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The Elementary Theory of the Category of Sets , or ETCS for short, is an axiomatic formulation of set theory in a category-theoretic spirit. As such, it is the prototypical structural set theory. Proposed shortly after ETCC in (Lawvere 65) it is also the paradigm for a categorical foundation of mathematics.^{1}
The theory intends to capture in an invariant way the notion of a (constant) ‘abstract set’ whose elements lack internal structure and whose only external property is cardinality with further external relations arising from mappings. The membership relation is local and relative i.e. membership is meaningful only between an element of a set and a subset of the very same set. (See Lawvere (1976, p.119) for a detailed description of the notion ‘abstract set’.^{2} ^{3} ^{4} ^{5})
More in detail, ETCS is a first-order theory axiomatizing elementary toposes and specifically those which are well-pointed, have a natural numbers object and satisfy the axiom of choice. The theory omits the axiom of replacement, however.
The idea is, first of all, that much of traditional mathematics naturally takes place “inside” such a topos of constant sets, and second that this perspective generalizes beyond ETCS proper to toposes of variable and cohesive sets by varying the axioms: for instance omitting the well-pointedness and the axiom of choice but adding the Kock-Lawvere axiom gives a smooth topos inside which synthetic differential geometry takes place.
That is, ETCS locates the category of sets by the well-pointedness axiom as the discrete zero point on a ‘continuous’ range of toposes eligible for foundations. In particular, whereas ZF mainly provides ‘substance’ for mathematics, ETCS lives as a special type of form within the continuum of mathematical form itself.
The axioms of ETCS can be summed up in one sentence as:
The category of sets is the topos which
is a well-pointed topos
has a natural numbers object
and satisfies the axiom of choice.
For more details see
Erik Palmgren (Palmgren 2012) has a constructive predicative variant of ETCS, which can be summarized as:
$Set$ is a well-pointed $\Pi$-pretopos with a NNO and enough projectives (i.e. COSHEP is satisfied). Here “well-pointed” must be taken in its constructive sense, as including that the terminal object is indecomposable and projective.
Modern mathematics with its emphasis on concepts from homotopy theory would more directly be founded in a similar spirit by an axiomatization not just of elementary toposes but of elementary (∞,1)-toposes. This is roughly what univalent homotopy type theory accomplishes – for more on this see at relation between type theory and category theory – Univalent HoTT and Elementary infinity-toposes.
Instead of increasing the higher categorical dimension (n,r) in the first argument, one may also, in this context of elementary foundations, consider raising the second argument. The case $(2,2)$ is the elementary theory of the 2-category of categories (ETCC).
Todd Trimble has a series of expositional writings on ETCS which provide a very careful introduction and at the same time a wealth of useful details.
Todd Trimble, ZFC and ETCS: Elementary Theory of the Category of Sets (nLab entry, original blog entry)
Todd Trimble, ETCS: Internalizing the logic (nLab entry, original blog entry)
Todd Trimble, ETCS: Building joins and coproducts (nLab entry, original blog entry)
ETCS grew out of Lawvere’s experiences of teaching undergraduate foundations of analysis at Reed college in 1963 and was originally published in
A more or less contemporary review is
A longer version of Lawvere’s 1965 paper appears in
An undergraduate set-theory textbook using it is
Lawvere explains in detail his views on constant and variable ‘abstract sets’ on pp.118-128 of
See also ch. 2,3 of
On the anticipation of ‘abstract sets’ in Cantor:
A short overview article on ETCS:
An insightful and non-partisan view of ETCS can be found in a section of:
An extended discussion from a philosophical perspective is in
For a more recent review from a critical perspective containing additional recent references see
An informative discussion of the pros and cons of Lawvere’s approach can be found in
Palmgren’s ideas can be found here:
For the relation between the theory of well-pointed toposes and weak Zermelo set theory as elucidated by work of J. Cole, Barry Mitchell, and G. Osius in the early 1970s see
Peter Johnstone, Topos Theory , Academic Press New York 1977 (Dover reprint 2014). (sections 9.2-3)
Barry Mitchell, Boolean Topoi and the Theory of Sets , JPAA 2 (1972) pp.261-274.
Gerhard Osius, Categorical Set Theory: A Characterization of the Category of Sets , JPAA 4 (1974) pp.79-119.
For a comparative discussion of its virtues as foundation see foundations of mathematics , the texts by Todd Trimble or the informative paper by McLarty (2004). ↩
It has been pointed out by John Myhill that Cantor’s concept of ‘cardinal’ as a set of abstract units should be viewed as a structural set theory and a precursor to Lawvere’s concept of an ‘abstract set’. This view is endorsed and expanded in Lawvere 1994. ↩
R. Dedekind's views are also anticipating ‘abstract sets’ e.g. Bernstein reports in Dedekind’s works vol.3 (1932, p.449) that Dedekind gave as his intuition of a set: “a closed bag, containing determinate things that one can not see and of which one knows nothing beyond their existence and determinateness”. ↩
The first axiomatic set theory without primitive membership relation $\in$ was presumably proposed by A. Schoenflies in 1920: he modeled elements of sets as indecomposable subsets. See A. Schoenflies, Zur Axiomatik der Mengenlehre , Math. Ann. 83 (1921) pp.173-200; and Bemerkung zur Axiomatik der Grössen und Mengen , Math. Ann. 85 (1922) pp.60-64. ↩
The first axiomatic set theory based on the notion of function was von Neumann’s 1925 version of what later became the set based NBG theory of classes. ↩