The irony of progress
Chesterton feels there is sacred worth in the Christian worldview, so he feels personally in his opinion that perhaps progress is ironic, because it seems that by moving past Christianity, his society is moving toward rationality, but Chesterton feels the existential weight of Christianity and it's many paradoxical riddles demands treatment. In other words, he feels to move forward might be begging the question of what is left behind.
The irony of paradox
The book begins with an astute observation about human nature. He notices that technically, the urge to go into the unknown, to explore, to be completely by adventure and journey in one's experience of life, it stands at odds with another perfectly human instinct, which is to stay and rest in one place, to be safe, and to have security. This paradox is the human experience of chaos and order, and the ironic nature of human experience could therefore be the consequence of a paradoxical reality.
Sacredness and irony
Of course if everything was sacred, that would be ironic, given the quest for sacredness seems to be object oriented in a more specific sense of the word, but the sacredness of human experience is ironically super-imposed on a normal, regular, narrative experience of life through time. This idea, that human experience necessitates religious beliefs because of its sacred nature—it is ironic.
The irony of rationality
To treat something with pure rationality seems like a valid approach, and Chesterton even defends rationality citing St Augustine and St Thomas as authorities on the subject. His opinion, however, is that rationality is an experience of reality that is too beautiful to not imply a higher design in the universe. He feels that "rationalism" is ironic if it forgets to treat the ontological question of why rationality is even possible in the first place. Of course, most Postmodernists disagree with him on this.
The irony of authority
Although he feels that the church has some valid authority, he admits that religious authority is an ironic mystery, given the true mystic nature of Christian philosophy. He feels that the human experience of religion is valid, so he feels church authority is also valid, because it preserves the traditional accounts of human experiences of the divine.
