Why aren't citizens who have given up looking for work not included in the unemployment numbers?
Various measures of the U.S. labor market are prepared and published by the Bureau of Labor Statistics (BLS), U.S. Department of Labor. The (civilian) labor force (L) is defined to be the summation of those in the non-institutionalized and non-military population who are 16 years of age and older who are employed (E) and those that are unemployed (U), L = E + U. Those not in the labor force (N) are persons who are in the civilian population 16 years of age an older and who are not employed or unemployed. It includes individuals who are discouraged workers (D), marginally attached workers (MA), and those who are attending school/college/university, engaged in unpaid housework and retired workers (R). The number of individuals in each of these categories is estimated from a sample of the U.S. population surveyed in the Current Population Survey (CPS) conducted monthly of roughly 50,000 individuals by the Bureau of the Census for the BLS. Survey respondents are interviewed to obtain information about the employment status of each member of the household 15 years of age and older. 1
(E) Employed persons are individuals who are 16 years of age and older in the civilian non-institutional population who, during the reference week: (a) did any work at all (at least 1 hour) as paid employees, worked in their own business, profession, or on their own farm, or worked 15 hours or more as unpaid workers in an enterprise operated by a member of the family (e.g., those engaged in part-time work are counted as employed), and (b) all those individuals who were not working but who had jobs or businesses from which they were temporarily absent because of vacation, illness, bad weather, child care problems, maternity or paternity leave, labor-management dispute, job training, or other family or personal reasons, where or not they were paid for the time off or were seeking other jobs.
(U) The standard definition of the number of unemployed individuals is all persons 16 years of age and older who had no employment during the reference week, were available for work, except for temporary illness, and had made specific efforts to find employment sometime during the 4-week period ending with the reference week. Persons who were waiting to be recalled to a job from which they had been laid off need not have been looking for work to be classified as unemployed. Hence, the standard and most popular definition of the U.S. unemployment rates, is computed as the number who are unemployed (U) divided by the number of individuals in the labor force, or U-3 = U/L. This has broad acceptance in that it is also the International Labor Organization (ILO) definition of unemployment and is widely reported for other countries.
(D) Discouraged workers are individuals who are not in the labor force who want and are available for a job and who have looked for work sometime in the past 12 months (or since the end of their last job if they held one within the past 12 months), but who are not currently looking because they believe there are not jobs available or there are none for which they would quality.
(MA) Marginally attached workers are individuals who want, and are available for work, and who have looked for a job sometime in the past 12 months (or since the end of their last job if they held one within the past 12 months), but were not counted as unemployed because they had not searched for work in the four weeks preceding the survey. Notice that the definition of discouraged worker are specifically not looking or work because they believe there are not jobs available, but marginally attached workers may not be looking for these and other reasons.
Broader measures of the unemployment than U and unemployment rate U-3 have some appeal. For example, some of those individuals who are discouraged or marginally attached workers would really like to work, i.e., they are involuntarily unemployed, and some of the part-time employed would like to work more, i.e., are under-employed. Since 1994, BLS has reported an extended set of unemployment statistics—U-4, U-5, and U-6. U-4 includes discouraged workers (D) as being unemployed and also adds them to the labor force base, so U-4 = (U+D)/(L+D). Hence, the number of discouraged workers is added to the numerator and denominator. In a similar fashion, U-4 includes marginally attached workers (MA) as being unemployed and also adds them into the labor force base, U-5 = (U+MA)/(L+MA). Finally, U-5 is modified to include as unemployed those employed individuals who are working part-time for economic reasons, i.e., those employed part-time who want full-time jobs (PTE). This measure of unemployment is computed as U-6 = (U+MA+PTE)/(L+MA). Hence, BLS has responded to earlier concerns and criticisms about the somewhat arbitrary nature of the U-3 definition of unemployment, and since 1994, it has been publishing data regularly on U-4, U-5, and U-6, in addition to U-3. Table A15 below provides recent data on these four measures of the U.S. unemployment rate so that one can compare them. Clearly U-6 is substantially larger than the other three unemployment rates measures. However, the media and most political discussions of unemployment are almost certainly using the conventional measure of unemployment, U-3. Recall that it does have ILO endorsement.
For more information, see Laing, D. Labor Economics, New York, NY: W.W. Norton & Co., 2011, pp. 34-38, 759-762.
 Some of the individuals identifies in the CPS may be non-citizens, both those with documents, meaning they are legally in the U.S., and others who are undocumented, meaning that they came illegally or overstayed their visas. Both of these non-citizen types may be included in BLS’s labor market statistics.