Bad proof!

[DrPizza]

Okay, we've all seen the proofs that 1=2, and generally, the reason is because the proof makes unobtrusive use of dividing by 0 or squares of i

Anyway, Let's prove that *every* counting number
(i.e. 1, 2, 3, 4, . . .)
*every* counting number can be uniquely described by 13 words or less.

Now, obviously this isn't true, because there are a finite number of words and an infinite number of counting numbers. i.e. if there are 1 million words, then there are (only) 1 million ^13th power unique combinations of 13 words, 1 million^12 unique combinations of 12 words, . . . A huge number, but a finite number nonetheless.

Proof anyway:
1. Lets assume that there *are* counting numbers that can't be uniquely described by 13 words or less.
2. Then, there must be a smallest such counting number that cannot be described in 13 words or less. Lets call it x
3. But now, x is "the smallest counting number which cannot be described in thirteen words or less" [count them... 13 words]
4. This is a contradiction. Therefore, the assumption in statement one must be incorrect. Thus, all counting numbers must be able to be uniquely described.


No googling, find the error.

 

[kogase]

No.

 

[akubi]

This statement is false!

 

[Anubis]

its too early for this

 

[quakefiend420]

Originally posted by: Anubis
its too early for this

qft

 

[mitmot]

Originally posted by: Anubis
its too early for this

 

[dullard]

#3 does not uniquely identify a number. That description describes many numbers.

For example, x is "the smallest counting number which cannot be described in thirteen words or less".

But wait, x was described in thirteen words or less. Thus x+1 is now "the smallest counting number which cannot be described in thirteen words or less".

But wait, x+1 was just described in thirteen words or less. We must look at x+2.

See how it isn't unique?

 

[DrPizza]

Originally posted by: dullard
#3 does not uniquely identify a number. That description describes many numbers.

For example, x is "the smallest counting number which cannot be described in thirteen words or less".

But wait, x was described in thirteen words or less. Thus x+1 is now "the smallest counting number which cannot be described in thirteen words or less".

But wait, x+1 was just described in thirteen words or less. We must look at x+2.

See how it isn't unique?

Which is why it's a contradiction. If you say that there are a group of numbers which cannot be uniquely described. Then, there has to be a smallest one of those numbers. There is a smallest member for any subset of counting numbers (except, I suppose, the empty set.) Thus x is the smallest that can't be described, yet x is uniquely described.

 

[DrPizza]

Originally posted by: akubi
This statement is false!

Bingo!

Anyone understand what akubi means? Or is akubi the blind squirrel that gets lucky and finds a nut occasionally?

 

[mdchesne]

uh... ehhh.... meh.... AHHHH! math... in the morning!.... too early!..... mehhhhhh

 

[ColdFusion718]

Originally posted by: DrPizza

Originally posted by: akubi
This statement is false!

Bingo!

Anyone understand what akubi means? Or is akubi the blind squirrel that gets lucky and finds a nut occasionally?

LOL

 

[dullard]

Originally posted by: DrPizza
Which is why it's a contradiction. If you say that there are a group of numbers which cannot be uniquely described. Then, there has to be a smallest one of those numbers.

But then you just uniquely described a number in a group of numbers that supposedly cannot be uniquely described. Therefore, it never belonged in that group. There is no smallest number in that specific group. Thus you cannot uniquely describe this number that does not exist.

Don't even get me started on questions like: how many possible languages are there? Is that an infinite number?

And blind squirrels do quite well I assume. Squirrels only remember 10% of their nuts and 90% of what they eat was just them randomly burrowing until they found a nut that was burried for another reason (maybe another squirrel).

 

[DrPizza]

Originally posted by: dullard

Originally posted by: DrPizza
Which is why it's a contradiction. If you say that there are a group of numbers which cannot be uniquely described. Then, there has to be a smallest one of those numbers.

But then you just uniquely described a number in a group of numbers that supposedly cannot be uniquely described. Therefore, it never belonged in that group. There is no smallest number in that specific group. Thus you cannot uniquely describe this number that does not exist.

Don't even get me started on questions like: how many possible languages are there? Is that an infinite number?

