What is wrong with this proof that 1 is the largest integer? What does it prove?

• Jun 1st 2007, 09:00 PM
tukeywilliams
What is wrong with this proof that 1 is the largest integer? What does it prove?
Let $\displaystyle n$ be the largest integer. Then since $\displaystyle 1$ is an integer we must have $\displaystyle 1 \leq n$. On the other hand, since $\displaystyle n^{2}$ is also an integer we must have $\displaystyle n^{2} \leq n$ from which it follows that $\displaystyle n \leq 1$. Thus, since $\displaystyle 1 \leq n$ and $\displaystyle n \leq 1$ we must have $\displaystyle n = 1$. Thus $\displaystyle 1$ is the largest integer.

I said that the problem was that if $\displaystyle 1$ is the largest integer, then $\displaystyle 1 < n$ and $\displaystyle n^{2} < n$. There is no $\displaystyle \leq$ or $\displaystyle \geq$. So $\displaystyle n < 1$ and $\displaystyle n > 1$ which contradicts the Trichotomy law.

The above argument that $\displaystyle 1$ is the largest integer proves that direct proofs arent the best way to approach a problem?

Am I correct?

Thanks
• Jun 1st 2007, 10:25 PM
JakeD
Quote:

Originally Posted by tukeywilliams
Let $\displaystyle n$ be the largest integer. Then since $\displaystyle 1$ is an integer we must have $\displaystyle 1 \leq n$. On the other hand, since $\displaystyle n^{2}$ is also an integer we must have $\displaystyle n^{2} \leq n$ from which it follows that $\displaystyle n \leq 1$. Thus, since $\displaystyle 1 \leq n$ and $\displaystyle n \leq 1$ we must have $\displaystyle n = 1$. Thus $\displaystyle 1$ is the largest integer.

I said that the problem was that if $\displaystyle 1$ is the largest integer, then $\displaystyle 1 < n$ and $\displaystyle n^{2} < n$. There is no $\displaystyle \leq$ or $\displaystyle \geq$. So $\displaystyle n < 1$ and $\displaystyle n > 1$ which contradicts the Trichotomy law.

The above argument that $\displaystyle 1$ is the largest integer proves that direct proofs arent the best way to approach a problem?

Am I correct?

Thanks

How would you respond if I said there is nothing wrong with the proof? It correctly proves the statement

"n is the largest integer implies n = 1"

and this is true. To see why, note that "P implies Q" is equivalent to "not P or Q." Then the implication is equivalent to

"n is not the largest integer or n = 1"

which is true because there is no largest integer, so n cannot be the largest integer.

So if "n is the largest integer implies n = 1" is a true statement, correctly proven, then what is wrong with that paragraph?
• Jun 2nd 2007, 01:49 AM
tukeywilliams
The above paragraph doesnt take into account a number like $\displaystyle n+1$ which will be larger than $\displaystyle n$?
• Jun 2nd 2007, 03:15 AM
CaptainBlack
Quote:

Originally Posted by tukeywilliams
The above paragraph doesnt take into account a number like $\displaystyle n+1$ which will be larger than $\displaystyle n$?

JakeD says what you have proven is:

"n the largest integer implies that n=1"

Which is true. Now what he wants you to think about is what conclusion you
can draw from this. Is "1 is the largest integer" true? If not what does that tell you

RonL
• Jun 2nd 2007, 03:22 AM
tukeywilliams
' 1 is the largest integer' is not true. The premiss is that we are assuming that there is a largest integer?
• Jun 2nd 2007, 03:29 AM
Glaysher
Quote:

Originally Posted by tukeywilliams
' 1 is the largest integer' is not true. The premiss is that we are assuming that there is a largest integer?

Yes. You proved IF there is a largest integer then it is equal to 1. You have not shown that such an integer exists.
• Jun 2nd 2007, 11:30 AM
Jhevon
Quote:

Originally Posted by tukeywilliams
' 1 is the largest integer' is not true. The premiss is that we are assuming that there is a largest integer?

apparently CaptainBlack's hint was too subtle. I won't give you the answer (everyone here might get upset with me if i do :p) but i will give you a hint that should make it starkingly obvious what is wrong with this proof.

