By Iron Crown Enterprises
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Extra info for Arms Companion (Rolemaster RPG)
It says: if you have a set of sets then there exists a set consisting of one member from each of these sets. And drawing it in picture form no-one can possibly argue with it, but a lot of things cannot be proved without it. For instance, if you have an infinite set it is a very old theorem that this contains a denumerable subset, and the way you count out an infinite subset is to take one member and since the set was infinite to start off with, there are more members left, so we can take another one; there are still some left, because it is infinite, so we can take a third one, and so on.
And which have function symbols and constant symbols as well. The formal system PC 1. The alphabet of PC: a denumerable set of individual variables: v1, v2, v3,… and a two-place predicate letter P. Two logical connectives: (not) and & (and). One quantifier symbol (there exists). Three improper symbols: (, ) (the left bracket, the comma, and the right bracket). 2. We use these symbols to build the (well-formed) formulae of PC, according to the following rules. (a) If x, y are individual variables, then P(x, y) is a formula of PC.
This means that we shall have to specify particular values for free variables when interpreting them. This problem does not arise for bound variables. So now here is the definition. An interpretation of the formal system PC is a structure , where U is a set (which must not be empty) the members of which are: a, a|, a||, …, and R is a relation on U. U is said to be the universe of the interpretation and we will eventually interpret the predicate letter P as R, so R will itself be two-place. Now let members of U be assigned to all the individual variables of PC in such a way that not more than one member of U is assigned to each variable of PC.
Arms Companion (Rolemaster RPG) by Iron Crown Enterprises