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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 12th 2010
   • (edited Jul 12th 2010)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexBegan [Freyd cover](http://ncatlab.org/nlab/show/Freyd+cover). What's it for?

   Began Freyd cover. What’s it for?

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 12th 2010
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI'm by no means an expert on this, but some of the original applications were to higher-order logic. The story goes that Lambek was giving a talk at a conference and was presenting results on the initial topos, such as the external projectivity of the terminal object. The proof was by an inductive syntactic analysis. Almost immediately after the presentation, Freyd provided a much shorter (two-line) model-theoretic proof using the gluing construction. For details on this, see section 7.7 of Paul Taylor's book, which ought to be Labbified.

   I’m by no means an expert on this, but some of the original applications were to higher-order logic. The story goes that Lambek was giving a talk at a conference and was presenting results on the initial topos, such as the external projectivity of the terminal object. The proof was by an inductive syntactic analysis. Almost immediately after the presentation, Freyd provided a much shorter (two-line) model-theoretic proof using the gluing construction. For details on this, see section 7.7 of Paul Taylor’s book, which ought to be Labbified.