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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 12th 2013
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   Author: Todd_Trimble
   Format: MarkdownItexWrote [[recursive subset]] and [[partial recursive function]]. Not much more than stubs.

   Wrote recursive subset and partial recursive function. Not much more than stubs.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeAug 12th 2013
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   Author: Urs
   Format: MarkdownItexThanks! I have added more hyperlinks, have cross-linked the two entries, and have cross linked with entries such as _[[recursion]]_, _[[set]]_, _[[function]]_, _[[definable set]]_.

   Thanks! I have added more hyperlinks, have cross-linked the two entries, and have cross linked with entries such as recursion, set, function, definable set.

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeAug 13th 2013
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   Author: TobyBartels
   Format: MarkdownItexNice! I added a couple of properties.

   Nice! I added a couple of properties.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 14th 2013
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   Author: Todd_Trimble
   Format: MarkdownItexThanks guys -- I'm still adding stuff. Partial recursive or primitive recursive functions $\mathbb{N}^k \to \mathbb{N}$ may be regarded as $k$-ary operations $k \to 1$ of a Lawvere theory. Is there any literature on algebras over these Lawvere theories?

   Thanks guys – I’m still adding stuff.

   Partial recursive or primitive recursive functions $\mathbb{N}^k \to \mathbb{N}$ may be regarded as $k$-ary operations $k \to 1$ of a Lawvere theory. Is there any literature on algebras over these Lawvere theories?

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 14th 2013
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   Author: Todd_Trimble
   Format: MarkdownItexWell -- maybe answering my own question -- this looks like it might be connected with Joyal's arithmetic universes or arithmetic pretoposes. As a first guess as to what's going on, perhaps it's true that in the Lawvere theory of primitive recursive functions (say), the generating object $1$ is a parametrized natural numbers object. And perhaps the Lawvere theory of primitive recursive functions is initial among such Lawvere theories. (Which would be a nice POV on things. But for now I'm guessing without calculating.)

   Well – maybe answering my own question – this looks like it might be connected with Joyal’s arithmetic universes or arithmetic pretoposes. As a first guess as to what’s going on, perhaps it’s true that in the Lawvere theory of primitive recursive functions (say), the generating object $1$ is a parametrized natural numbers object. And perhaps the Lawvere theory of primitive recursive functions is initial among such Lawvere theories. (Which would be a nice POV on things. But for now I’m guessing without calculating.)

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeAug 14th 2013
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   Author: Mike Shulman
   Format: MarkdownItexWild guess: could this be related to the Cockett-Hofstra notion of "Turing category"?

   Wild guess: could this be related to the Cockett-Hofstra notion of “Turing category”?

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 14th 2013
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   Author: Todd_Trimble
   Format: MarkdownItexThanks, Mike. I'm having a look.

   Thanks, Mike. I’m having a look.