Your favorite paradoxes
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You are restating my point and calling me wrong. The phrase "the smallest not namable under ten words" describes precisely one number. Once that number has been dealt with there is nowa second number that meets the criteria and it now is the smallest. If this second number is now referred to as the smallest number, you'll find that your original answer is now the smallest.Maze1125 said:
Let's just have a quick look at our string of words being used as a name
It starts out "The smallest number". By starting with the finite article we know that this sentence refers to a single object or concept, in this case a number. The adjective of smallest lets us know what essentiall quality this number is going to posess. The closing portin "not nameable in under ten words" gives the remainder of our standard. Built into this sentence is the fact that we cannot use this string to refer to more than a single value.
Thus, the assertion that the prhase "the smallest number not namable in under ten words" can be used to shorten any number to ten words is not a paradox. The problem can be resolved through simple logic and demonstrated to be false. It's structure can refer to ony one value and yet it's expected to refer to an inifinite number. If this fact is ignored, the name itself has no meaning precisely because it refers to an infinite set of numbers.
Faulty logic doesn't mean you have a paradox. A true paradox doesn't need such a crutch as it defies all logical attempts to resolve it.
That's where you're misunderstanding.Eclectic Dreck said:
We aren't trying to apply the phrase to an infinite number of values, we're attempting to apply to each value in turn and finding that it fails to apply to any value.
Here's a more formal proof using strong induction.
Every Natural Number can be Unambiguously Described in Fourteen Words or Less:
Base case, 0:
Zero is one word which is less than 14 words.
Inductive step:
Assume every number up to and including n can be unambiguously described in 14 words or less.
Look at n+1, either n+1 can be unambiguously described in 14 words or less or it can't be.
If it can't be then, as every previous natural number can be, this is the smallest natural number that can't be, therefore the phrase "The smallest natural number that cannot be unambiguously described in fourteen words or less." unambiguously describes n+1 in fourteen words. This is a contradiction.
Hence, as the other option results in contradiction, n+1 must be able to be unambiguously described in fourteen words or less.
Therefore, by strong induction, every natural number can be unambiguously described in 14 words or less.
Less an actual paradox than an amusing observation: All human beings have one thing in common (species aside): We are all individuals.
For a proof to apply, it must apply to ALL possible outcomes. The addition of the word "unambiguously" is the doom of this proof. As you go through the numbers in turn, yes you find that each number does in fact meet the qualifications and can be correctly described using the given phrase. But, just because this phrase applies to numbers taken in turn does not imply that the proof holds. It must likewise apply to all numbers in an infinite series to which the term may apply. The phrase can obviously only apply to one number at a time - thus why it works when taken in series. But since you have demonstrated that it applies to ALL numbers when taken in a series we find the problem.Maze1125 said:
Essentially, the problem is thus:
Value n is the first instance of a natural number that cannot be described in foureen words or less. Thus it is now described as "the smallest number that cannot be unambiguously described in 14 words or less".
Value n+1 is the second instance of a natural number that cannot be described in fourteen words or less. Because the value n currently has teh description "the smallest natural number that cannot be unambiguously described in 14 words or less", a description containing fewer than 14 words" we find that n+1 is now the smallest value. As such, it becomes worthy of holding the phrase.
Value n + a is an arbitrarily instance of a natural number that cannot be described in fourteen words or less. Because the value of n + (a-1) currently has the description "the smallest natural number that cannot be unambiguously described in 14 words or less", the value n + n now holds the title.
Having demonstrated that the concept works in sequence, the set n >= n + a must be examined as a whole. During the process we have found an arbitrary number of values that have assumed the title "the smallest natural number that cannot be unambiguously described in fourteen words or less". Because the title must, by it's own definition, only refer to a single value, the proof that works when taken at a sequence fails when applied to the set as a whole. By referring to all values in turn from n to n + a, you find that it cannot apply to anything because to do so introduces ambiguity.
The flaw is a common one in inductive proof - because the proof works in precisely one capacity it is assumed to work in all capacities. This is precisely why an inductive proof is considered less rigorous than an decuctive one.
Since cats allways land on there feet.
1.Get two cats
2.Tie them together (at the feet)
3. and throw them up in the air.
Logically there never going to hit the ground.
There just going to spin around just 6 inches above the ground.
1.Get two cats
2.Tie them together (at the feet)
3. and throw them up in the air.
Logically there never going to hit the ground.
There just going to spin around just 6 inches above the ground.
