22 Jun 08
As I watch a rerun of Cosmos, I think Carl Sagan was wrong. In desribing very large numbers, he discussed a googol, and said that there were only 10 to the 81st poer number of protons, neutrons, and electrons in all the "accessable universe," still less than "googol." No problem-I'll take his word for that. But THEN, he said that infinity is so large that it is "absolutely as far from googol as it is from the number one." Now that makes no sense. Infinity is distant from googol by Infinity - googol. One is that, PLUS googol minus 1. Right?
22 Jun 08
Originally posted by PinkFloydNo. Infinity is not a set value. You cannot add or substract with infinity.
As I watch a rerun of Cosmos, I think Carl Sagan was wrong. In desribing very large numbers, he discussed a googol, and said that there were only 10 to the 81st poer number of protons, neutrons, and electrons in all the "accessable universe," still less than "googol." No problem-I'll take his word for that. But THEN, he said that infinity is so large th ...[text shortened]... is distant from googol by Infinity - googol. One is that, PLUS googol minus 1. Right?
22 Jun 08
3 edits
Originally posted by PinkFloydinfinity is as far from 1 as it is from googol.
As I watch a rerun of Cosmos, I think Carl Sagan was wrong. In desribing very large numbers, he discussed a googol, and said that there were only 10 to the 81st poer number of protons, neutrons, and electrons in all the "accessable universe," still less than "googol." No problem-I'll take his word for that. But THEN, he said that infinity is so large th is distant from googol by Infinity - googol. One is that, PLUS googol minus 1. Right?
algebraically:
infinity = infinity - googol
= infinity - 1
The familiar rules of addition and subtraction are no longer valid when infinity becomes involved.
When the rules are set up, if you take the time to read the fineprint, most mathematicians put in some sort of proviso that the numbers are finite.
The rules can be extended to cover use of infinite ... but it tends to be fairly unuseful, and lead to controversial "proofs". People applying maths often get themselves in a knot when they do this ... for an example of using infinite you might like to look up "renormalisation".
23 Jun 08
Originally posted by flexmoreAs a novice, is this something that only comes into play with calculus and other higher mathematics? I don't recall algebra class dealing much with this problem 🙂
infinity is as far from 1 as it is from googol.
algebraically:
infinity = infinity - googol
= infinity - 1
The familiar rules of addition and subtraction are no longer valid when infinity becomes involved.
When the rules are set up, if you take the time to read the fineprint, most mathematicians put in some sort of proviso that the numb ...[text shortened]... ey do this ... for an example of using infinite you might like to look up "renormalisation".
23 Jun 08
Originally posted by flexmoreI checked out renormalozation sites, but all refer to something called the "Lagrangian". What would that be?
infinity is as far from 1 as it is from googol.
algebraically:
infinity = infinity - googol
= infinity - 1
The familiar rules of addition and subtraction are no longer valid when infinity becomes involved.
When the rules are set up, if you take the time to read the fineprint, most mathematicians put in some sort of proviso that the numb ...[text shortened]... ey do this ... for an example of using infinite you might like to look up "renormalisation".
23 Jun 08
Infinity is not a number. It is a concept. To use it in an equation, or to use it in the sense used in your first post is in effect assuming that there exists a real number in the set of real numbers that is maximum in that set, and call that number 'infinity'. However, no such number exists, so the whole think simply breaks down.
26 Jun 08
3 edits
Originally posted by PinkFloydIn pure mathematics there is a field called "algebra", which I fear is very different from the school-taught concept of algebra. Essentially, it deals with a set (for instance, the set of integers) under an operation (for instance, addition). This, I believe, deals with the concept of infinity much more than calculus does. 😉
As a novice, is this something that only comes into play with calculus and other higher mathematics? I don't recall algebra class dealing much with this problem 🙂
Anyway, Infinity. The concept of infinity, I believe, has been looked at in it's purist form by set theorists. They introduced the "aleph" notation for sets of cardinality (size) infinity. Now, here is the wied bit... There is more than one infinity.
Without getting into too much detail, as I learned this during honours mathematics courses and only really started to understand it when I came across it at masters-level, but basically there are infinetly many infinities. The natural numbers is the smallest set of cardinality infinity, and we say this has cardinallity "aleph null". Curiously, this had the same cardinality (size) as the integers and also of the the rational numbers. This is because aleph_0 * aleph_0=aleph_0 (and of course the rational numbers are the set of all pairs of integers)*.
