Principia is nowadays only of historical interest, since the subject has developed in quite different directions from those initiated by Russell and Whitehead. The idea of basing mathematics (including the development of the usual integers, reals, function spaces) purely on “logic” has largely been abandoned in favour of set-theory based formulations. And Principia does not have a clear separation between syntax and semantics. Such a separation is essential to the development of Model Theory in the past 80 years.

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— Mathematics – Stack Exchange

— Andre Nicolas

2012.09.25 Tuesday ACHK

Universal claims about empty sets are all true, because there are no falsifying instances.

NOTE: Claims about empty sets are trivially true. Sure, “all irrational prime numbers are odd” because there are no irrational prime numbers, but it is equally true that “all irrational prime numbers are even”.

— Tutorials, PL 120 Symbolic Logic I, Fall 2011

— Professor H. Hamner Hill

2012.09.22 Saturday ACHK

Using the compactness and completeness theorems

Godel’s completeness theorem (not to be confused with his incompleteness theorems) says that a theory has a model if and only if it is consistent, i.e. no contradiction is proved by the theory. This is the heart of model theory as it lets us answer questions about theories by looking at models and vice-versa.

One should not confuse the completeness theorem with the notion of a complete theory. A complete theory is a theory that contains every sentence or its negation.

— Wikipedia on Model theory

2012.09.20 Thursday ACHK

In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms.

The semantic definition states that a theory is consistent if and only if it has a model, i.e. there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead.

The syntactic definition states that a theory is consistent if and only if there is no formula P such that both P and its negation are provable from the axioms of the theory under its associated deductive system.

If these semantic and syntactic definitions are equivalent for a particular logic, the logic is complete.[clarification needed][citation needed]

— Wikipedia on Consistency

2012.09.17 Monday ACHK