"You're unique. Just like everyone else."
I find the best paradoxes are the ones that have practical application... like...
You need the experience to get the job, but you need the job to get the experience...
Or, you eat because you're unhappy, and you're unhappy because you eat...
I find the best paradoxes are the ones that have practical application... like...
You need the experience to get the job, but you need the job to get the experience...
Or, you eat because you're unhappy, and you're unhappy because you eat...
That is not what we are doing.Eclectic Dreck said:
What we are doing is attempting to apply the phrase "The smallest natural number that cannot be unambiguously described in fourteen words or less." and showing this results in a contradiction and thereby concluding that the phrase "The smallest natural number that cannot be unambiguously described in fourteen words or less." cannot apply to that number.
We are never ever ever trying to apply the phrase to more than one number at once, so the fact that it cannot be applied to more then one number at once is never a problem.
Please try and read over the proof again, because as far as I can tell you're so set on the idea that I'm wrong, and the specific reason that you think I've got wrong, that you're not actually reading what I'm saying.
Yes, it's quite possible there is a flaw, but that is not it.
That's not true at all. In fact, the proof that induction works on the natural numbers is a deductive one.
My version of that is this:coxafloppin said:
Cat lands on feet
Toast lands butter side up
1. Strap toast on cat, butter facing upwards
2. Drop off cliff
3. Perpetual Motion Engine created
4. Use to fuel the world forever
So, if I'm to understand the proof given a second (more careful) reading, it states in effect:Maze1125 said:
Value n is the LARGEST natural number that CAN be described, unambiguously in 14 worsd or less.
Value n + 1 is the first (and smallest) natural number that does NOT meet the criteria of being a natural number unambigiously described in 14 words or less.
This value, n + 1 can however be described as "the smallest. . . resulting in a contradiction as the description now applied is shorter than 14 words.
As this logic can hold true for all values of n where the given criteria is met it can be concluded that all natural numbers MUST have a description shorter than 14 words that is non-ambiguous.
Assuming THIS is indeed correct, the proof still demonstrates a logical fallacy.
We are given a condition with two concievable outcomes - true or false. This inductive proof essentially states because the condition is true, it is false; therefore the condition is false for all false.
The logical flaw is not hidden the the mathematical process of the proof itself but rather than the LANGUAGE of the statement. The phrase "the smallest natural number that cannot be unambiguously described in fourteen words or less" is a sentence that makes a statement about itself. Any time a sentence does this it runs the risk of being logically inconsistant.
In effect, the sentence refers to itself because the sentence makes a claim about the description of a number while itself being a description of a number.
The logical flaw is simply this: when you try to apply any value to this sentence, the sentance states in effect that it does not apply to this value. Therefore, the sentence cannot possibly be self-consistently asserted about any number.
The flaw in your proof lies here:
Because your descriptive sentence is not consistant with respect to itself, it is assumed that the mathematical nature of n is in fact in contradiction. This is very similar to the "Liar's paradox" of "this sentence is false". The sentence itself is a paradox because it is self referential and contradictary.
Because you begin with a statement that cannot apply to any value of n does not imply that this value of N does not exist. This value of N certainly does exist, but when you attempt to apply a description that states in effect "this description does not EVER apply" you obviously run into a problem.
EDIT:
After a bit of snooping about on the internet, I dug up a note that describes this flaw better than I can:
[/quote]
It is very difficult to pin down this fallacy. It has more to do with the informal nature of the English language, and the paradoxes that can arise from this, than with any mathematical error.
Any sentence that tries to make claims about itself runs the risk of being logically inconsistent. The classic example is the liar's paradox "this sentence is false".
The problem in this case is the following phrase (let's call it S):
the smallest natural number that cannot be unambiguously described in fourteen words or less.
S refers to itself (because it makes claims about descriptions of numbers, and S is such a description). Moreover, it does so in a logically inconsistent fashion: if you try to apply the description S to a number, then it ends up stating that S does not apply to that number.
This means that S cannot be considered as a self-consistent description of any natural number. This, however, does not mean that the number n (in the proof) does not exist! There is such a number n, and n is the smallest natural number that cannot be unambiguously described in fourteen words or less; it's just that the phrase "the smallest natural number that cannot be unambiguously described in fourteen words or less" is not a description of it (in the sense that is being used in the proof), because it is a phrase that cannot be self-consistently asserted about any number.
Therefore, step 4 of the proof (which mistakes the self-inconsistent nature of S with a mathematical contradiction arising from the existence of n) is at fault.
