Your favorite paradoxes

[Eclectic Dreck]

Why do you assume there are only a finite number of words in a language? Languages evolve, which leaves the potential as infinite.

And, before, you said that hyphenated words counted as one word.
So if we say 1,003,201 is one-million-three-thousand-two-hundred-and-one that is a system that can describe every single natural number using only one word that looks very much like English.

Not that that is really relevant as proving the conclusion correct by other means does not prove the argument previously used was valid, and this discussion was about the validity of the argument, and not so much the conclusion.

In all the worlds languages combined at any given moment you have a finite number of words. New words are in fact created and nothing stops the creation of said words to fill our given criteria. I said precisely this in one of the opening posts in this chain.

The problem is, now you're delving deeply into semantics for little reason. At the end of the day, the <a href http://home.earthlink.net/~djbach/paradox.html>general conensus <a href=http://www.math.toronto.edu/mathnet/falseProofs/fallacies_46.html>is that <a href =http://komplexify.com/math/humor_pure/BadProofs.html>what you have given us <a href=http://everything2.com/title/False+mathematical+proofs>is a bad proof based on the reasons I have already outlined, quoted and restated exhaustively.

As such, until I get a proof that demonstrates this claim without relying on misuse of the english language, I cannot nor will I accept the theorm.

 

[Maze1125]

The problem is, now you're delving deeply into semantics for little reason. At the end of the day, the <a href http://home.earthlink.net/~djbach/paradox.html>general conensus <a href=http://www.math.toronto.edu/mathnet/falseProofs/fallacies_46.html>is that <a href =http://komplexify.com/math/humor_pure/BadProofs.html>what you have given us <a href=http://everything2.com/title/False+mathematical+proofs>is a bad proof based on the reasons I have already outlined, quoted and restated exhaustively.

As someone who claims to have had a formal logic education I'm surprised you commit an Appeal to Authority fallacy combined with and Appeal to Popularity. Especially as the two of those links that work provide the same point I've already refuted.

The claim "The smallest natural number that cannot be unambiguously described in fourteen words or less." is not logically inconsistent. You only reach a contradiction if you take it to be true. It can be false with no problems. Logic does not conclude such statements are inconsistent, it concludes they are false.

As such, until I get a proof that demonstrates this claim without relying on misuse of the english language,

Now your adding conditions that were not in the original problem.

You're also refusing to back up a lot of your points, such as:

"you can construct any number of examples where your criteria cannot possibly be true and then apply any of a number of tests designed to reach an arbitrary conclusion."

 

[Eclectic Dreck]

As someone who claims to have had a formal logic education I'm surprised you commit an Appeal to Authority fallacy combined with and Appeal to Popularity. Especially as the two of those links that work provide the same point I've already refuted.

The claim "The smallest natural number that cannot be unambiguously described in fourteen words or less." is not logically inconsistent. You only reach a contradiction if you take it to be true. It can be false with no problems. Logic does not conclude such statements are inconsistent, it concludes they are false.

As such, until I get a proof that demonstrates this claim without relying on misuse of the english language,

Now your adding conditions that were not in the original problem.

You're also refusing to back up a lot of your points, such as:

"you can construct any number of examples where your criteria cannot possibly be true and then apply any of a number of tests designed to reach an arbitrary conclusion."

And I'm equally surprised that you continue to insist a fallcy born from poor use of english is anything but. As I stated previously: until you can demonstrate a proof that does not require the use of such a fallacy to work, I will not accept your theorm. Nor, it would appear, does any other source out of a dozen that I checked.

I have demonstrated my end of the argument to the length that it can possibly be demonstrated. So long as you continue to insist that your PROOF does not in and of itself rely on a FALLACY and I continue to insist that it DOES, we can not possibly make any headway.

And, for the final time, your PROOF is FALLACIOUS because your CONDITION states, in effect that "any value that this condition may apply to cannot possibly meet this condition". Such a statement is self referential and self contradictary. This, in spite of what you seem to want to demonstrate, is a logical fallacy born from the MISUSE of english.

In order for your proof to hold, there can be no such fallacy at any point of the process. The proof process you are using is a common one that can indeed be quite rigorous.

What you seem to misunderstand here is not that I am unwilling to believe that there is an logical argument to the effect that an infinite set of objects can be unambiguously described by a finite set of descriptions. What I am stating is that your proof neither demonstrates this property nor proves it's existance.

