I just thought of an analogy of the concept of infinity to the concept of relativity.
In relativity, it makes no difference if we stand still or run up an escalator moving at the speed of light. We always reach the top at the same time.
In infinity, it makes no difference what quantity of numbers or sets of numbers are represented. We always end up with an infinitely large number.
Infinity, like relativity, cannot be broken into parts as each part will always be infinitely large.
Cantor’s “harmless little argument”, if it is accepted uncritically, merrily leads one to accepting a hierarchy of infinities culminating in an “infinity of infinites” which to him, was God.
This “harmless little argument” is theology masquerading as mathematical insight.
There is no crusade against infinity. Cantor’s argument, however, is not “accepted by nearly all mathematicians” because its implications are outrageous.
So it is a crusade, a crusade against God, driving him away from every possible (logical, metaphysical) corner of the universe by combating the corner. Well, I can only say, your mathematical ‘god’ has not much to do with any picture of god I heard.
May I ask you for the outrageous implications? (Again, no citations and links.)
And, to second Doug, also afaik there are not other kinds of mathematical infinities. A set is infinite as N (i.e. you must count endlessly to count them all), or it is infinite as R (i.e. you don’t even know how you should order the elements to count them).
Cantor’s “harmless little argument”, if it is accepted uncritically, merrily leads one to accepting a hierarchy of infinities culminating in an “infinity of infinites” which to him, was God.
Firstly, I recall hearing a proof that there is no largest infinity, e.g., that encompasses all the other infinities. Hence there is no such thing as the infinity of infinities, as though it were a thing of defined size and character. If this recollection is correct, then there is nothing for God to be, since there is no such thing as “the infinity of infinities”.
Secondly, even if there were such a thing as “the infinity of infinities”, it would not be God. God is a person who responds to prayer and created the world. An abstractum like a number or an infinity is none of those things.
Thirdly, although I have not personally polled all mathematicians, my sometime dealings with them leave me quite confident that the vast majority, indeed “nearly all”, accept Cantor’s argument. It is taught as absolutely standard mathematics.
Perhaps an infinity of infinities: where each infinity set is equal to all other sets and equal to the whole…..?
Perhaps an infinity of infinities: where each infinity set is equal to all other sets and equal to the whole…..?
I don’t think so, no. Since there are different orders of infinities (an infinite number of different orders, in fact), “each infinity” is not going to equal “all other sets”, nor is going to equal “the whole”. Again, I forget the details, but IIRC it was impossible to construct a meaningful symbol that would encompass “the whole” or “all the infinities”, in the same way that people have constructed symbols for some of the lower orders of infinities.
Here is a youtube video from the BBC - “Dangerous Knowledge” about Cantor, Bolzmann, Godel and Alan Turing and their investigations into the realms of the infinite, thermodynamics, mathematical completeness and computability. Their ideas were too advanced for their time and met fierce opposition from their contemporaries. Tragically, it drove them insane or to commit suicide. It is in 10 parts:
Here is a youtube video from the BBC - “Dangerous Knowledge” about Cantor, Bolzmann, Godel and Alan Turing and their investigations into the realms of the infinite, thermodynamics, mathematical completeness and computability. Their ideas were too advanced for their time and met fierce opposition from their contemporaries. Tragically, it drove them insane or to commit suicide. It is in 10 parts: http://www.youtube.com/watch?v=Cw-zNRNcF90
I guess you give an outline now? But if I understand the idea from your posting, numerous mathematicians and physicists would have committed suicide. Turing was driven to suicide by mainly by his not accepted homosexuality, Gödel was paranoid, afraid that he might be poisoned. If there is a connection between their scientific discoveries (if!) and their suicides, it is mainly that their genial insights were not accepted.
As you do now, obviously. What would you have said to Cantor? What would you have said to Galileo? Or Giordano Bruno?
Turing was driven to suicide by mainly by his not accepted homosexuality, Gödel was paranoid, afraid that he might be poisoned. If there is a connection between their scientific discoveries (if!) and their suicides, it is mainly that their genial insights were not accepted.
Right, their suicides had nothing to do with the math. Gödel was mentally ill.
Perhaps an infinity of infinities: where each infinity set is equal to all other sets and equal to the whole…..?
I don’t think so, no. Since there are different orders of infinities (an infinite number of different orders, in fact), “each infinity” is not going to equal “all other sets”, nor is going to equal “the whole”. Again, I forget the details, but IIRC it was impossible to construct a meaningful symbol that would encompass “the whole” or “all the infinities”, in the same way that people have constructed symbols for some of the lower orders of infinities.
1,2,3,4,5….......(the set of natural numbers) (NN)
1.3.5.7,9….......(the set of odd numbers) (ON)
There is a one-to-one correspondence for every number in both sets (if the number of numbers in both sets are assumed to be inexhaustible), therefore NN=ON and both sets are infinite.
Mathematically brilliant, except for the oddness of the outcome which arises from the definition of an infinite set whereby a subset (ON) is equal to the whole set (NN).
Seems that would confirm that a subset (ON) can be and is equal to the whole (NN).
Question: is an infinite set of inches shorter than an infinite set of feet, or does the one-to-one correspondence apply here at all?
The word density keeps popping into my mind. Is it possible that each infinity is equally large, but each set has a different “density of properties”, which accounts for any apparent unequality.
Again using relativity whereby, at the speed of light, the speed at which the object itself travels does not add to the overall speed. The object may become more dense or energetic, but it cannot travel faster than the speed of light.