This is also related to Russell's Paradox in set theory: there is no such thing as the "set of all sets" (if there were, you could look at "the set of all sets that do not contain themselves". Let S be this set. Does S contain itself, or not? Either way leads to a contradiction).
Finally, although this particular proof is fallacious, it illustrates a common proof technique which, when used correctly, is very powerful: the well-ordering principle.
If you want to show that something is true for all natural numbers n, one way to do it (which is mathematicall equivalent to a proof by induction but is sometimes more convenient than it) is to reason as follows:
Suppose it's not the case that the statement in question is true for all n. Then there is a smallest n for which it fails. But this leads to a contradiction because . . . . Therefore, it must indeed be the case that the statement in question is true for all n.
.
Proofs that follow this pattern are using the well-ordering principle (which says that any non-empty set of natural numbers must have a smallest element), and this is a very common and powerful pattern of proof. When used correctly, that is.
[/quote]
Those of actualy physical presence: as in, the double slit experiment:
There are two slits in a wall, side by side. A single partice is fired at them. The particle goes trough two different holes, trough only one of them, trough the other one as well while not going trough the one it went trough previously and trough none of them at all. And it does this simultaneously. Even more curiously, measuring trough which slit the particle goes trough means the above paradox does not happen: By some bizarre means, it makes it impossible.
Schrodinger's cat had it easy, at least it was only alive and dead simultaneously, instead of being alive ANDOR dead both at Miami and London at once, while still being only a single cat.
Quantum superpositions. Gotta love them.
Who needs metaphysical paradoxes, when the universe around us is more strange than we can suppose and filled to the brim with apparent contradictions.
There are two slits in a wall, side by side. A single partice is fired at them. The particle goes trough two different holes, trough only one of them, trough the other one as well while not going trough the one it went trough previously and trough none of them at all. And it does this simultaneously. Even more curiously, measuring trough which slit the particle goes trough means the above paradox does not happen: By some bizarre means, it makes it impossible.
Schrodinger's cat had it easy, at least it was only alive and dead simultaneously, instead of being alive ANDOR dead both at Miami and London at once, while still being only a single cat.
Quantum superpositions. Gotta love them.
Who needs metaphysical paradoxes, when the universe around us is more strange than we can suppose and filled to the brim with apparent contradictions.
Not quite.Eclectic Dreck said:
n is just a number that satisfies the condition that itself and all previous numbers can be unambiguously describe in 14 words or less.
It may or may not be the largest. But claiming it is the largest is assuming there are some numbers where it doesn't hold, which we have no reason to assume at this point.
Exactly, that's not a flaw, that is the proof.
The statement is not intrinsically contradictory, it only becomes so when it is applied to a number.
Take any number and the statement "The smallest natural number that cannot be unambiguously described in fourteen words or less." cannot consistently apply to it, and therefore it can't apply to any number.
Prove it.
Yes, that was actually posted before by someone else and, as I said to them, it's wrong, for the reason I've just given you.
The proof has been demonstrated: the question CANNOT apply to ANY value regardless of it's status because, the statement is a statement that says, in effect, the value represented by this statement cannot be represented by this statement.Maze1125 said:
You do NOT have a paradox here, you have an error in logic. The flaw you are consistantly taking is assuming that because your statement cannot possibly apply to any number because it is worded in a self referential way that is intentionally self-consistant that you have a demonstrable theorem. The logic simply does not hold, because if followed precisely here is what you end up with.
Value n meets the crita of the first number that requires more than fourteen words to describe. It is therefore the "the smallest...". This name immedieately becomes inconsequential as this number may now be referred to by a name that is fewer than 14 words.
Before the name is assigned, it meets the critera, after the name is assigned it no longer meets the criteria thanks to the self-inconsistant nature of the description. Thus far we have nothing but a logical flaw in the description and nothing more.
Here's a similar example:
Given the sentence "The first number not equal to one that is not described by this sentence"
The value of 1 is found to not apply. The value of 2 is examined and found to meet the criteria, thus it may be described as the first number not equal to one... The same conundrum arrives. What conclusion do you draw from this situation? Following the EXACT SAME logic we would find that since this cannot apply to any number, all numbers MUST therefore be 1.
The same fallacious argument has been made in my example as your more famous one. Yet, mathematics will generally agree that there are a great many numbers that are in fact not equal to one. The closest argument you can have is that one is a prime factor of any natural number.