In short, let's examine the facts:

If the language being used as the source for the words used in our unambiguous description contains a finite number of words, then your proof cannot possibly be true. A finite set of options to a finite power will yield a finite number, and even elementary math students ought to understand that any finite number is less than infinity.

If the language being used as the source for words in our unambiguous description containes an infinite number of words , then the proof can be demonstratably true precisely because you have an infinite set of options to apply to your infinite set.

However, even when leveraging the hypenated word trick, you'll find that your total word count only goes up to 28 (a single hypenated word, such as rat-faced, counts as one word, where rat-faced-jerk-bag counts as four). While any language adds words over time, the number of words added over a given interval have proven finite, as every language at a particular moment as a finite number of accepted words. One could argue that, given infinite time a language would concievably spawn infinite words, but one must begin to wonder in what spirit is the question asked?

Certainly one can randomly generate words in order to demonstrate that any number can be described unambigiously in fourteen words or less, but then there would be no need for a proof. The statement of "Given an infinite set of words, one can unamiguously describe any number using a phrase or arbitray lenght" would suffice. The reason is simple. Infinity = Infinity. Infinity/finite number = infinity. Infiniity +/- finite number = infinity. Infinity * infinity = infinity and so forth. There are few mathematical operations dealing with infinity that do not yield infinity in return. If we, for example had an infinite number set and an infinite word set and were asked to prove that you could unambiguously apply a name of four words in length, all you need to demonstrate is that name set >= number set.

Thus:

4^infinity >= infinity
infinity >= infinity

BUT: none of these scenarios are described in the problem. As I said, there are theoritical scenarios that can be generated that allow your thesis to be true. But, since these theoritical scenaios do not exist in reality and your proof itself suffers from a logical fallacy the only conclusion I can draw is that your theorm is, in fact, wrong.

As I have said countless times (not exactly countless but I'm certainly not going to do the counting), I will accept a proof even if it is of the same style so long as the proof does not rely on a fallacy to be correct.

 

[101194]

I dare you to click my name

This guys quote is hilarious.

 

[Maze1125]

And, for the final time, your PROOF is FALLACIOUS because your CONDITION states, in effect that "any value that this condition may apply to cannot possibly meet this condition". Such a statement is self referential and self contradictary. This, in spite of what you seem to want to demonstrate, is a logical fallacy born from the MISUSE of english.

Yes, you've said that lots and lots of times.
Yet, despite claiming there are loads of examples where such a "fallacy" gives a false result, you have yet to provide a single example that has the same "fallacy" and gives a false result.

Another thing I find hilarious is how you claim to be a student of formal logic, yet you fall pray to so many fallacies yourself. Such as appealing to authority or moving the goalposts, or how being unwilling to accept a conclusion leads you to decide the conclusion is false.

But hey, let me help you out, because you're constantly insisting that the words in the language must be finite despite the fact the proof in no way calls for that and with no real justification why a proof that doesn't call for a finite language ought to work with one, all you've given is hand-waving about the spirit of the problem, which is hardly mathematically rigorous.

It ought to work with a finite language, because changing the sentence "The smallest natural number that cannot be unambiguously described in fourteen words or less." to "The smallest natural number that cannot be unambiguously described in twenty two words that exist in the Oxford English dictionary or less." or even "The smallest natural number that cannot be unambiguously described in nineteen words that exist in this sentence or less." does not change the proof in any way.

Also, the number of words allowed to describe the numbers is fairly arbitrary too, as more concise languages may be able to accurately convey the sentence in only 5 or 6 words, or maybe even less.

So it may be possible to find a language where to proof can still be made yet the potential number of different sentences of the appropriate length is only 6^6. Yet I claim it is still possible for each natural number to be unambiguously described with that few options. This is clearly inane by any common-sense. But maths has rarely followed common-sense.

If you read carefully, understand the proof, and don't presume anything, the answer is actually quite simple.
The conclusion states that "Any natural number can be unambiguously described using fourteen words or less."
That is not the same things as saying "There exists a system of labelling numbers where every natural number can be unambiguously described in fourteen words or less with-in that system."

So, with that in mind, I will show the following theorem is true:
"Any natural number can be unambiguously described using only one word with-in a language that has only one word."

