Sentence examples for the wellfounded from inspiring English sources

Indeed as far as we know we can safely add to NF some axioms to say that the wellfounded sets form a model of ZFC.

Perhaps surprisingly, given that constructions like this provide complements for all sets, they give a central rôle to the concept of wellfounded set in that the wellfounded sets of the new model can be made to be an isomorphic copy of the wellfounded sets in the structure one starts with.

(Replacement for, say, wellordered sets doesn't come free in the same way) ZF arose as an attempt to axiomatise the theory of wellfounded sets, and this principle is kept constantly in mind by the ZF-istes as they struggle to find new and more informative axioms.

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.

To the extent that mathematical concepts can be implemented in the theory of wellfounded sets (aka ZF) they can also be interpreted in NF – since NF has a theory of wellfounded sets.

The axioms of ZF can usefully be thought of as arising from an attempt to axiomatise the theory of wellfounded sets.

This section offers a quick introduction to the central parts of the theory non-wellfounded sets: what one would need to know to use the theory and to read papers on it.

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

This project has been so successful that nowadays many people believe that wellfounded sets are the only kind of set there is if it isn't wellfounded it isn't a set but a hyperset perhaps.

There is no doubt that restricting attention to wellfounded sets (in the sense that it is them alone that one tries to axiomatise) concentrated the minds of 20th century mathematicians to great effect.

Our work on bisimulation above can be used to effect a reduction of the of non-wellfounded sets to that of ordinary sets, much in the spirit of what we saw for streams and functions in Section 1.1.