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Results tagged with type-theory
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user 11640
16
votes
Accepted
Why the reflection rule trivializes higher paths in Martin-Löf Extensional Type theory?
The point is that the reflection rule makes $p = \mathsf{refl}_x$ a well-formed expression. This turns out to be incredibly dangerous: now we can prove it by induction on equality.
More precisely:
…
- 15.9k
10
votes
Accepted
How should I be thinking about object classifiers / universal fibrations / universes?
Yes, the universe is the classifying space for small homotopy types. For various reasons, the universe is not itself a small homotopy type; so it fails to be an object classifier for the trivial reaso …
- 15.9k
8
votes
1
answer
638
views
The independence of path induction
In §1.12 of the Homotopy type theory book, it is mentioned that indiscernibility of identicals is a consequence of path induction. More precisely, for each type $C$ dependent over a type $A$, there is …
- 15.9k
7
votes
1
answer
204
views
Finitistic interpretation of Nelson's internal set theory
What does “standard” in internal set theory really mean?
Is it secretly a way of reconciling conventional mathematics with (ultra)finitism?
Until recently I thought “standard” was just a way of talkin …
- 15.9k