Dear Jeffrey,

You assume "kadosh" means holy. However, the Torah calls a harlot "kadesha" (see Gen. 38:15, 21), even though a street-walker doesnt exactly fit most peoples definition of holy. The commentators explain that kadosh means separate or set aside for a specific purpose. In the example above, then, kadesha refers to a woman who has separated herself from moral behavior and has designated herself for prostitution.

In the examples you brought, the third blessing of the shemone esrei (called kedushat Hashem) and Kedusha during the repetition of the shemone esrei, we are expressing the fact that G-d is entirely separate from the mundane world, as we attempt to raise ourselves out of materialism and cleave to Him. In this way we intend to mimic the angels, who are themselves separate from this world and designated to praise G-d.

The Jewish people are also called kadosh, as in "For you are an Am Kadosh to the Lord your G-d who has chosen you to be a special people to Himself, above all peoples of the earth" (Deut. 7:6). Here too, as explicit in the verse, the Jewish people are separate and set aside from the other nations to be G-ds special, chosen people.

Similarly, the Land of Israel is called Eretz HaKodesh, the city of Jerusalem is called Ir HaKodesh, the Temple is called Beit HaMikdash, and the most restricted area therein was called the Kadosh HaKadoshim. In all of these instances, the term kadosh expresses that these places are separate and more special than other places of their kind.

In commanding us to refrain from eating forbidden foods, G-d says hitkadashtem (withdraw yourselves from it) and you shall be kedoshim because I am kadosh (Lev. 11:44). Finally, one who sacrifices his life for his belief in God, the Torah or the commandments is also called kadosh. Such an extreme expression of faith repudiates the false beliefs of his tormentors, distinguishes him from among his people and relegates him to a special status in Heaven.