Final answer:
Proof-theoretic ordinals are significant because they express the strength of mathematical theories, indicating the level of infinitary reasoning that a system can support.
Step-by-step explanation:
According to the passage, the significance of proof-theoretic ordinals in mathematics is that they express the strength of mathematical theories. Proof-theoretic ordinals provide a way to understand the power of a mathematical system, essentially indicating the levels of infinitary reasoning that the system can support. This can be seen as the system’s ability to define and prove the existence of certain mathematical entities.
The concept of ordinal analysis in proof theory evaluates how complex or strong a system is based on the ordinals that can be constructed within that system. This doesn't directly measure the difficulty of mathematical problems, determine the relevance of mathematical concepts, nor does it establish the foundational logic of mathematics. However, it plays a crucial role in logic and the foundations of mathematics by providing insight into the capabilities and limitations of various mathematical systems.