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.
Mathematical proof that there are a finite number of natural numbers
Started by ●September 4, 2015
Reply by ●September 4, 20152015-09-04
In article <55e9dd77$0$32363$4c5ecfc7@frugalusenet.com>, bitrex <bitrex@de.lete.earthlink.net> 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.If N is the largest natural number that can be described in your 20 words, then N+1 can be expressed in 22 words, the additional words being "plus one". Since that final word need not be "one", but the representation of any natural number, there is no upper bound on natural numbers.
Reply by ●September 4, 20152015-09-04
On 09/04/2015 02:05 PM, 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.That's sort of a variant of Russell's paradox. You have to know how big L is, in order to sort it out from all the other indescribable rationals, but you have to be able to describe it to do that. It's an ill-posed problem, because no 1:1 correspondence is proposed between sentences and rationals--I can arbitrarily move names to different numbers, which means that the number L does not exist. Cheers Phil Hobbs -- Dr Philip C D Hobbs Principal Consultant ElectroOptical Innovations LLC Optics, Electro-optics, Photonics, Analog Electronics 160 North State Road #203 Briarcliff Manor NY 10510 hobbs at electrooptical dot net http://electrooptical.net
Reply by ●September 4, 20152015-09-04
On Friday, September 4, 2015 at 2:05:47 PM UTC-4, 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.I don't see any contradiction. You've just added one more to the list of numbers describable by 20 words or fewer. There still remains natural number L+1, ...etc. (But then I'm not a math guy either.) George H.
Reply by ●September 4, 20152015-09-04
On 04/09/2015 20:11, Phil Hobbs wrote:> On 09/04/2015 02:05 PM, 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.You immediately have a problem here in your initial assumptions because the prefix "one plus" or suffix "plus one" generates a successor. Mathematicians working with transfinite numbers worry about the order of the operands but for the purposes of this basic argument it suffices that there is always another next number available. See https://en.wikipedia.org/wiki/Transfinite_number>> >> 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. > > That's sort of a variant of Russell's paradox. You have to know how big > L is, in order to sort it out from all the other indescribable > rationals, but you have to be able to describe it to do that. > > It's an ill-posed problem, because no 1:1 correspondence is proposed > between sentences and rationals--I can arbitrarily move names to > different numbers, which means that the number L does not exist.It is a variant on the reasoning used to "prove" for example that Alexander the Great's horse had an infinite number of legs. (see Random Walk in Science for details). -- Regards, Martin Brown
Reply by ●September 4, 20152015-09-04
On 9/4/2015 3:11 PM, Phil Hobbs wrote:> On 09/04/2015 02:05 PM, 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. > > That's sort of a variant of Russell's paradox. You have to know how big > L is, in order to sort it out from all the other indescribable > rationals, but you have to be able to describe it to do that. > > It's an ill-posed problem, because no 1:1 correspondence is proposed > between sentences and rationals--I can arbitrarily move names to > different numbers, which means that the number L does not exist. > > Cheers > > Phil Hobbs >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...
Reply by ●September 4, 20152015-09-04
On 9/4/2015 5:34 PM, bitrex wrote:> On 9/4/2015 3:11 PM, Phil Hobbs wrote: >> On 09/04/2015 02:05 PM, 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. >> >> That's sort of a variant of Russell's paradox. You have to know how big >> L is, in order to sort it out from all the other indescribable >> rationals, but you have to be able to describe it to do that. >> >> It's an ill-posed problem, because no 1:1 correspondence is proposed >> between sentences and rationals--I can arbitrarily move names to >> different numbers, which means that the number L does not exist. >> >> Cheers >> >> Phil Hobbs >> > > 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...You forgot expletive and invective. ;) Cheers Phil Hobbs -- Dr Philip C D Hobbs Principal Consultant ElectroOptical Innovations LLC Optics, Electro-optics, Photonics, Analog Electronics 160 North State Road #203 Briarcliff Manor NY 10510 hobbs at electrooptical dot net http://electrooptical.net
Reply by ●September 5, 20152015-09-05
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.
Reply by ●September 5, 20152015-09-05
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.
Reply by ●September 5, 20152015-09-05
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.
