: > > > > > > > > > > > On Oct 7, 9:39 pm, Alfonso<Alfo...@duffadd.com> wrote: > >> On 05/10/11 10:27, Alen wrote: > > >>> The purpose of this post is to try to understand what > >>> might be meant by the concept of 'infinity'. This appears > >>> necessary because it is not clear as to how even the > >>> limitless and finite, or measurable, can be linked to the > >>> concept of the infinite, since there is, for example, no > >>> number called infinity, no location that has a coordinate > >>> of infinity, and so on. > > >>> There are different kinds of infinities which, I think, > >>> are not all equally easy to identify and understand. > >>> In what follows, I use the example of space, because > >>> this is, perhaps, the easiest kind of infinity to deal > >>> with. I think it should be possible, however, to find > >>> ways to apply the following general definition of an > >>> infinity to infinities of all kinds. > > >>> GENERAL DEFINITION OF AN INFINITY !! TA DA !! :) > > >>> An infinity is a reality which forms a basis that > >>> enables the existence of the finite, the countable > >>> or measurable, but is not in itself either countable or > >>> measurable. > > >>> SPACE AS AN EXAMPLE > > >>> As an example of what I mean by this, take the > >>> measurement of a distance AB, between spatial > >>> locations A and B. The locations exist in space, and > >>> are identified by some content of space, such as the > >>> ends of a ruler, as a convenient example. The > >>> measurement of the distance requires the definition > >>> of a unit of the measurement, which can be arbitrarily > >>> small. What this means is that, however small the unit > >>> for the measurement of distance might be, it can > >>> always, forever, be made yet smaller. > > >>> This process identifies the existence of an underlying > >>> reality, which supports this possibility, but is necessarily > >>> forever beyond it, and unreachable to it. This underlying > >>> reality can be identified by what we refer to as 'continuity'. > >>> The nature of continuity is, therefore, that it supports and > >>> enables the process of measurement, as something > >>> finite, expressed in countable form, but is not itself > >>> intrinsically measurable, or countable. It is not merely > >>> an extension of the limitlessly small - there is a > >>> QUALITATIVE difference between continuity and the > >>> endlessly small. > > >>> Continuity is thus an infinity, as a reality that is not > >>> in itself measurable, but underlies, or supports and > >>> adopts, though distinct in itself, the finite measure that > >>> it enables. > > >>> In the opposite direction, that of greatness of extent, > >>> we have a similar consideration. By extent I mean a > >>> distance, such as AB which, however, is increasing, > >>> in the sense that the total measure is becoming greater, > >>> without the unit for the measurement being altered. > > >>> It is the case that, however great this measure may > >>> become, it can forever be made greater than before. > >>> This, again, identifies an underlying reality that is > >>> necessarily forever beyond the process, and > >>> unreachable to it. I refer to this underlying reality as > >>> simply 'space', for want of a better word. The nature > >>> of space is, therefore, like continuity, that it supports > >>> and enables the process of measurement, as something > >>> finite, expressed in countable form, but is not itself > >>> intrinisically measurable, or countable. (I assume a flat, > >>> open space, and not one that is closed, like the surface > >>> of a sphere) > > >>> Space, like continuity, is thus an infinity, as a > >>> qualitatively distinct reality that is not in itself > >>> measurable, but supports and adopts, though > >>> remaining distinct in itself, the finite > >>> measure that it enables. > > >>> The result is that the size of space is currently that > >>> of the size of its contents. If one goes to the edge of > >>> the contents of space, one can be said to go to the > >>> edge of space. If one goes beyond this, one expands > >>> the contents of space, as oneself a content of space, > >>> and expands the space. It does not mean that the > >>> extra extent, into which one goes, is an extent that > >>> already exists. It means that space automatically > >>> provides whatever extent its contents require, but > >>> does not have to be said to provide any extent they > >>> do not require. One thus does not speak of an > >>> 'infinite extent', since the infinity of space only > >>> provides 'extent' in so far as its contents require it. > >>> We thus do not ever have to deal with a concept of > >>> an 'infinitely distant' location, which is really a > >>> contradiction in terms. > > >>> That is the nature of an infinity, that it supports and > >>> enables the finite and measureable, while not, in itself, > >>> being finite or measurable. > > >>> Thus continuity and space are aspects of one reality, > >>> called space (I don't like using the same word twice, > >>> but the language is deficient), which is, in itself, an > >>> infinity. It has no intrinsic size or number of dimensions, > >>> but supports and enables whatever size and number > >>> of dimensions its contents may require. > > >>> Alen > > > What I have tried to define as infinite is, I believe, > > a real, actual infinity, and the only kind that is real. > > > But I agree entirely with what you say about the usual > > kinds of uses of the word infinity. > > >> I have come to the conclusion that infinity should be considered as > >> mathematical shorthand the meaning of which is context specific. > > > Yes, I think that the way it has normally been used is > > understandable only as a kind of shorthand. > > >> When one sums a converging sequence "to infinity" it actually means > >> "keep adding terms until they no longer have a significant effect" > > > Exactly right > > >> When one talks of "the sum of all integers is infinity" what one means > >> is that "the sum just gets bigger and bigger without end" not that it > >> reaches something called "infinity". > > > Again, exactly right. Whenever there is any kind of > > 'sum to infinity', it can, in reality, only be a sum to a > > sufficiently large number that will achieve the desired result. > > If one tries to think of infinity as something real - even in the > mathematical sense of real, "of use in mathematics" then - as > mathematicians have found - you end up with an infinite number of > infinities each infinitely larger than the other. > > "Take the natural numbers 0,1,2,3,4.... You can go on counting these > till kingdom come, so there is no doubting that the set of natural > numbers is infinity. But this "countable" infinity occupies only the > lowest rung of an infinite ladder. Ironically larger infinities arise > when you break down the natural number into subsets..... How many > subsets are there altogether? An infinite number of course. Cantor was > able to prove that this infinity is bigger than the original countable > set. This second level is the "continuum".....And so it goes on. By > looking at the collection of all possible subsets of real numbers, you > find a still higher level of infinity, and so on ad infinitum. Infinity > is not a single entity, but an infinite ladder of infinities with each > rung infinitely higher than the one before." 'It doesn't add up' *New > Scientist Aug 2010 P38* > > This "ladder" is not a useful concept.
Why not?
> > As I have said before If you start the process of say adding integers > then no matter how many you have added together the result is always 0% > of infinity. The only way of reaching infinity it to assume you have > added an infinite number together. Thus there is a discontinuity between > the process - which never gets any nearer to the destination and the > destination which can only be reached by an assumption that you have got > there. > > I'm not sure what you call the nth integer when n is infinity the > Infinit-th integer perhaps. The Infinit-th integer is of course infinity > so to reach infinity there was no point in adding all those other > integers but if you have done then the sum must be greater than the > infinity contributed by the infinit-th digit. After all the (infinit-th > - 1) integer is only one less than infinity and the next previous > integer is infinity - 2 .... etc. > > The only definition of "infinity" which makes logical sense is as a > shorthand description of a process without a limit. "Infinity" does not > exist as the limit of that process as it has no specific meaning.
