Thread: Beaty infinite

If this kind of topic already sent, this topic can be removed...

I want to ask a simple (and again classical) question:
What is the highest value of a number?

Many say it is Googleplex (if I remember right 10^10^1000)
but how can it be so? there is still possibility to
add one "1" after it, or is this just a question of
"sensible limitations of mathematical capabilitys"
(as some might say)?

The reason I wanted to ask this, is that I am allways been
rather bothered by the definition of "infinite"...

According to it, there is infinite amount of numbers,

e.g. -oo ...-5,-4,-3.-2,-1,01,2,3,4,5... oo (oo = infinite)

and between each numbers there are another
infinites,

e.g. -oo ...,0,1,1.1,1.2,1.3,1.4... oo

and even between them can be found more infinetes,

e.g. - oo ...,0,1,1.1,1.11,1.12,1.13,1.14... oo

and even between them are infinite amount of combinations...

So is it reasonable to ask the highest number (that would
be "near +/- oo", as you know) or should we better
to ask, what is the " most sensible" number to reach?

Well, first you should look into Cantor's trans-infinites. For example there is countable infinity, which is the number of items in any set which can be placed into a one to one correspondence with the natural numbers 1, 2, 3, 4, 5, .... Cantor proved that the set of rational numbers (fractions, or numbers which can be expressed as ratios of two whole numbers) is also countable, but that the set of irrational numbers which include pi and the square root of two is not countable. So the set of irrational numbers represents a higher order of infinity, into which all the points on a finite line segment and all the points of an infinite plane can be put into a 1-1 correspondence. However there are even higher orders of infinity, for example, the number of possible functions on a real lines segment is believed to represent the next higher order of infinity.

is an infinite set, of say integers, the universal set of all integer based subsets.

Originally Posted by wallaby

is an infinite set, of say integers, the universal set of all integer based subsets.

Let me rephrase that to see if I understand you. Are you asking whether the set of all countable sets is itself a countable set?

I think it is not countable and here is my arguement.

For every irrational number between 0 and 1 I can construct a unique countable set by the following method: take the tenths digit and add 10 as the first number in the set, take the next two digits and add 100 to this two digit number and make this another member of the set, take the next three digits and add 1000 as another member of the set and so on. The set is clearly countable (1: the 2 digit member, 2: the three digit member, 3: the four digit member, and so on). So the set created for each irrational number is in the set of all countable sets. The addition of the power of 10 assures that digits like 00 and 000 become different members of the set, and helps insure that the set which is created is in fact unique. This set is different for every irrational number in the interval from 0 to 1, therefore there is a 1 to 1 correspondence between the irrationals from 0 and 1 and a subset of the set of all countable sets. But this means that the set of all countable sets must have at least as many members as the number of irrationals between 0 and 1. Since the latter is not countable then the set of all countable sets cannot be countable.

Now here is a challenge. If you can come up with a way of making a 1 to 1 correspondence between the natural numbers and all the members of the set of all countable sets then you will have proven that the irrationals between 0 and 1 are countable. Which will prove that Cantor was wrong.

i'm sorry i can't tell if this answers my question or not... words are too confusing for me.