"It was first suggested that a googolplex should be 1, followed by writing zeros until you got tired."
http://www-users.cs.york.ac.uk/~susan/cyc/g/googol.htm
http://www-users.cs.york.ac.uk/~susan/cyc/g/googol.htm
abyssoft said:And for 3 more insainly large numbers
Myriad is Googol^Googol
Myriadol is Googol^Googolplex
Myriadaplex is Googolplex^Googolplex
I remember my Calc prof telling my class about these and said that these have no real practical value.
I understand this argument but it seems unproveable. I have demonstrated that rationality is a sufficient but not necessary condition for countability. Based upon history, each time we expand mathematics we stand a good chance in the process of developing a way to list a new set of irrational numbers that includes one or more of the sets already covered. Therefore we cannot say that the possibility does not exist that we will discover a way to list all numbers.Absane said:The reason rational numbers are countable is because, roughly speaking, there exists a way to list them in a set that includes all rational numbers. Irrational numbers are not countable because there does not exist a way to list them in a set (when one thinks they listed all the irrational numbers, you can always create a new one).
Isn't any set of numbers defined by a function of A, B, and C countable if A, B, and C are countable?How is A^(B/C) countable if you let B/C = 1/2 and K = {A^(1/2) : A is in [0,1]}? It includes an infinite number of irrational numbers and A is not even countable. Aleph-0 is any set having the same cardinality as the set of integers, like rational numbers... Let Aleph-0 = X
X<sup>X</sup> = c, which c is the nondenumerable set of real numbers, called the continuum. Also, X + c = c.
Fraggle Rocker said:I understand this argument but it seems unproveable. I have demonstrated that rationality is a sufficient but not necessary condition for countability. Based upon history, each time we expand mathematics we stand a good chance in the process of developing a way to list a new set of irrational numbers that includes one or more of the sets already covered. Therefore we cannot say that the possibility does not exist that we will discover a way to list all numbers.
Isn't any set of numbers defined by a function of A, B, and C countable if A, B, and C are countable?
That certainly seems to be the thesis upon which the countability of rational numbers is based: All numbers of the form A/B are countable so long as A and B are integers.
So I count
1 ^ (1/1)/; 1 ^ (1/2); 1 ^ (2/1); 1 ^ (2/2); 2 ^ (1/1); 2 ^ (1/2); 2 ^ (2/1); 2 ^ (2/2);
1 ^ (1/3); 1 ^ (2/3); 2 ^ (1/3); 2 ^ (2/3); 1 ^ (3/2). . . .
or any other way of mapping three variables onto a one-dimensional vector. I get all their values and this particular mapping allows them all to approach infinity at the same decreasing rate.
A ^ (B/C) is indeed countable so long as A, B, and C are integers.
Giambattista said:Complete BS. I use these all the time. In fact, none of us would be here if it weren't for myriadaplex.
I understand that "to count" by definition is the ordination of integers. Therefore we can only count numbers that can be expressed as formulas that are combinations of cardinal numbers. I do indeed intend to play by those rules.Absane said:But your list is only a proper subset of the positive reals. Tell me when A^(B/C) = Pi or e or Euler's Constant.
Fraggle Rocker said:Is this an ordered list of all numbers, rational and irrational?
The integer part is trivial. The fractional parts, in binary notation:
.1, .01, .11, .001, .101, .011, .111, .0001, .1001, etc.
It's just counting from one to infinity, reversing the order of the bits, and putting a binary point in front of it. The leading zeros become trailing zeros and by the same convention are not written.
There cannot be any number which is not in that series or the integrity of our numbering system is called into question. Its infinitude is exactly the same as that of integers because they map one-to-one.
All numbers can be expressed as (A, B) where both are positive integers. B is the fractional part written backwards and its place in my list is defined by counting.
I don't understand the notation but I agree with the words.Absane said:Are you asking if the set {.1, .01, .11, .001, .101, .011, .111, .0001, .1001, ...} Forms the interval (0,1) . . . R?
And also, if that is the case, since we found a way to line up the <b>all</b> the numbers in (0,1), we can expand on that for all R and therefore R is countable. Corollary, irrationals are countable?
I hope I understand this right
Hi Fraggle Rocker,Fraggle Rocker said:Is this an ordered list of all numbers, rational and irrational?
The integer part is trivial. The fractional parts, in binary notation:
.1, .01, .11, .001, .101, .011, .111, .0001, .1001, etc.
It's just counting from one to infinity, reversing the order of the bits, and putting a binary point in front of it. The leading zeros become trailing zeros and by the same convention are not written.
There cannot be any number which is not in that series or the integrity of our numbering system is called into question. Its infinitude is exactly the same as that of integers because they map one-to-one.
All numbers can be expressed as (A, B) where both are positive integers. B is the fractional part written backwards and its place in my list is defined by counting.
This list contains all numbers if the integrity of our numbering system is taken for granted. Any number can be expressed in binary notation. (I could haved done it in decimal but I assume that we're all scientists here and binary was easier.) I think the problem you're driving at is that some finite numbers have an infinite number of digits. I can describe to you with complete precision how to navigate down my list to the number 1/3, which is .01010101... in binary and therefore has a sequence number of 10101010... And if you take my instructions you will indeed find it. Except of course for the fact that finding it will take an infinite amount of time.Pete said:This list contains no irrational numbers, and not all rational numbers (what is the sequence number of 1/3, for example).