Hint: What is the difference between a valid and a sound argument? Is the proof given valid? Is it sound?
• Jun 2nd 2007, 12:51 PM
tukeywilliams
A valid argument holds true for some cases. A sound argument holds true for all cases. I think the problem is that we chose n = 1. We could also have chosen n = 2, n =3 and so on. Is this correct?

Thanks
• Jun 2nd 2007, 01:01 PM
Jhevon
Quote:

Originally Posted by tukeywilliams
A valid argument holds true for some cases. A sound argument holds true for all cases. I think the problem is that we chose n = 1. We could also have chosen n = 2, n =3 and so on. Is this correct?

Thanks

a valid argument is one in which the conclusion follows logically from the premises and the conclusion is true (or very likely to be true) if all the premises are true. a sound argument is a valid argument that has true premises, and therefore, the conclusions that follow logically must be true (or must be highly likely to be true).

there is nothing wrong with the proof itself, it is a valid proof. however, it is not sound, since the premise is not true (there is no largest integer). therefore, we can assume that the conclusion is false, since, even though it follows logically from what we originally assumed, what we originally assumed was false.

you should also realize, an implication is only false, if we have a true statement implying a false statement, otherwise the implication is true. a false statement implies whatever, literally. the statement "if n is the largest integer then pigs can fly" is true as far as the truth table definition of an implication is concerned.

therefore, the only thing wrong with this proof is that its premises are false, and therefore any conclusion from it, even though that conclusion may be derived logically, is highly likely to be false--which in this case, we can clearly see that it is.
• Jun 2nd 2007, 01:08 PM
tukeywilliams
dangit, I should have realized that. The first chapter dealt with the following: if you have a false conclusion, then $\displaystyle P \Rightarrow Q$ will be false if $\displaystyle P$ is true. However if the hypothesis is false $\displaystyle P \Rightarrow Q$ can still be true.

Thanks for the help.
• Jun 2nd 2007, 01:13 PM
Jhevon
Quote:

Originally Posted by tukeywilliams
if you have a false conclusion, then $\displaystyle P \Rightarrow Q$ will always be false.

no, $\displaystyle P \Rightarrow Q$ is only false if we have a true statement implying a false statement, otherwise it is true...but you get the idea. good job :D
• Jun 2nd 2007, 01:34 PM
CaptainBlack
Quote:

Originally Posted by tukeywilliams
dangit, I should have realized that. The first chapter dealt with the following: if you have a false conclusion, then $\displaystyle P \Rightarrow Q$ will be false if $\displaystyle P$ is true. However if the hypothesis is false $\displaystyle P \Rightarrow Q$ can still be true.

Thanks for the help.

What you have is a false conclusion following from your assumption, so
yor assumption is false, there is no largest integer.

RonL
• Jun 2nd 2007, 02:48 PM
JakeD
Quote:

Originally Posted by Jhevon
a valid argument is one in which the conclusion follows logically from the premises and the conclusion is true (or very likely to be true) if all the premises are true. a sound argument is a valid argument that has true premises, and therefore, the conclusions that follow logically must be true (or must be highly likely to be true).

there is nothing wrong with the proof itself, it is a valid proof. however, it is not sound, since the premise is not true (there is no largest integer). therefore, we can assume that the conclusion is false, since, even though it follows logically from what we originally assumed, what we originally assumed was false.

you should also realize, an implication is only false, if we have a true statement implying a false statement, otherwise the implication is true. a false statement implies whatever, literally. the statement "if n is the largest integer then pigs can fly" is true as far as the truth table definition of an implication is concerned.

therefore, the only thing wrong with this proof is that its premises are false, and therefore any conclusion from it, even though that conclusion may be derived logically, is highly likely to be false--which in this case, we can clearly see that it is.

Thanks to everyone for this thread and to Jhevon for this post. It has been instructive for me, both in how to explain things (not my way) and in the concepts. Here is what I think I've learned.

The key concepts are the validity of the argument and the truth values of the resulting implication P => Q, premise P and conclusion Q. In the problem, we had a valid argument establishing P => Q. But the premise P was false, so the conclusion Q could not be drawn.

Now I first brought up the fact that because P was false, P => Q was true; however, that is really of no importance. What is important is that P => Q be true when P is true, implying Q is true, and that is guaranteed by the argument being valid.