Here's a terminological question: The Łoś–Vaught theorem makes a claim about models whose cardinality is at least the cardinality of the language. In class today we talked about a language whose signature is \((\emptyset,\{<,=\}, \{(<,2), (=,2)\}\), and so I would have thought that its cardinality was \(\aleph_{0}\). However, Prof. Buss said that its cardinality was 1. I can't find the definition of a language's cardinality in any of the notes, so could someone clear this up for me?
The cardinality of a language L is the number of function symbols, predicate symbols, and constant symbols. We frequently worked with \(\max\{ |L|, \aleph_0 \} \), since this is the cardinality of the set of L-formulas. (At least, if you restrict to a set of variables of this cardinality.)
Here's a terminological question: The Łoś–Vaught theorem makes a claim about models whose cardinality is at least the cardinality of the language. In class today we talked about a language whose signature is \((\emptyset,\{<,=\}, \{(<,2), (=,2)\}\), and so I would have thought that its cardinality was \(\aleph_{0}\). However, Prof. Buss said that its cardinality was 1. I can't find the definition of a language's cardinality in any of the notes, so could someone clear this up for me?
ReplyDeleteThe cardinality of a language L is the number of function symbols, predicate symbols, and constant symbols. We frequently worked with \(\max\{ |L|, \aleph_0 \} \), since this is the cardinality of the set of L-formulas. (At least, if you restrict to a set of variables
ReplyDeleteof this cardinality.)