Skip to main content Accessibility help
×
×
Home

Lawvere's basic theory of the category of categories

  • Georges Blanc (a1) and Anne Preller (a1)

Extract

It is long known that Lawvere's theory in The category of categories as foundations of mathematics A[1] does not work, as indicated in Ishell's review [0]. Isbell there gives a counterexample that CDT—Category Description Theorem—[1, p. 15] is in fact not a theorem of BT (the Basic Theory of [1]) and suggests adding CDT to the axioms.

Our starting point was the claim in [1] that “the basic theory needs no explicit axiom of infinity.” We define a model ℳ of BT in which all categories are finite. In particular, the “monoid of nonnegative integers N” coincides in ℳ with the terminal object 1. We study ℳ in some detail in order to establish the true status of various “theorems” or “metatheorems” of BT: The metatheorem of [1, p. 11] saying that the discrete categories form a category of sets, CDT, the theorem on p. 15, and the theorem on p. 16 of [1] are all nontheorems. The remaining results indicated in [1] concerning BT are provable. However, as the Predicative Functor Construction Schema—PFCS—are justified in [1] by using the “metatheorem” and CDT, we provide a proof of these two schemata by showing that the discrete categories of BT (or of convenient extensions of BT) form a two-valued Boolean topos.

Copyright

References

Hide All
[0]Isbell, J. R., Review of [1], Mathematical Reviews, vol. 34 (1967), #7332.
[1]Lawvere, F. W., The category of categories as a foundation for mathematics, Proceedings of a Conference on Categorical Algebra (La Jolla, California, 1965), Springer, New York, 1966, pp. 120.
[2]Lawvere, F. W., An elementary theory of the category of sets, Proceedings of the National Academy of Sciences of the U.S.A., vol. 52 (1964), pp. 15061511.
[3]MacLane, S., Categories for the working mathematician, Springer, New York, 1971.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed