“The statistician should set the probability of Type I error as low as possible” Do you agree with the following statement? Why or why not?
I disagree with this statement but i have no reasons to back up. Anyone can help?
 A: As low as possible is ill defined. You can always make a test with the probability of a Type 1 error 0 but then this test has no power at all. (Think what happens if you take a p-value of 0). You will have to make a trade off between Type 1 and Type 2 errors and what the impact is of each of these errors.
A: In an ideal world, the probability of type I error would be adjusted on a case by case basis, rather than being set to .05 or .01 or whatever by fiat.
Less type I error means more type II error. In many fields, defaults are 0.05 and 0.20 for these two errors (that is, power of 0.8 is sought with p 0.05). Are these choices reasonable? It depends.
Suppose you are testing a drug that may cure a quickly terminal disease. Then type I error would mean people with the disease die when they could be cured, while type II error means dying people take a drug that does not work.  
I'd be willing to have a higher type I error in such a case.
Over and above this, I always prefer looking at effect sizes rather than significance levels. What people care about is how much of an effect a (treatment, drug, condition) is likely to have; not whether a test statistic as extreme as the one we got could have easily arisen from a population where there was no effect. 
A: It's possible to set it to zero. 
In general it would make for a test with no power.
So I disagree with the statement because - except in a few very special instances - setting it as low as possible makes for a pointless test.*
* [In fact, in the case of point nulls, given that point nulls are almost never exactly true, it may arguably make more sense to set it as high as possible.]
A sensible approach would look at the tradeoff between type I error rate and power - you can get more power (correct rejections) by accepting a higher rate of type I error (false rejections) -- and conversely you can lower type I error by giving up some power. The choice of significance level more properly relates to how you weigh the relative importance of the two in a particular situation.
Here's the rejection rate for the two-sample t-test (i.e. power everywhere but at $\delta=0$) at n=14:

the black curve shows $\alpha=0.10$, the red curve shows $\alpha=0.01$. When the true means are $\sigma$ apart ($\delta=1$ above), the power is $82.4\%$ for $\alpha=0.10$ and $46.1\%$ for $\alpha=0.01$. Obviously you'd prefer the higher power in this case; but** you can only get it at the cost of accepting an increase in type I errors. Of course there's a whole range of intermediate values of $\alpha$ that will give power in between. Your exercise is to choose which curve you want, which is always a tradeoff.
** (assuming your $n$ is fixed -- and if your budget is finite there will be pretty strong limits on how far you can play with $n$)
A: Usually the statement is false.
The statistician should make sure that the chance of false inference is as low as possible. Sometimes that would involve minimising the type I error rate, but mostly is involves looking at the problem differently. In other words, there are many circumstances where the type I error rate should be of no interest, and in those circumstances the statement is wrong.
The type I error rate relates to the global false positive error rate in a notional series of experiments, and it is useful in planning experiments and for testing hypotheses with an automatic decision rule. In many cases (most scientific cases, in my opinion) the statistician should be concerned with the more local question of what the data allow him to conclude about the state of the world in the context of this experiment. That may involve consideration of the actual p-value (as opposed to determining if it is above or below a threshold in a manner that is consistent with quantitation of long term type I errors) or inspection of a likelihood function or a Bayesian posterior probability distribution.
Even if one is determined to work within a strictly frequentist framework it is sensible to examine the trade-of between type I and type II errors. It is unfortunate that we describe tests in terms that privilege type I errors over all other types of error.
A: I disagree. Consider a petty crime trial. A prosecutor's null-hypothesis is that you committed a crime. The court considers this hypothesis, otherwise there wouldn't be a trial to start with. 
The easiest way to minimize Type I error is to simply accept prosecutor's case, i.e. find you guilty without deliberations. In this case probability of Type I error would be minimal: you will not be acquitted if you did it. 
Unfortunately, this will increase the probability of Type II error: you'll be be found guilty even if you didn't do it.
That's why we have presumption of innocence and a trial with competing prosecutor and defense, because sometimes Type II errors cost too much. You have to consider the costs of both errors when deciding on the balance between them.
UPDATE: this is somewhat off-topic, but still interesting.
whuber thinks that presumption of innocence applies to a prosecutor. I'm not a lawyer myself, but I know that it's clearly not the case in USA. 
Consider the following quotes from Laudan, L. (2006). Truth, Error, and Criminal Law: An Essay in Legal Epistemology. Cambridge: Cambridge University Press, pp.93-96:

One might have supposed, naively as it turns out, that a person is
  presumed innocent from the moment police first suspect him until his
  trial ends and a verdict has been pronounced (supposing things
  progress that far). One might have supposed, equally erroneously, that
  all those charged to make decisions about the defendant (judges, grand
  juries, prosecutors, and so on) are under some duty to presume his
  innocence. 
While such a broad construal of the meaning of PI is common
  in several countries (at least in theory), the Supreme Court has made
  it very clear that in American criminal procedure, the PI applies only
  to the trial itself and, within the trial, only to the jury or other
  trier of fact.
The Louisiana Supreme Court put the same argument in blunter terms:
The presumption of innocence is a guide to the jury. If it were
  absolute and operative at every stage of a prosecution, the defendant
  could never be jailed until conviction. It does not prevent arrest.
  ... There is little relationship between the right to bail and the
  presumption of innocence. The presumption of innocence is operative
  and protects against conviction, not against arrest (which is taking
  into custody). 
State v. Green, 275 So.2d 184, at 186 (La., 1973).

Apparently, Lord doesn't like both Type I and II errors:

Proverbs 17:15
He that justifieth the wicked, and he that condemneth the just, even
  they both are abomination to the LORD.

A: In general, there is no consensus on how large one should set the type 1 error. In slightly high-powered language, it doesn't matter when the set of best tests forms a complete class. In such cases, all best tests are admissible, one may choose the type 1 error (size of the test) freely and the choice of type 1 error has no effect on the credibility of inference. 
It is traditionally set at lower values, 0.01, 0.05, 0.1 for good reason: in science, rejection is more pronounced than acceptance because it supports the falsification. Rejecting the true is fatal, accepting the false is undesirable, but tolerable. Setting at too low leads to, as demonstrated by @Glen_b, frequent acceptance of the false, so we never know what is true or false.
