Mathematical proof that there are a finite number of natural numbers

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.

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

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.

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

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).

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

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.

The contradiction is the statement itself, which claims to describe a numbe r which was defined as being not describable. The pseudo proof fails becaus e you have not shown how those N words describe the finite set of natural n umbers 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 natur al numbers.

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 n! which is for N==20:

make the set size unbounded (countably many).

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 des cribable in 20 words or less through permutations of the N words in the dic tionary." So first it is and then it isn't and not because of anything havi ng 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), snipped-for-privacy@gmail.com Gave us:

snip

Start a new organization...

N words matter.

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

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

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

What D? There's no D in the original post.

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