Logic

Come, Let Us Reason: An Introduction to Logical Thinking by Norman L. Geisler, Ronald M. Brooks

By Norman L. Geisler, Ronald M. Brooks

The suitable introductory textbook, this simplified learn of good judgment prepares readers to cause thoughtfully and to identify illogic in a controversy.

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But no metaphysics is involved, only ordinary mathematics. ” Note that this use of “true” extends to the axioms of a theory. It is sometimes thought, when “true” is used in some philosophical sense, that the axioms of a theory cannot be described as true, since they constitute the starting point that determines what is meant by “true” in later discourse. All such philosophical ideas are irrelevant to the mathematical use of the word “true” explained above, which will be adhered to throughout the book when speaking of mathematical statements as true or false.

The Incompleteness Theorem: An Overview arithmetical statements as being true or false. Very often in discussions of the incompleteness theorem it is regarded as unclear what might be meant by saying that an arithmetical statement which is undecidable, say in PA, is true. What, for example, are we to make of the reflection that the twin prime conjecture may be true, but undecidable in PA? In saying that the twin prime conjecture may be true, do we mean that it may be provable in some other theory, and if so which one?

The most tedious part of a formal treatment of the incompleteness theorem consists in defining a G¨ odel numbering and in showing that “n is the G¨odel number of a proof in S of the sentence with G¨odel number m” can be defined in the language of arithmetic. It is required that the G¨ odel number of any sentence or sequence of sentences can be mechanically computed, and that computable properties of syntactic objects correspond to computable properties of G¨ odel numbers. Details regarding G¨ odel numberings are just the kind of technicality that will be avoided in this book, although an example of a G¨ odel numbering will be given in Chapter 3.

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