[FOM] Alternative foundations?

[henk]

In this thread one question has not been asked and has some urgency.

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What is the intended aim of a foundation of mathematics?
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Only then (from the answers) we can formulate the degree of their
successes.

For example, Girard sees the role of set theory as a way of showing
that the different niches of mathematics together are consistent.

The proof-checking community sees a foundation as a system
in which the different niches of mathematics can be actually
formalized in order to provide a machine verification of results.

There are surely more of such intended aims.

Henk Barendregt