Is it possible that such a principle could be applied to the concept of infinity as well? Where in relativity the speed of light is a barrier which cannot be exceeded, perhaps infinity, because of its lack of any barriers, condenses or expands the “density” of each set of infinities to always yield the same result of infinity, regardless of how these sets are partioned. Could there be a basic modifier or qualifier at work here, which has not yet been identified?
Question: is an infinite set of inches shorter than an infinite set of feet, or does the one-to-one correspondence apply here at all?
You’re asking if 12 * ∞ = ∞
The answer is, “Yes”.
If you don’t understand this intuitively, you don’t have a grasp on what infinity is.
No, I’m asking if 1 * 00 = 00 is equal to 12 * 00 = 00 (sorry I have no infinity sign to use).
If the answer is still yes, then what is the fuss all about?
My problem is not a lack of intuition, but a lack of clarity in the proposition that there can be larger and smaller infinities.
If you don’t understand this intuitively, you don’t have a grasp on what infinity is.
No, I’m asking if 1 * 00 = 00 is equal to 12 * 00 = 00 (sorry I have no infinity sign to use).
If the answer is still yes, then what is the fuss all about?
My problem is not a lack of intuition, but a lack of clarity in the proposition that there can be larger and smaller infinities.
Obviously you have. On first sight it is counter intuitiv.
To expand a little, in the hope you get some intuition:
The set of N can be counted, i.e. there is an order in the elements: 1,2,3,4,5,...
Of course this goes on endlessly, but at least you did not jump over any element: if you got to ‘5’, you know there are no other elements somewhere in between.
This goes for Z (natural numbers, and negative numbers):
0,1,-1,2,-2,3,-3… etc.
Again I know, I did not leave out any element.
Of course it is the same for even numbers 2,4,6,8 etc. It is even true for 2,4,9,16,25,36… It is endless, and I know I did not leave any element out.
It is even true for Q (broken numbers): I can find an order where I can count diagonally:
3/3 3/2 3/3 2/1 2/2 2/3 1/1 1/2 1/3 0/1 0/2 0/3
Start from 0/1, then go from 0/2 to 1/1, then from 0/3 to 2/1 etc. So I count certain elements twice or more (0/1 = 0/2, 1/1 = 2/2), but again I am sure I leave no element out. Said in another way, I can number them:
1: 0/1
2: 0/2
3: 1/1
4: 0/3
5: 1/2
6: 2/1
So there is a one to one relationship between N and Q, which means their infinities are the same, even if you count certain elements twice. You must distinguish between the number of elements in a set, and possible orders. Even if we feel that we leave out elements from the sets (where are 1,3,5, etc in the set of 2,4,6?), their count is still the same because for every element in the even numbers I can assign 1 element in the natural numbers (and the other way round).
Cantors argument (use google to find it, it is definitely too long to explain here) clearly shows that there is no such relationship between R (the real numbers, containing pi, sqrt(2) etc) and N. And afaik there is no still bigger infinity in mathematics.
If you don’t understand this intuitively, you don’t have a grasp on what infinity is.
No, I’m asking if 1 * 00 = 00 is equal to 12 * 00 = 00 (sorry I have no infinity sign to use).
If the answer is still yes, then what is the fuss all about?
Er, you say “No” and then you go ahead and provide two equations that are precisely the same as the equation I just provided.
For starters, equations can’t equal one another, so what you wrote (“if 1 * 00 = 00 is equal to 12 * 00 = 00”) is confused; unless you simply mean: “if 1 * 00 = 00 = 12 * 00 = 00, but then you have 00 twice, and they can be removed. In which case you’re left with 1 * 00 = 12 * 00.
Or, said another way:
(A) 1 * ∞ = ∞
(B) 12 * ∞ = ∞
Replacing the ∞ on the right side of equation (A) with the left side of equation (B):
(C) 1 * ∞ = 12 * ∞
Since 1 * X = X, for any value X,
(D) ∞ = 12 * ∞
This is precisely what I wrote in my last post, above. If you don’t understand this, there’s no point in continuing because you aren’t understanding the most basic mathematical operations.
Write4U - 31 January 2010 09:12 PM
My problem is not a lack of intuition, but a lack of clarity in the proposition that there can be larger and smaller infinities.
Larger and smaller orders of infinity do not involve addition, subtraction, multiplication or division operations.
Two picture the first two orders of infinity, we can start with the integers:
1, 2, 3, 4, 5 ...
This is called a “countable” infinity, because all the items in it can be perused by a discrete counting operation.
The next order of infinity is the real number line, including all points on it.
I can’t write this out, because it is an uncountable infinity. But think of all the points along a ruler. For every two integers, there are an infinite number of points in between: an uncountable number of points.
[ Edited: 01 February 2010 07:30 AM by dougsmith ]
If you don’t understand this intuitively, you don’t have a grasp on what infinity is.
No, I’m asking if 1 * 00 = 00 is equal to 12 * 00 = 00 (sorry I have no infinity sign to use).
If the answer is still yes, then what is the fuss all about?
Er, you say “No” and then you go ahead and provide two equations that are precisely the same as the equation I just provided.
For starters, equations can’t equal one another, so what you wrote (“if 1 * 00 = 00 is equal to 12 * 00 = 00”) is confused; unless you simply mean: “if 1 * 00 = 00 = 12 * 00 = 00, but then you have 00 twice, and they can be removed. In which case you’re left with 1 * 00 = 12 * 00.
Or, said another way:
(A) 1 * ∞ = ∞
(B) 12 * ∞ = ∞
Thank you, that answers my question as I posed it.
p.s. as a retired accountant I do have some experience with sets of numbers.
