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If one gets a phd in math, is it able to be admitted to another phd program in math? It is not for me, for someone else.

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Depends on the University. Some specifically say you should not have a Ph.D. or equivalent in the same area in order to be admitted; others do not. It also depends on what the "equivalent" may mean (e.g., some European Doctorates used to be considered the equivalent of an advanced Master's degree in the U.S., so they would not be considered a bar for admission). – Arturo Magidin Feb 7 '11 at 14:54
You might be able to get PHDs in pure math and applied math. – Unreasonable Sin Feb 7 '11 at 15:45
Are we allowed to ask why someone would want to get two PhDs in math, say both at US universities? That is certainly what the admissions committee will be wondering... – Pete L. Clark Feb 7 '11 at 16:01
+1 for Pete.... – Mariano Suárez-Alvarez Feb 7 '11 at 16:23
@Unreasonable Sin: It may be possible in places like Waterloo, where there are distinct departments of Pure Maths, Applied Maths, Combinatorics and Optimization, etc., or in places where Stats are separate from Maths (if the "applied" degree is in statistics). But in most US Depts, the two are both in a "Mathematics" department, and they would be indistinguishable from the outside (say, to the Graduate School/Division), making it just as difficult as two PhDs in mathematics. – Arturo Magidin Feb 7 '11 at 16:27

Here is the statement from the Graduate Division at Berkeley:

Duplication of Degree.

Students who already hold a doctoral level degree are not admitted and duplication of degree or admission to a lesser degree is not permitted. However, in extraordinary circumstances, the faculty of the department may request an exception from the Dean of the Graduate Division. The department must demonstrate that the second degree field of study and program are distinctly different from that of the original degree, and that there is a professional or scholarly purpose that requires this second degree.

Similar restrictions apply throughout the UC system. Harvard likewise states:

Persons holding a PhD or its equivalent, or who have completed most of the work required to earn the PhD elsewhere, may apply to a PhD program in the Graduate School only if it is an unrelated field of study.

I know that at least until the late 1990's MIT's Computer Science Department had a rule that nobody with a Ph.D. (in any field) would be admitted to their Ph.D. program; the rule was changed in the early 2000s so that a Ph.D. in a field other than Computer Science was no longer a bar for admission, but it still placed applicants at a disadvantage. I would expect similar rules to exist in other Departments at MIT, but I did not spot any explicit rule in the mathematics admission page.

Similar policies are explicit in many US universities; even in those where it is not explicit, space limitations will usually lead admission committees to consider such applications very skeptically, prefering to give spots to new students, and if they are very keen on the applicant, will likely suggest a post-doc instead.

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