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Addition of initial segments. I. (English) Zbl 0658.03030
The author investigates an extension of addition and subtraction to cuts (complete parts of the class of all natural numbers N) in the Alternative Set Theory. This extension is similarly useful as the extension of arithmetical operations to ordinal and cardinal numbers in classical set theory. The main theorem of the paper asserts that every real cut \(\sigma\) is of the form \(a+\rho\) or a-\(\rho\), where \(\rho\) is a real cut closed under addition and \(a\in N\). The cut \(\rho\) is defined from \(\sigma\) and uniquely determined, a is uniquely determined but the difference \(\in \rho\). (Remember that the system of real classes contains all definable classes - e.g. the class FN of finite natural numbers; real classes may be used in definitions as parameters, too, and all sets are real classes. On the other hand some classes obtained due to the axiom of choice as well-ordering of the universal class V or one-one mapping of a set onto V cannot be real.) An example of a (non-real) cut not having the given property is constructed. Due to the main theorem addition and subtraction of real cuts are quite easy. The author investigates the extended operations in more detail and describes some examples of cuts and formulas which do not hold, in spite of the fact that for natural numbers the formulas hold. (From another branch, let us remember that semiregular cuts introduced by Paris and Kirby are closed under addition.)
Reviewer: K.Čuda

03E70 Nonclassical and second-order set theories
03H15 Nonstandard models of arithmetic
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