wellfounded

He thought that non-wellfounded sets should be called by a name that reflected the change in conception, and he proposed calling them hypersets in parallel to the hyperreal numbers of non-standard analysis.

Non-wellfounded Set Theory Moss, Lawrence S. 2008

A hyperset or non-wellfounded set is a set that is obtained by decorating an arbitrary graph.

Non-wellfounded Set Theory Moss, Lawrence S. 2008

The study of non-wellfounded sets proposes to treat every graph as a picture of a unique set.

Non-wellfounded Set Theory Moss, Lawrence S. 2008

The term non-wellfounded set refers to sets which contain themselves as members, and more generally which are part of an infinite sequence of sets each term of which is an element of the preceding set.

Non-wellfounded Set Theory Moss, Lawrence S. 2008

Discussion of such sets is very old in the history of set theory, but non-wellfounded sets are ruled out of Zermelo-Fraenkel set theory (the standard theory) due to the Foundation Axiom (FA).

Non-wellfounded Set Theory Moss, Lawrence S. 2008

With wellfounded extensional relations one does not need to make the pointing explicit, since the wellfoundedness ensures that at most one element of the carrier set can satisfy the accessibility condition.

Quine's New Foundations Forster, Thomas 2006

Further, these models satisfy a scheme of replacement for wellfounded sets: anything the same size as a wellfounded set is a set.

Quine's New Foundations Forster, Thomas 2006

This idea is a modern rewriting of history: Set theory did not start with the idea that all sets should be wellfounded (had it done so, the axiom scheme of naïve comprehension would never have been entertained!)

Quine's New Foundations Forster, Thomas 2006

The fact that NF refutes the axiom of choice sharpens the consistency problem greatly, and ought to sharpen public interest, but the widely-held view that sets are, of their essence, wellfounded represents a huge rhetorical obstacle to NF attracting the attention that its foundational interest merits.

Quine's New Foundations Forster, Thomas 2006

The answer is that, although it may be a mistake to think that all sets are wellfounded, it is not a mistake to think that the concept of wellfounded set is worth axiomatising.

Quine's New Foundations Forster, Thomas 2006