Constructive
Let S(n) be the nth word in the sequence of words from ship to dock
and C(n) denotes the number of vowels in the nth word

speculative
If C(n)‚2 and
If C(n)=1, when you change the letter to obtain S(n+1), you can change a letter which is not a vowel or the letter which is the vowel, if you change the vowel, you must change it to another vowel as no nonvowel words exist; if you change another letter, it must not become a vowel as we have assumed C(S(n)) is not 2. In both cases the word contains one vowel, in the same position. So C(n+1)=1 and C(1)=1 so by induction C(n)=1.

constructive
Since there is only one vowel in S(n)
let P(n) be the position of that vowel in the word

continued speculation
Again when you change the letter to obtain S(n+1), you can change a letter which is not a vowel or the letter which is the vowel, if you change the vowel, you must change it to another vowel as no nonvowel words exist; if you change another letter, it must not become a vowel as we have assumed C(S(n)) is not 2. In both cases the word contains one vowel, in the same position. So P(n+1)=P(n)=P(1)

But P(f)‚P(1);a contradiction

So one of our assumptions is wrong and we assume it isn't consistency (that would be highly circular) or 4 letter vowelless words

resolution
Therefore the assumption C(n)‚2 is false
So C(m)=2 for some m less than f
 
All proofs can be said to alternate between 3 phases: what i shall call axiomatic, contructive and imaginative; maybe 4 for contradiction proofs. As we see the axiomatic stage is the statement of the relevant axioms, the constructive stage is where some properties pertaining to the proof's statement are defined, and the imaginative phase is where the consequences on the properties of a sequence containing no double vowels. And then given a contradiction, the imagining of p was wrong hence not p. In this proof the flow starts, as it must, in the axiomatic phase and proceeds through to the constructive, followed by the speculative and then given a certain consequence, more construction and further speculation and a final statement as required. A good proof, at least aesthetically, depends on the quality of the imaginative phase.

Maybe also you could talk about types of proofs (induction, etc) as well as the flow of a general proof as discussed above more as well, as these are highly relevant topics. But the site's a great idea and generally suceeds in interesting.

Thank you.

>But i think for the purposes of such a massively accessed
>site, your proofs could be perhaps a bit better formatted,

Long, long ago I thought of presenting proofs (as explaining argumentation) in different formats, say, more/less formal or more/less descriptive. This is awfully hard. Wherever possible, I still prefer to give several proofs, but, as they come, all or none may be rigorous. The proofs shift a view point, not necessarily the style.

To rework the existent proofs is unpractical. I however welcome any assitance. Submissions are welcome. The authors preserve the copyright, but give me the right to modify at will and include the material in any distribution. I do not promise to include any submission.

>All proofs can be said to alternate between 3 phases: what i
>shall call axiomatic, contructive and imaginative; maybe 4
>for contradiction proofs. As we see the axiomatic stage is
>the statement of the relevant axioms, the constructive stage
>is where some properties pertaining to the proof's statement
>are defined, and the imaginative phase is where the
>consequences on the properties of a sequence containing no
>double vowels. And then given a contradiction, the
>imagining of p was wrong hence not p. In this proof the
>flow starts, as it must, in the axiomatic phase and proceeds
>through to the constructive, followed by the speculative and
>then given a certain consequence, more construction and
>further speculation and a final statement as required. A
>good proof, at least aesthetically, depends on the quality
>of the imaginative phase.

This is nice and interesting, but I do not believe very universal. As I mentioned before, I welcome thoughtful and relevant articles. The above paragraph, if amplified with several examples, would be a very good introduction into the idea of proof. If you decide to do that, please, try doing that in HTML.

>Maybe also you could talk about types of proofs (induction,
>etc) as well as the flow of a general proof as discussed
>above more as well, as these are highly relevant topics.

This is a good idea, except that a single person can do just so much.

>But the site's a great idea and generally suceeds in
>interesting.

Thank you again.