I think the interesting question is what does the function on natural 

      n → shortestDescriptionOf(n)/n

look like.

- pt

On Friday, March 1, 2019 at 3:22:59 PM UTC-6, Lawrence Crowell wrote:
> There are numbers that have no description in a practical sense. The 
> numbers 10^{10^{10^{10}}} and 10^{10^{10^{10^{10}}}} have a vast number of 
> numbers that have no description with any information theoretic sense. This 
> is even if all the particles in the some 10^{500}cosmologies in the 
> multiverse or with 12-dimensions some 10^{10100} cosmos's were employed as 
> information bits. It is also not hard to construct a Berry paradox for a 
> vast number of numbers here. 
> LC
> On Monday, February 25, 2019 at 1:35:49 PM UTC-6, Philip Thrift wrote:
>> via
>> https://twitter.com/JDHamkins/status/1100090709527408640
>> Joel David Hamkins   @JDHamkins
>> *Must there be numbers we cannot describe or define? Definability in 
>> mathematics and the Math Tea argument*
>> Pure Mathematics Research Seminar at the University of East Anglia in 
>> Norwich on Monday, 25 February, 2019.
>> Abstract:
>> *An old argument, heard perhaps at a good math tea, proceeds: “there must 
>> be some real numbers that we can neither describe nor define, since there 
>> are uncountably many real numbers, but only countably many definitions.” 
>> Does it withstand scrutiny? In this talk, I will discuss the phenomenon of 
>> pointwise definable structures in mathematics, structures in which every 
>> object has a property that only it exhibits. A mathematical structure is 
>> Leibnizian, in contrast, if any pair of distinct objects in it exhibit 
>> different properties. Is there a Leibnizian structure with no definable 
>> elements? Must indiscernible elements in a mathematical structure be 
>> automorphic images of one another? We shall discuss many elementary yet 
>> interesting examples, eventually working up to the proof that every 
>> countable model of set theory has a pointwise definable extension, in which 
>> every mathematical object is definable.*
>> http://jdh.hamkins.org/must-there-be-number-we-cannot-define-norwich-february-2019/
>> Lecture notes:
>> http://jdh.hamkins.org/wp-content/uploads/2019/02/Must-every-number-be-definable_-Norwich-Feb-2019.pdf
>> - pt

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