In this context, axioms contradicting the axiom of regularity are known as "'anti-foundation axioms "', and a set that is not necessarily well-founded is called a "'hyperset " '.
22.
The axiom of regularity prevents this from happening . ) The minimal set " X " satisfying the axiom of infinity is the von Neumann ordinal ?, which can also be thought of as the set of natural numbers \ mathbb { N }.
23.
The axiom of regularity ( with the axiom of pairing ) also prohibits such a universal set, however this prohibition is redundant when added to the rest of ZF . If the ZF axioms without regularity were already inconsistent, then adding regularity would not make them consistent.
24.
Comparing ZF with type theory, Alasdair Urquhart wrote that " Zermelo's system has the notational advantage of not containing any explicitly typed variables, although in fact it can be seen as having an implicit type structure built into it, at least if the axiom of regularity is included.
25.
This results from the axiom of foundation or the axiom of regularity which enacts such a prohibition ( cf . p . 190 in " Being and Event " ) . ( This axiom states that every non-empty set A contains an element y that is atheist.
26.
He also adopted the axiom of regularity, and replaced the axiom of limitation of size with the axioms of replacement and von Neumann's choice axiom . ( Von Neumann's work shows that the last two changes allow Bernays'axioms to prove the axiom of limitation of size .)
27.
The existence of Quine atoms ( sets that satisfy the formula equation " x " = { " x " }, i . e . have themselves as their only elements ) is consistent with the theory obtained by removing the axiom of regularity from ZFC . Various non-wellfounded set theories allow " safe " circular sets, such as Quine atoms, without becoming inconsistent by means of Russell's paradox.
28.
Some of " mainstream mathematics " ( mathematics not directly connected with axiomatic set theory ) is beyond Peano arithmetic and second order arithmetic, but still, all such mathematics can be carried out in ZC ( Zermelo set theory with choice ), another theory weaker than ZFC . Much of the power of ZFC, including the axiom of regularity and the axiom schema of replacement, is included primarily to facilitate the study of the set theory itself.
29.
Most of the other Zermelo Fraenkel axioms ( but not the axiom of extensionality or the axiom of regularity ) then became necessary to make up for some of what was lost by changing the axiom schema of comprehension to the axiom schema of specification each of these axioms states that a certain set exists, and defines that set by giving a predicate for its members to satisfy, i . e . it is a special case of the axiom schema of comprehension.