And blind squirrels do quite well I assume. Squirrels only remember 10% of their nuts and 90% of what they eat was just them randomly burrowing until they found a nut that was burried for another reason (maybe another squirrel).

Ughhhh!
Yeah, that's what I said!
It has to be, but it can't be, that's what the contradiction is.

What Akubi said is correct. When you have a sentence that refers to itself, it leads to logical inconsistencies. i.e. Russell's paradox (Russell, right??) about sets.

 

[Whitecloak]

why should there be a finite set of words?

 

[DrPizza]

Originally posted by: whitecloak
why should there be a finite set of words?

Because I can't lift a dictionary with an infinite set of words?

 

[dullard]

Originally posted by: DrPizza
Ughhhh!
Yeah, that's what I said!

Ok, I was uncertain as to why you disliked my first post when we were agreeing.

"You uniquely identified a number that you said could not be uniquely identified."

vs

"It has to be, but it can't be, that's what the contradiction is."

Same thing in my book.

 

[Jzero]

Originally posted by: DrPizza

Originally posted by: whitecloak
why should there be a finite set of words?

Because I can't lift a dictionary with an infinite set of words?

The lexicographer documents words, but he does not create them. There could be infinite words that haven't been invented yet, but since they are not in use, they cannot be added to any dictionary. In fact, I would hazard that if there were a number that exceeds the words we have to name it, we would just come up with a new word for it, which would begin to appear in dictionaries (like googol, for example).

 

[Reel]

Originally posted by: DrPizza

Originally posted by: whitecloak
why should there be a finite set of words?

Because I can't lift a dictionary with an infinite set of words?

Sounds like someone needs to hit the gym...

 

[jessicak]

 

[StormRider]

The theorem is true.

We have an infinite number words.

The word for 10001 is 10001, the word for 10002 is 10002 and so on.

 

[StormRider]

Originally posted by: DrPizza

Originally posted by: dullard

Originally posted by: DrPizza
Which is why it's a contradiction. If you say that there are a group of numbers which cannot be uniquely described. Then, there has to be a smallest one of those numbers.

But then you just uniquely described a number in a group of numbers that supposedly cannot be uniquely described. Therefore, it never belonged in that group. There is no smallest number in that specific group. Thus you cannot uniquely describe this number that does not exist.

Don't even get me started on questions like: how many possible languages are there? Is that an infinite number?

And blind squirrels do quite well I assume. Squirrels only remember 10% of their nuts and 90% of what they eat was just them randomly burrowing until they found a nut that was burried for another reason (maybe another squirrel).

Ughhhh!
Yeah, that's what I said!
It has to be, but it can't be, that's what the contradiction is.

What Akubi said is correct. When you have a sentence that refers to itself, it leads to logical inconsistencies. i.e. Russell's paradox (Russell, right??) about sets.

Not all statements that refer to themselves lead to logical inconsistencies. What about this statement:

This statement is true!

 

[Billzie7718]

Not just words, but there would be an infinite number of possible combinations of letters in the alphabet as you make the words longerzabcd.

 

[Billzie7718]

Originally posted by: StormRider

Originally posted by: DrPizza

Originally posted by: dullard

Originally posted by: DrPizza
Which is why it's a contradiction. If you say that there are a group of numbers which cannot be uniquely described. Then, there has to be a smallest one of those numbers.

But then you just uniquely described a number in a group of numbers that supposedly cannot be uniquely described. Therefore, it never belonged in that group. There is no smallest number in that specific group. Thus you cannot uniquely describe this number that does not exist.

Don't even get me started on questions like: how many possible languages are there? Is that an infinite number?

And blind squirrels do quite well I assume. Squirrels only remember 10% of their nuts and 90% of what they eat was just them randomly burrowing until they found a nut that was burried for another reason (maybe another squirrel).

Ughhhh!
Yeah, that's what I said!
It has to be, but it can't be, that's what the contradiction is.

What Akubi said is correct. When you have a sentence that refers to itself, it leads to logical inconsistencies. i.e. Russell's paradox (Russell, right??) about sets.

Not all statements that refer to themselves lead to logical inconsistencies. What about this statement:

This statement is true!

How bout this one?

This statement is false!

 

[arcenite]

Both statements are both true and false

 

[Titan]

what if numbers are words? One word each!