Let the single word in the language be "noo" w.l.o.g.
Let S_n be a system of labelling natural numbers, such that 0 is labelled "noo noo" 1 is labelled "noo noo noo" and the number m is labelled using m+2 noos. The only exception to this is n, which will be labelled with a single "noo".

Now, if I want to describe 1 with a single word, I will pick the system S_1, where 1 is unambiguously described with a single "noo".
Equally, if I want to describe 12950328 with a single word, I will pick the system S_12950328 where 12950328 is unambiguously described with a single "noo".

Therefore, as required, I have shown how every single natural number can be unambiguously described using a single word with-in a language with only a single word.

Of course, this may seem like cheating, but in truth an understanding of the processes of the proof clearly indicates that this is actually what the conclusion means, because if you add the constraint that it is done with-in a single system, the proof doesn't work.

 

[Eclectic Dreck]

Yes, you've said that lots and lots of times.
Yet, despite claiming there are loads of examples where such a "fallacy" gives a false result, you have yet to provide a single example that has the same "fallacy" and gives a false result.

And you have yet to disprove my assertion that you have a fallacy. Your argument is circular and my argument is circular. The reason is simple: we disagree about the facts of the argument. You believe you proof is sound. I believe your proof is not.

Another thing I find hilarious is how you claim to be a student of formal logic, yet you fall pray to so many fallacies yourself. Such as appealing to authority or moving the goalposts, or how being unwilling to accept a conclusion leads you to decide the conclusion is false.

There is no moving of the goal posts happening. Follow me a moment:

Theorm: all numbers can be unambiguously described using an arbitrary number of words

If your proof is TRUE, then it follows that

There is a system that can allow us to unambiguously describe this set of numbers.

The second part MUST be true if the first part is true. If we assert that it is possible (the entire point of your proof) then it follows that it MUST be possible.

But hey, let me help you out, because you're constantly insisting that the words in the language must be finite despite the fact the proof in no way calls for that and with no real justification why a proof that doesn't call for a finite language ought to work with one, all you've given is hand-waving about the spirit of the problem, which is hardly mathematically rigorous.

The words in a language ARE finite at any given moment. I am not speaking strictly of the contents of a dictionary. At any instant, the sum total words in a language are finite but increasing at a variable rate. Given infinite time, it stands to reason that the total words approach infinity. This proof, however, does not ever state infinity in it's presmise. If it did, there is no need of proof.

The reason such a proof is commonly demonstrated (and ever commonly cited as being fallacious) is because it demonstrates something that runs counter to logic. As I have stated time and again, the proof is essentially stating that you will have a finite number divided by a number that approaches infinity. if you take look at the function f(x) = 1/x and take the limit as x - infinity do you know what you arrive at? If you guessed zero, you'd be correct.

Now, the implication of such a function is simple: you have proven that any finite number divided by an arbitrarily large number that approaches infinity approaches zero. Or, to put it another way, if you use finite words, no matter how large your word set is at a given instant, you can only assign unique (or, as the proof calls it, unambiguous) names to precisely 0% of your infinite number set.

It ought to work with a finite language, because changing the sentence "The smallest natural number that cannot be unambiguously described in fourteen words or less." to "The smallest natural number that cannot be unambiguously described in twenty two words that exist in the Oxford English dictionary or less." or even "The smallest natural number that cannot be unambiguously described in nineteen words that exist in this sentence or less." does not change the proof in any way.

You're indeed correct - the precise number is irrelevent in your proof. My qualm is not with the number fourteen, it's with the claim that a finite number can be divided by infinity and yield anything other than zero.

Also, the number of words allowed to describe the numbers is fairly arbitrary too, as more concise languages may be able to accurately convey the sentence in only 5 or 6 words, or maybe even less.

I can agree with your assertion thus far.

So it may be possible to find a language where to proof can still be made yet the potential number of different sentences of the appropriate length is only 6^6. Yet I claim it is still possible for each natural number to be unambiguously described with that few options. This is clearly inane by any common-sense. But maths has rarely followed common-sense.

Let's bring your argument into the realm of the finite. We know we are dealing with an infinite set, so substitue that for a number - say 1,000

Then say we have a langage of words that represent a smaller set - let's say 10 words. If you have 3 word limit for description can you unambiguously name all the words?

The answer is, in fact YES (10^3 = 1000) you have precisely enough words and spaces to assign a unique name to every single member of our finite set.

