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".

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 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 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 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 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 Friday, September 4, 2015 at 2:05:47 PM UTC-4, bitrex wrote:

>  "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 contradiction is the statement itself, which claims to describe a number which was defined as being not describable. The pseudo proof fails because you have not shown how those N words describe the finite set of natural numbers that are known. All this is aside from the obvious observation that whatever finite set you settle on as being the totality of natural numbers, it can always be enlarged, meaning there is no largest finite set of natural numbers.

On 9/5/2015 12:41 AM, Johann Klammer wrote:
> On 09/04/2015 11:34 PM, bitrex wrote:
>> On 9/4/2015 3:11 PM, Phil Hobbs wrote:
>>
>> Indeed, the objection I had when I first encountered this "proof" was
>> one of aliasing - say the number of natural numbers uniquely
>> describable by all the unique permutations of the N words of the
>> dictionary is X, then if one uses the phrase "The least such number
>> not describable in 20 words or less through permutations of the N
>> words in the dictionary" to describe X+1 it means one no longer has
>> uniqueness as to have X natural numbers you must have already used
>> that phrase.  Or something.
>>
>> By assuming that the natural numbers are finite you've tacitly
>> assumed that there must be a bijection between them and the
>> permutations of words in the dictionary, but the contradiction step
>> takes a step where this mapping is actually surjective but not
>> injective.
>>
>> To have a contradiction one must show that P and not(P) is true where
>> P is a result which follows from your initial assumption, but P must
>> be logically consistent within the framework of the problem or else
>> the proof doesn't work.
>>
>> At least that's how I see it...
>
> It is /not/ a permutation. you can have unlimited number of repetitions in there.
>
> eg: "one hundred and twenty one"
>
> has "one" in there twice.
> The `proof' is completely #bull.
>

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.

On 09/04/2015 11:34 PM, bitrex wrote:
> On 9/4/2015 3:11 PM, Phil Hobbs wrote:
> 
> Indeed, the objection I had when I first encountered this "proof" was
> one of aliasing - say the number of natural numbers uniquely
> describable by all the unique permutations of the N words of the
> dictionary is X, then if one uses the phrase "The least such number
> not describable in 20 words or less through permutations of the N
> words in the dictionary" to describe X+1 it means one no longer has
> uniqueness as to have X natural numbers you must have already used
> that phrase.  Or something.
> 
> By assuming that the natural numbers are finite you've tacitly
> assumed that there must be a bijection between them and the
> permutations of words in the dictionary, but the contradiction step
> takes a step where this mapping is actually surjective but not
> injective.
> 
> To have a contradiction one must show that P and not(P) is true where
> P is a result which follows from your initial assumption, but P must
> be logically consistent within the framework of the problem or else
> the proof doesn't work.
> 
> At least that's how I see it...

It is /not/ a permutation. you can have unlimited number of repetitions in there. 

eg: "one hundred and twenty one"

has "one" in there twice. 
The `proof' is completely #bull.