Introduction to Metalogic: With an Appendix on by Imre Ruzsa

By Imre Ruzsa

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Extra resources for Introduction to Metalogic: With an Appendix on Type-Theoretical Extensional and Intensional Logic

Example text

3): it applies (individual) variables, names, name functors, predicates, and quantifiers. ) First-order languages mayuse different stocks of name functors and 40 predicates. We shalldealherewith the maximal first-order language which has an infinite supplyof name functors for all possible numbers of argument places, and, similarly, an infinite supplyof predicates for all possible numbers of argument places (and, of course, an infinite supplyofvariables). Thus,the alphabet we needmustbe muchricherthan JlPL.

Our next goal is to find out some new devices of presenting inductive definitions that will pave the way for some generalizations as well. We approach this goal via some examples. 3. The simplest example of an inductive definition is as follows. Given an alphabet 51, let us define the subclass A * of Base: 0 E ~0 by induction as follows: A*. Inductive rule: (x E A * & (X E51) => "xa" E A 32 *. 91.. 91. o, a v ... , an}, then we can enumerate them: X E A* :::::) X E A * : : :) "xao" E A *. "xa( E A *.

The letters are objects, thus, in enumerating them, we have to name them. For example, the two-letter alphabet whose letters are '0' and '1', is to be displayed as {CO', '1' }. Could be omitted here the quotation marks? We can answer this question by YES, agreeing that inside the brackets, the letters stand in an autonymous sense as names of themselves. However, we can give a deeper "theoretical" argument in doing so. Namely, if we do not want to use a language for communication, if we are only interested in the structure of that language then we can totally avoid the use of the letters and words of the language in question - by introducing metalanguage names for the letters and words of the object language.

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