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    • I added a proposition to this subsection which seems valid intuitionistically, but I wouldn’t mind a reality check from someone.

    • Indicated where to find (in the nLab) a proof of the equivalence with the axiom of choice and with Zorn’s lemma (<– it’s there).

      diff, v21, current

    • also created axiom UIP, just for completeness. But the entry still needs some reference or else some further details.

    • Added a bit to Hartogs number. Including the curiosity that GCH implies AC. :-)

    • Somebody should add a word on categorification to this page (just a note in case anyone cares, maybe a note to myself, but right now I am on my phone…)

      diff, v2, current

    • We don’t have anything on this, I think, but there is mention of “intrinsic” and “à la Church” at coercion. From Type refinement and monoidal closed bifibrations:

      One of the difficulties in giving a clear mathematical definition of the “topic” of type theory is that the word “type” is actually used with two very different intuitive meanings and technical purposes in mind:

      1. Like the syntactician’s parts of speech, as a way of defining the grammar of well-formed expressions.
      2. Like the semanticist’s predicates, as a way of identifying subsets of expressions with certain desirable properties.

      These two different views of types are often associated respectively with Alonzo Church and Haskell Curry (hence “types à la Church” and “types à la Curry”), while the late John Reynolds referred to these as the intrinsic and the extrinsic interpretations of types [11]. In the intrinsic view, all expressions carry a type, and there is no need (or even sense) to consider the meaning of “untyped” expressions; while in the extrinsic view, every expression carries an independent meaning, and typing judgments serve to assert some property of that meaning.

      [11] is John C. Reynolds. The Meaning of Types: from Intrinsic to Extrinsic Semantics. BRICS Report RS-00-32, Aarhus University, December 2000. pdf

      There are two very different ways of giving denotational semantics to a programming language (or other formal language) with a nontrivial type system. In an intrinsic semantics, only phrases that satisfy typing judgements have meanings. Indeed, meanings are assigned to the typing judgements, rather than to the phrases themselves, so that a phrase that satisfies several judgements will have several meanings.

      In contrast, in an extrinsic semantics, the meaning of each phrase is the same as it would be in a untyped language, regardless of its typing properties. In this view, a typing judgement is an assertion that the meaning of a phrase possesses some property.

      The terms “intrinsic” and “extrinsic” are recent coinages by the author [1, Chapter 15], but the concepts are much older. The intrinsic view is associated with Alonzo Church, and has been called “ontological” by Leivant [2]. The extrinsic view is associated with Haskell Curry, and has been called “semantical” by Leivant.

      [1] John C. Reynolds. Theories of Programming Languages. Cambridge University Press, Cambridge, England, 1998. [2] Daniel Leivant. Typing and computational properties of lambda expressions. Theoretical Computer Science, 44(1):51–68, 1986.

      Anyone have a preferred name for this distinction?

    • Page created, but author did not leave any comments.

      v1, current

    • added to identity type a mentioning of the alternative definition in terms of inductive types (paths).

    • gave this material, taken from the Examples-section at filtered topological space, its own entry

      Didn’t touch the material beyond some trivial formatting and adding more references.

      v1, current

    • Minor corrections in the statement and proof of Example 3.1.

      diff, v22, current

    • Added link to book treating the proof, by Haesemeyer and Weibel

      diff, v5, current

    • Observation that the metric topology and product topology agree.

      diff, v10, current

    • Page created, but author did not leave any comments.

      v1, current

    • starting something, not done yet, but need to save

      v1, current

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