By G. B Keene
This textual content unites the logical and philosophical facets of set thought in a fashion intelligible either to mathematicians with out education in formal good judgment and to logicians and not using a mathematical historical past. It combines an uncomplicated point of remedy with the top attainable measure of logical rigor and precision. 1961 variation.
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Additional resources for Abstract Sets and Finite Ordinals. An Introduction to the Study of Set Theory
Some” We use the sign “(∃x)” (read “for some x”) immediately to the left of a propositional function, to mean that there exists at least one value in the range of appropriate values of the variable x in that propositional function, which gives rise to a true proposition when substituted for x. For example, if we use Pr to name the class of prime numbers we can construct the propositional function: (x ε Nn) ⋅ (x ε Pr) Here, the following individuals are in the range of values concerned: 1 2 3 4 … etc.
If one is a science student and the other an arts student, the two classes will probably be entirely distinct. If both are reading the same or related subjects, on the other hand, the classes are likely to overlap; and it is just conceivable, if both are attending the same course, that the two classes may be identical. But in each case the identity or otherwise of the two classes is determined by whether or not the conditions-for-membership of the one class is fulfilled by at least one book which fails to fulfil the conditions-for-membership of the other class.
Thus, where k = 1, we have: For example, the assertion: means: Or, where k = 2, for instance, the assertion: means: In each such case, since coextensionality determines identity between classes, every authorized predicate determines a unique class. 22. Rule for Defining Individual Constants So far we have a means of indicating that a given predicate is a class condition, but no means of permanent identification of the unique class determined by that predicate. In order to allow for this, we introduce the following Rule For Defining Individual Constants4.