The threshold voltages of a carbon nanotube (CNT) field-effect transistor (FET) are derived and compared with those of the metal-oxide-semiconductor (MOS) FETs. The CNT channel is so thin that there is no voltage drop in a CNT diameter direction perpendicular to the gate electrode plane, and this makes the CNTFET characteristics different from those in MOSFETs. The relation between the voltage and the electrochemical potentials, and the mass action law for electrons and holes are examined in the context of CNTs, and it is shown that the familiar relations are still valid because of the macroscopic number of states available in the CNTs. This situation is significantly different from that of quantum dots. Using these relations, we derive an inversion threshold voltage V@sub Ti@ and an accumulation threshold voltage V@sub Ta@ as a function of the Fermi level E@sub F@ in the CNT channel, where E@sub F@ is a measure of doping. V@sub Ti@ of the CNTFETs has a much stronger dependence on E@sub F@ than that of MOSFETs, while V@sub Ta@s of both CNTFETs and MOSFETs depend quite weakly on E@sub F@ with the same functional form. This means that the transition from normally-off mode to normally-on mode is much sharper in CNTFETs as E@sub F@ is modulated through doping, and this property has to be taken into account in circuit design.