Now, it can be shown that aleph_0 /= aleph_0 ^ aleph_0=2^aleph_0 = the cardinality of the real numbers = the cardinality of the powerset of the natural numbers. Infact, you can always take the power set of a set to get a set of size strictly greater than the original set. This is why there are infinetly many infinities.
Now, I know this is all confusing, and explained badly, but I shall confuse you some more. Because it's an interesting result. Now, there are more than one "levels" of infinity. They go, "aleph_0, aleph_1, aleph_2,...". Gregor cantor hypothesised in his "continuum hypothesis" that the cardinality of the real numbers was equal to aleph_1. That it is the next level of infinity. In 1939, Kurt Godel showed that this could not be disproved. In 1963, Paul Cohen showed that it could not be proved. In short, it is a result outside of our general axioms of set theory (called ZF, or, more controversially, ZFC).
And for the icing on the cake, and a vague explanation of the ZF/ZFC controversial ataement? http://en.wikipedia.org/wiki/Axiom_of_choice
NOTE: /= is does not equal.
*Actually, just to confuse you more, this is the wrong way round. aleph_0 * aleph_0 = aleph_0 because the rational numbers is the set of all pairs of integers, and because we can find a bijection between the set of pairs of integers and the set of integers.
http://en.wikipedia.org/wiki/Continuum_hypothesis
http://en.wikipedia.org/wiki/Power_set
26 Jun 08
Originally posted by SwlabrOkay, usual questions about infinities are:
Anyway, Infinity. The concept of infinity, I believe, has been looked at in it's purist form by set theorists. They introduced the "aleph" notation for sets of cardinality (size) infinity. Now, here is the wied bit... There is more than one infinity.
(And the answers are quite contra-intiutive.)
(1) There are infinitly many natural numbers (positive whole numbers). Right?
(2) But if you bring in even the negative whole numbers into that, there should be the double amount of numbers, right?
(3) How many more (double?, triple?) are there rational nubers (p/q and p and q are whole numbers) than natural numbers?
(4) Allright, what about real numbers, they should be many more than natural naumbers, right?
(5) Take two rational numbers that are very close to each other. How many real numbers are there in between? Even if they are very very near to eachother?
(6) Take two real numbers that are very close to each other. How many rational numbers are there in between? Even if they are very very near to eachother?
(7) From the two answers pf (5) and (6) can we then make any deductions about what number system that contains more numbers in them?
Are we talking about late mathematics here? Like maths invented by Kantor and those? Then I remind you that Gallilei knew the answer of this last question:
(8) How many more squares (from the natural numbers) is there compared with the qubes?
26 Jun 08
Originally posted by FabianFnas3 .. same number
Okay, usual questions about infinities are:
(And the answers are quite contra-intiutive.)
(1) There are infinitly many natural numbers (positive whole numbers). Right?
(2) But if you bring in even the negative whole numbers into that, there should be the double amount of numbers, right?
(3) How many more (double?, triple?) are there rational nubers ...[text shortened]... uestion:
(8) How many more squares (from the natural numbers) is there compared with the qubes?
4 .. infinitely more (step up one notch)
5.. infinite ... of the second smallest infinite type
6 .. infinite ... of the smallest infinite type
7.. the real numbers have a few more ... but not many more only infintely more 😉
26 Jun 08
Originally posted by flexmoreCan you explain further?
3 .. same number
4 .. infinitely more (step up one notch)
5.. infinite ... of the second smallest infinite type
6 .. infinite ... of the smallest infinite type
7.. the real numbers have a few more ... but not many more only infintely more 😉
Or else, (as a math teacher) I cannot give you more than a minimum of points.
Extra points if you can get my grand ma to understand your explanation.
26 Jun 08
Originally posted by flexmoreActually they are of the same cardinality.
3 .. same number
It is incorrect to say they are the same number as infinity is not a number.
It is true to say that you can create a one-to-one mapping between the two sets, but again it is incorrect to say that they have the same number - because there is no number that represents the number of elements in the set!
The sets members are countable, ie can be put in a sequence which can then be mapped to the set of positive integers. For us to correctly say 'the number of elements in the set are:" then the sequence must have a final element which requires that the sequence is finite.