But, what if our number set becomes 1001? Suddenly you run into a problem - you have run out of unique word combinations to use. From where I stand, if to continue this process my set of word options or my word length needs to increase if I want to continue unambiguously naming numbers.

Interestinly, this is precisely the process used in the number naming process. Each digit in each decimil place has a unique name. Thus, for a number of nine digits long, it may take as may as nine words to describe. For example, 999,999,999
Is nine-hundred ninenty-nine million, nine-hundred ninety-nine thousand, nine-hundred and ninety-nine. This isn't the most efficent way perhaps in terms of numbers of words but it is incredibly efficient in terms of the total numbers of words required to describe a number of given length.

If you read carefully, understand the proof, and don't presume anything, the answer is actually quite simple.
The conclusion states that "Any natural number can be unambiguously described using fourteen words or less."
That is not the same things as saying "There exists a system of labelling numbers where every natural number can be unambiguously described in fourteen words or less with-in that system."

If the your infintite set can be described using the given conditions then there MUST be a system in which one can generate the responses. What you are essentially saying is "While a unique name certainly exists for an infinite set using my conditions, it is not necessarily possible to generate this name". At this point, you have fallen deeply into paradox. Such scenarios do exist within mathematics and it could be that a proof with rigor is rooted somewhere in the realm of trans-finite numbers and other areas of wierdness.

However, while it appears to be mathematically impossible to generate a set of names from a finite set of words used in a phrase of finite length, I have given you a perfect way to resolve this dilemma: use an infinite set of words.

So, with that in mind, I will show the following theorem is true:
"Any natural number can be unambiguously described using only one word with-in a language that has only one word."

Let the single word in the language be "noo" w.l.o.g.
Let S_n be a system of labelling natural numbers, such that 0 is labelled "noo noo" 1 is labelled "noo noo noo" and the number m is labelled using m+2 noos. The only exception to this is n, which will be labelled with a single "noo".

Now, if I want to describe 1 with a single word, I will pick the system S_1, where 1 is unambiguously described with a single "noo".
Equally, if I want to describe 12950328 with a single word, I will pick the system S_12950328 where 12950328 is unambiguously described with a single "noo".

Therefore, as required, I have shown how every single natural number can be unambiguously described using a single word with-in a language with only a single word.

Of course, this may seem like cheating, but in truth an understanding of the processes of the proof clearly indicates that this is actually what the conclusion means, because if you add the constraint that it is done with-in a single system, the proof doesn't work.

This proof demonstrates that your origian proof CAN be true if and ONLY if you have an infinite set of words.

Numbers are represented in two seperate ways as you are undoubtedly aware: they can be a symbol, such a 1, or they can be words such as one. In this case, you have attached a unique symbol to a word. This symbol changes the meaning of the word. What you have generated as a result is a new word. This is much the same as adding a suffix to any normal word. The word normal for example has a different (but related) meaning to the word normally yet they are still counted as distinctly different. One cannot correctly use the word "normally" in place of the word "normal" because doing so changes the meaning of the sentence.

In your very specific case, you resolved the problem not by using a single set of infinite words but rather an infinite set of finite words. What you have essentially done is proven that I can describe the number 1,000,0000 as 1,000,000. A number is, afterall, nothing more than a string of symbols that impart meaning both individually and as a whole.

That said, this discussion has taken up an immense amount of our respective time. Thus, I am willing to conceed the argument on the following grounds:

Your second proof demonstrates that given a proper word set, one can generate an unambiguous descricription of arbitrary length for each member of an infinite set. This proper word set must be infinite.

Your original proof implies that a finite set can unambiguously refer to an infinite set, a clear violation of mathematical law. Any finite set divided across infinity yields zero, or rather yields a number that approaches zero.

That said, I still don't LIKE the wording of your proof, but the second one at least holds up to it's own implications.

EDIT: A bit of research actually supports one of your assertions I had discarded out of hand: that is, by technical definition, a regular language has infinite words. However, only a finite number of these words have associated meanings at any given moment. Thus, if you want to assign words with known meanings at the moment you are dealing with a finite set (rendering your initial assertion impossible). If you, however do not feel the need to restrict yourself and instead generate new words and assign them a meaning (as you do in your second proof) then you have an infinte set. Infinity/Infinity is undefined as it can literally yield any value. While this value could be zero, it could also be any non-zero number such as 1, which implies that we can assign precisely 1 (or none or more) members from one infinite set to the other.