By G. B Keene
ISBN-10: 0486155005
ISBN-13: 9780486155005
This textual content unites the logical and philosophical points of set concept in a fashion intelligible either to mathematicians with out education in formal common sense and to logicians with out a mathematical historical past. It combines an easy point of remedy with the top attainable measure of logical rigor and precision. 1961 variation.
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Extra resources for Abstract Sets and Finite Ordinals. An Introduction to the Study of Set Theory
Sample text
V) In case v2 ε v1 sim : (iv). (vi) In case v1= v2 (T14, p. 50). (vii) In case v2 = v1 sim: (vi). (viii) In case v1 ε r (T15, p. 51). ) Where r ε v1, the required class can be proved to exist as follows: Sub-proof This completes part I of the proof of the class theorem. Proof: part II We now consider all possible cases of non-primary standard expressions. But disjunction and implication are both expressible in terms of negation and conjunction, and universal quantification is expressible in terms of negation and existential quantification.
The way in which this paradox is avoided in the present system is, in outline, as follows. We first of all reject the assumption that every predicate determines a class. But since there are many classes for which we can specify a determining predicate, a distinction is introduced between a class—the use of this term presupposing that a determining predicate is available, and a set—meaning by this, the only type of entity which a class can have as a member. e. assertions whose only variables are variables for sets) shall, in our system, operate as determining conditions for classes.
T4. For any sets c, d, the set (cd) exists. 12. Immediate Consequences of AxIII T5. Every axiom in AxIII admits a unique class. ) T6. The class ∧ exists. (The empty class) Proof T7. The class St exists. (The class of all sets) Proof T8. The class Pr exists. (The class of all pairs) Proof T9. The class /A∪B/ exists. (The sum of A and B)5 Proof T10. The class mem2B exists. (The class of pairs whose second member is in B) Proof Tll. The class exists. (The class of pairs whose first member is in A and whose second member is in B)6 Proof T12.
Abstract Sets and Finite Ordinals. An Introduction to the Study of Set Theory by G. B Keene
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