It's not saying you need rewards that are infinite in value per se. It's saying that you might have more situations that need to be differentiated from each other than there are real numbers. For instance, you might need to have one set of rewards that map to the real numbers, and then another set of rewards that also map to the real numbers when compared to themselves, but are all considered strictly greater than the first set.
It's assserting that there's a need to differentiate more situations from each other than there are real numbers, and basing its conclusions on the assertion, but it is not providing a convincing basis why that assertion/assumption/hypothesis is true.
The author demonstrates that a system without such a capability would not be able to solve a certain set of problems. However, going from that claim to a claim that this capability is needed for human level AGI is a non-sequitur - there is no evidence that such a capability is needed for human-level intelligence, and there's no evidence (at least not mentioned in the paper) that humans have the exact capability described.
It's not saying you need rewards that are infinite in value per se. It's saying that you might have more situations that need to be differentiated from each other than there are real numbers. For instance, you might need to have one set of rewards that map to the real numbers, and then another set of rewards that also map to the real numbers when compared to themselves, but are all considered strictly greater than the first set.