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 May 24th, 2010, 11:16 AM #1 Member   Joined: Apr 2010 Posts: 47 Thanks: 0 Biggest possible number??? would Infinity to the power of Infinty to the power of Infinity and writing Infinity to the power of Infinity and power infinity to it Infinite Number of times, then power Infinity to the result and do that an Infinite number of times, and then do that power to Infinity and Infintie Number of Times an Infintie amount of time....would that be the highest possible number you could imagine, since there is a higher arky of Infinities. I know its mind bogglingly big, but could you call this number the largest number, also if you didn't get it Imagine it in Groups. Group I: Infinity to the power of Infinity (X) Group II: X, to the power of Infinity, to the power of Infinity, and do that an Infinite Number of times (Y) Group III: Do the same to Y as you did to X, and every result from then on do the same and finally after all that Infinite amount of mathematical acts (Z) Group IV: Could we conclude Z is the highest possible Infinity? Thus in a sense a largest amount of numbers.
 May 24th, 2010, 11:51 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Biggest possible number??? You'll need to define what you mean by "number" before we can answer the question "What is the biggest possible number?". For example: There is no largest integer: for every integer there exists a larger integer (e.g., its successor).[/*:m:2bv12m56] There is no largest real number: for every real number there exists a larger real number (a special case is the so-called "Archimedian principle", where in particular the larger number is an integer).[/*:m:2bv12m56] There is no largest complex number: complex numbers have no order.[/*:m:2bv12m56] There is no largest integer mod 5: integers mod 5 have no order.[/*:m:2bv12m56] There is no largest projective real: projective real have no order.[/*:m:2bv12m56] The largest extended real is +?.[/*:m:2bv12m56] There is no largest ordinal in ZF: for every ordinal there is a larger ordinal (e.g., its successor).[/*:m:2bv12m56] There is no largest cardinal in ZF: for every ordinal there is a larger cardinal (e.g., its power set).[/*:m:2bv12m56] The largest set (by inclusion) in NF is the universal set V.[/*:m:2bv12m56] I'm pretty sure there is no largest hyperreal or surreal, but I'd have to check.
 May 24th, 2010, 12:53 PM #3 Senior Member   Joined: Apr 2010 Posts: 215 Thanks: 0 Re: Biggest possible number??? Whatever operation you do, you can always do it once more. Thus there's no biggest possible number. If you do infinity to the power of infinity, repeated infinity times, you can still do infinity to the power of infinity, repeated infinite times, repeated infinite times. You could define a new "number", and say that it is always bigger than any other number. However it cannot follow the current laws of mathematics, as they simply don't allow for it. There will be inconsistencies.
 May 25th, 2010, 09:14 AM #4 Senior Member   Joined: Apr 2008 Posts: 435 Thanks: 0 Re: Biggest possible number??? So, we should really take a step back and consider that 'infinity' is not a number, and so 'infinity to the infinity' isn't defined in it's own right. So, as Charles has mentioned, what do you mean by 'number?' I think a useful consideration would be the following three questions: Which are there 'more of': integers like 1,2,3 OR rationals like 1/2, 4/7, etc. Which are there 'more of': rationals, or all real numbers besides rationals (like pi, e, and the sort) Are there 'more' numbers between 10 and 20 than there are between 1 and 2? I think these are good introduction questions to concepts of 'infinity.'
May 27th, 2010, 03:51 PM   #5
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Re: Biggest possible number???

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 Originally Posted by deanmullen10 ...a higher arky of Infinities. ...
I love this guy!

 May 28th, 2010, 02:22 AM #6 Senior Member   Joined: Apr 2010 Posts: 215 Thanks: 0 Re: Biggest possible number??? There is an infinite number of integers. There is an infinite number of odd integers. There are twice as many integers as odd integers. I believe that infinite numbers can be used in a similar fashion as the imaginary unit, as they have no real number representation.
 May 28th, 2010, 04:38 AM #7 Global Moderator     Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4 Re: Biggest possible number???
May 30th, 2010, 12:50 PM   #8
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Re: Biggest possible number???

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 Originally Posted by brangelito There is an infinite number of integers. There is an infinite number of odd integers. There are twice as many integers as odd integers. I believe that infinite numbers can be used in a similar fashion as the imaginary unit, as they have no real number representation.
What does this mean? A good way to decide whether or not you have 'more' of one thing than another is to see if you can put them into bijection, that is to see if fr every element of one you can associate an element of the other. For example, for ever integer, I can associate it's double. So 1 corresponds to 2, 2 to 4, 3 to 6, and so on. In this case, we see that each integer corresponds to exactly one even integer, and so there are 'about the same' in number. Or, as the above post implies, they have the same cardinality.'

May 30th, 2010, 01:11 PM   #9
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Re: Biggest possible number???

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 Originally Posted by jason.spade What does this mean? A good way to decide whether or not you have 'more' of one thing than another is to see if you can put them into bijection, that is to see if fr every element of one you can associate an element of the other. For example, for ever integer, I can associate it's double. So 1 corresponds to 2, 2 to 4, 3 to 6, and so on. In this case, we see that each integer corresponds to exactly one even integer, and so there are 'about the same' in number. Or, as the above post implies, they have the same cardinality.'
There are lots of different ways to compare sizes. As ordinals, $\omega+1<\omega$, but both have cardinality $\aleph_0$.

In the case of comparing subsets of the natural numbers, one measurement that seems appropriate is natural density:
$D(S)=\lim_{N\to\infty\frac{|S\cap\{\1,2,\ldots,N}| }{N}$

In that sense, the integers have density 1 and the evens have density 1/2, making the integers 'twice as big'. In the sense of cardinality they're the same size.

 May 31st, 2010, 08:36 AM #10 Senior Member   Joined: Apr 2010 Posts: 215 Thanks: 0 Re: Biggest possible number??? Do primes have the same cardinality as positive integers? Every prime could be indexed as p_n, where n is a positive integer. Then every positive integer corresponds to a prime, right? Also, if I understood the definition correctly, the natural density of primes is 0.

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