“showed him all the kingdoms.” Matthew and Luke both record the 3 temptations that the Adversary tempted Jesus with, but worded slightly differently and in a different order. Commentators differ as to whether Matthew or Luke has the order of events as they actually happened. We believe Luke has the order correct because Luke says he recorded things “in order” (Luke 1:3). Furthermore, another reason we believe Luke has the correct order is that to the Devil, the most desirable outcome would be to have Jesus worship him, but if he could not accomplish that, to kill Jesus and be done with him. The order of temptations in Luke accomplishes that goal. The second temptation would result in Jesus worshipping the Devil, and if that failed the third temptation would have resulted in Jesus’ death.
“inhabited world.” There are different words translated “world” or “earth,” and the differences in the meanings are important. Unfortunately, most versions translated both oikoumene and kosmos as “world,” leaving the English reader with no way to see the differences. The Greek word in Luke 4:5 is oikoumene (#3625 οἰκουμένη), and it means 1. The earth as inhabited area, exclusive of the heavens above and nether regions, the inhabited earth, the world. 2. The world as administrative unit, the Roman Empire (in the hyperbolic diction commonly used in reference to emperors, the Roman Empire equaled the whole world). 3. All the inhabitants of the earth, then, figuratively humankind (cp. Acts 17:31; Luke 2:1, 4). When it means the whole world so far as living beings inhabiting it, it seems to include the realm of transcendent beings as well. The inhabited world is different from kosmos, the world as a creation.
The Greek world kosmos (#2889 κόσμος) has several different definitions (from BDAG). The basic idea is one of order or orderliness. 1. That which serves to beautify through decoration, adornment, adorning (1 Peter 3:3). 2. Condition of orderliness, orderly arrangement, order. 3. The sum total of everything here and now, the world, the (orderly) universe (John 17:5). 4. The sum total of all beings above the level of the animals, the world (1 Cor. 4:9). 5. The planet earth as a place of inhabitation, the world, the world in contrast to heaven. 6. By metonymy: humanity in general, the world. 7. The world, and everything that belongs to it, appears as that which is hostile to God. 8. Collective aspect of an entity, totality, sum total the tongue becomes (or proves to be) the sum total of iniquity (James 3:6).
In Matthew 4:8 the Slanderer showed Jesus the kingdoms of the kosmos, in Luke, the Slanderer shows Jesus the kingdoms of the oikoumene. Putting the two together shows that the Devil was offering Jesus everything in his dominion, the physical earth and the inhabitants of it.