Replacing Axiom Five by an axiom schema one obtains a ( weaker ) first-order theory called " Peano Arithmetic ."
42.
In the presence of the axiom schema of separation, Russell's paradox becomes a proof that there is no set of all sets.
43.
With the axiom schema of specification instead of unrestricted comprehension, the conclusion doesn't hold and, hence is not a logical consequence.
44.
A different objection put forth by Henri Poincar?is that defining sets using the axiom schemas of circularity, into the definitions of mathematical objects.
45.
An appealing but somewhat mysterious feature of NBG is that its axiom schema of Class Comprehension is equivalent to the conjunction of a finite number of its instances.
46.
If the new axiom schema is not a tautology, then every formula becomes a theorem ( which makes the concept of a theorem useless in this case ).
47.
Now, axioms are particular formul?( or, for axiom schemas, rules for constructing formul?) while inference rules describe how to transform formul?into other formul?
48.
Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results.
49.
If no element E of A satisfies \ theta ( E ) then the set B desired by the relevant instance of the axiom schema of separation is the empty set.
50.
The axiom schema of specification states that this subset always exists ( it is an axiom " schema " because there is one axiom for each \ phi ).
