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.
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.
