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July 30th, 2017, 07:00 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 371 Thanks: 26 Math Focus: Number theory  Descriptions of numbers vs. their magnitudes
Does the mix of meanings for numbers generally increase with their value?

July 30th, 2017, 07:35 PM  #2 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,600 Thanks: 546 Math Focus: Yet to find out. 
What do you mean "mix of meanings"?

July 30th, 2017, 08:38 PM  #3  
Senior Member Joined: May 2015 From: Arlington, VA Posts: 371 Thanks: 26 Math Focus: Number theory  Quote:
 
July 31st, 2017, 03:40 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,327 Thanks: 2451 Math Focus: Mainly analysis and algebra 
What do you mean by "descriptors". I'm not at all sure that the "quantity of descriptors of a number" is well defined mathematically.

July 31st, 2017, 04:58 AM  #5  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,600 Thanks: 546 Math Focus: Yet to find out.  Quote:
If this is the case, then when won't the 'descriptor' be the same as the magnitude? And what does magnitude even mean in this context. In your example, won't the description for both the number, and the magnitude of the number be the same for each value of n?  
July 31st, 2017, 10:55 AM  #6 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 371 Thanks: 26 Math Focus: Number theory 
If a number is smaller, for instance, would it necessarily be described by fewer "sequences," and if larger, always by more "sequences"? Extend "fewer sequences" and "more sequences" to every way a number can be defined. Descriptors are the set of all permutations which comprehensively describe a number. Finite sets of descriptors of numbers are unbounded, but transfinite sets of descriptors of numbers exceed them even more. Do all sets of descriptors include the transfinite numbers? Joppy, instead of "magnitude of numbers," do consider "magnitude of the set of [their] descriptors." Please excuse that I am having to invent and define some of these terms myself. Last edited by Loren; July 31st, 2017 at 10:58 AM. 
July 31st, 2017, 12:36 PM  #7 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 371 Thanks: 26 Math Focus: Number theory 
I.e., are there as many ways to describe one number as another?

July 31st, 2017, 02:37 PM  #8 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,237 Thanks: 884 
In order for that to have an answer, you will need to give a very precise definition of "ways to describe a number".

July 31st, 2017, 07:34 PM  #9  
Senior Member Joined: Aug 2012 Posts: 1,956 Thanks: 547  Quote:
First, what can it mean to describe a number? It means to write down a finitelength string of symbols that represent the given number via some rules that tell you whether it's a legal string, and some way of mapping strings to numbers (like "12345" to 12345. But that's not the only description. For example we could say "the number after 12344" and that would uniquely characterize 12345 just as well. How many descriptions can there be? If we limit our alphabet to a finite or countably infinite set, there are only countably many finite strings. (Standard proof). But now I claim that there are countably many descriptions of any given number. For example the number 5 can be described as, "The number that if you add 1, you get 6." Or "The number that if you add 2, you get 7." You can see that there are countably many of these statements, one for each number you can add. Those are all statements that could be expressed in firstorder arithmetic, using nothing more than the Peano axioms. In other words each description is a legal string in a formal system. Which is what we mean by "description." [I'm using countable to mean countably infinite here]. But there can't be any more than countably many descriptions of a single number, since we already know that there are only countably many descriptions possible at all. By this interpretation, each natural number can be expressed the same number of ways as any other; and this number of ways is countable infinity, $\aleph_0$. Last edited by Maschke; July 31st, 2017 at 07:42 PM.  
July 31st, 2017, 07:48 PM  #10 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,327 Thanks: 2451 Math Focus: Mainly analysis and algebra  I was right in my interpretation. You should read up on the Berry paradox, Definable numbers, and computable numbers (see previous link). Essentially, I suspect that since the phrase "definable in fewer than $n$ characters" is not a welldefined phrase, we find that "the number of ways to describe a number" is not welldefined either  not in every case anyway. My gut feeling is that the number of ways to define a number (whatever that means) is invariant up to cardinality. This is because you can define every number in terms of every other number E.g. a given number $n$ can be written as $n = a + b$ for every $a$. This gives us a multiplicity of descriptions for $n$ that exceeds that for $a$ but this is independent of the magnitude of $a$ and $n$. In particular, if $n \ll a$ we see that the number of descriptions of $n$ is not necessarily smaller than that for $a$. Of course, this is not a proof, nor is it watertight as an argument, not least because of selfreference, but it does indicate that no number should be massively less describable than any other by virtue only of magnitude. Having said that, computability is an important concept here. But equally, the integer part of any noncomputable number is necessarily finite and thus computable (I claim, on no solid basis), and so again the magnitude of a number should not affect computability. 

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