Mathematical proof that there are a finite number of natural numbers

On 09/05/2015 02:46 PM, bitrex wrote:
> 
> Whether something is a permutation or not doesn't have anything to
> do with how many repetitions there are, only on whether the order of
> the items in question matters.  "one hundred and twenty one" is
> certainly distinct from "twenty one hundred and one" - order matters
> here.

You posted:

By way of contradiction, suppose that the natural n
umbers are indeed unbounded and there is always some natural number 
which cannot be described by permutations of the N words in the dictionary
 consisting of 20 words or fewer.  Call the least such number of this 
type L.  However, one may then describe this least number L by the 
designation "The least such number not describable in 20 words or less
 through permutations of the N words in the dictionary."  This is a 
contradiction - hence the natural numbers are indeed finite. 


And the: `cannot be described by permutations of the N words'  
implies that there are no repetitions, as possibility of repetition would make the set size unbounded (countably many). 

A permutation of N words however is always of size <looks it up in wikipedia>
n! which is for N==20: 
2.43290200818e+18
(says galculator)

<https://en.wikipedia.org/wiki/Permutation>

On Saturday, September 5, 2015 at 2:56:37 PM UTC-4, Johann Klammer wrote:
> On 09/05/2015 02:46 PM, bitrex wrote:
> > 
> > Whether something is a permutation or not doesn't have anything to
> > do with how many repetitions there are, only on whether the order of
> > the items in question matters.  "one hundred and twenty one" is
> > certainly distinct from "twenty one hundred and one" - order matters
> > here.
> 
> You posted:
> 
> By way of contradiction, suppose that the natural n
> umbers are indeed unbounded and there is always some natural number 
> which cannot be described by permutations of the N words in the dictionary
>  consisting of 20 words or fewer.  Call the least such number of this 
> type L.  However, one may then describe this least number L by the 
> designation "The least such number not describable in 20 words or less
>  through permutations of the N words in the dictionary."  This is a 
> contradiction - hence the natural numbers are indeed finite. 
> 
> 
> And the: `cannot be described by permutations of the N words'  
> implies that there are no repetitions, as possibility of repetition would make the set size unbounded (countably many). 
> 
> A permutation of N words however is always of size <looks it up in wikipedia>
> n! which is for N==20: 
> 2.43290200818e+18
> (says galculator)
> 
> <https://en.wikipedia.org/wiki/Permutation>

That's not as dumb as hypothesizing "there is always some natural number 
which cannot be described by permutations of the N words in the dictionary" and then turns around and describes it with "The least such number not describable in 20 words or less through permutations of the N words in the dictionary." So first it is and then it isn't and not because of anything having to do with the assumption of infinity. A community college level course on proofs could do better than that.

On Sat, 5 Sep 2015 12:55:50 -0700 (PDT),
bloggs.fredbloggs.fred@gmail.com Gave us:

snip
>
>That's not as dumb as hypothesizing "there is always some natural number 
>which cannot be described by permutations of the N words in the dictionary"
> and then turns around and describes it with "The least such number not
> describable in 20 words or less through permutations of the N words in the
> dictionary." So first it is and then it isn't and not because of anything having
> to do with the assumption of infinity. A community college level course on
> proofs could do better than that.

  Start a new organization...

  N words matter.

  Populated by 100% gNappy headed bros and their gNappy head giving hos.

On 05/09/2015 19:56, Johann Klammer wrote:
> On 09/05/2015 02:46 PM, bitrex wrote:
>>
>> Whether something is a permutation or not doesn't have anything to
>> do with how many repetitions there are, only on whether the order of
>> the items in question matters.  "one hundred and twenty one" is
>> certainly distinct from "twenty one hundred and one" - order matters
>> here.
>
> You posted:
>
> By way of contradiction, suppose that the natural n
> umbers are indeed unbounded and there is always some natural number
> which cannot be described by permutations of the N words in the dictionary
>   consisting of 20 words or fewer.  Call the least such number of this
> type L.  However, one may then describe this least number L by the
> designation "The least such number not describable in 20 words or less
>   through permutations of the N words in the dictionary."  This is a
> contradiction - hence the natural numbers are indeed finite.

The problem arises from the assertion that all natural numbers can be 
described by at most twenty words taken from a standard dictionary.

> And the: `cannot be described by permutations of the N words'
> implies that there are no repetitions, as possibility of repetition would make the set size unbounded (countably many).
>
> A permutation of N words however is always of size <looks it up in wikipedia>
> n! which is for N==20:
> 2.43290200818e+18
> (says galculator)
>
> <https://en.wikipedia.org/wiki/Permutation>

Except that it is perm N=20 from the entire dictionary of size D (~10^8)

D!/(D-N)! ~ D^N

Which is of the order of 10^160 but still tiny compared to the first 
countable infinity, Aleph-0. See

http://mathworld.wolfram.com/CountablyInfinite.html

-- 
Regards,
Martin Brown

On 09/06/2015 10:02 AM, Martin Brown wrote:
> On 05/09/2015 19:56, Johann Klammer wrote:
> 
> Except that it is perm N=20 from the entire dictionary of size D (~10^8)
> 
What D? There's no D in the original post.

> D!/(D-N)! ~ D^N
> 
> Which is of the order of 10^160 but still tiny compared to the first countable infinity, Aleph-0. See
> 
> http://mathworld.wolfram.com/CountablyInfinite.html
> 

On 04/09/15 20:05, bitrex wrote:
> Suppose that there are finitely many natural number for some maximum
> natural number M.  Since there finitely many numbers, this finite amount
> M of numbers may be described by sentences in the English dictionary,
> composed of permutations of the N words in the dictionary, containing
> say, 20 words or fewer.
> 
> By way of contradiction, suppose that the natural numbers are indeed
> unbounded and there is always some natural number which cannot be
> described by permutations of the N words in the dictionary consisting of
> 20 words or fewer.  Call the least such number of this type L.  However,
> one may then describe this least number L by the designation "The least
> such number not describable in 20 words or less through permutations of
> the N words in the dictionary."  This is a contradiction - hence the
> natural numbers are indeed finite.

All you are doing here is showing that you can construct sentences that
are neither true nor false - Russell's paradox.  The simplest version
AFAIK is "This statement is false".

>All you are doing here is showing that you can construct sentences that 
>are neither true nor false - Russell's paradox. �The simplest version 
>AFAIK is "This statement is false". 

If Russell only knew...he could have invented the crystal oscillator. ;)

Cheers

Phil Hobbs