What parts of the Inferno prove the Medieval worldview of the 12th and 13th century, specifically the "synthetic" and hierarchal ideas of this age?
The synthetic ideas of the 12th and 13th centuries focused on synthesizing Christianity with reason. Up to this point, reason and faith had been seen as antithetical, as were philosophy and theology. Thomas Aquinas began to change this assumption, arguing that reason and faith were both paths to the single truth that God exists. Dante incorporates this idea outright in Canto III, lines 1-6. The gates read:
Through me you pass into the city of woe:
Through me you pass into eternal pain:
Through me among the people lost for aye.
Justice the founder of my fabric moved:
To rear me was the task of Power divine,
Supremest Wisdom, and primeval Love.
In other words, the inscription to Hell says that God, through his divine power and supreme wisdom, created Hell because he was moved by justice. The notion of justice is based in reason: The person who wrongs another should receive appropriate punishment for that wrong. (For example, the wrathful in the fourth circle live with their anger forever--Canto VII.)
Then, throughout the book, Dante combines reason and faith in a similar way: the worse the sinner (according to Dante's religious beliefs), the "higher" the circle of hell and the worse the punishments. For example, there are the greedy and their opposites, the prodigals, in the fourth circle; the wrathful in the fifth; blasphemers and sodomites in the seventh under a constant rain of fire; thieves in the eighth circle, being bitten by serpents, and so on. He leaves those who have never heard of Christianity in the Ante-Inferno where comparatively almost no punishment exists. According to his theology, the punishment allotted to each group of sinners is in direct correlation to the sins they committed, an idea derived from the marriage of reason and faith.
The hierarchical ideas of the time included a moral heirarchy (already mentioned), a structured notion of how the universe was ordered, and the placement of secular and religious figures, including pagan. The idea that the universe was spherical (composed of concentric spheres) is reflected in Dante's layout of hell as equally concentric spheres.
Strangely, Dante seems to give precedence in his satanic hierarchy to public and often pagan figures--not Christian ones, although his Hell is largely populated with monks and bishops and popes. First, a pagan figure, Virgil, is sent from Heaven to guide him through hell. A bit of thought shows how odd this is: for a Christian writer discussing a Christian subject and essentially creating a treatise on his personal theological ideas, the inclusion of a pre-Christian pagan figure is truly strange. (Why not an angel?) However, he ranks high enough to warrant being a wise guide.
The pagans whose philosophical thought formed a basis for the Renaissance are only in Limbo, indicating their rank is high above those of the inner circles. Compare these to the popes trapped in Circle 8 (Canto XIX) for simony (abuse of power within the church) and hypocrisy (Canto XXIII). In Circle 9, we find Satan himself (imbedded in ice), chewing on three figures: Judas Iscariot, Brutus, and Cassius. The fact that there is only one religious figure here but two political ones (betrayers of Julius Caesar), suggests that Dante was just as concerned with political corruption--maybe more--as he was with religious corruption.