From: "Robert E. Beaudoin" Subject: Re: LINDENBAUM Algebra Date: Fri, 12 Nov 1999 21:46:37 -0500 Newsgroups: sci.math.symbolic The Lindenbaum algebra A of a theory T is the set of all equivalence classes of sentences of the language of T, under the relation that makes two sentences P and Q equivalent if and only if T |- (P <-> Q). Using the notation [P] for the equivalence class of P, one defines operations /\, \/, and ' on A by [P] /\ [Q] = [P /\ Q], [P] \/ [Q] = [P \/ Q], and [P]' = [~P] (where the /\ and \/ inside the brackets are the logical connectives "and" and "or", and ~ is the connective "not"). This makes A into a Boolean algebra whose maximum element is [P \/ ~P] and whose minimum element is [P /\ ~P] (for any P). Details should be easy to track down in most standard texts on first-order logic. For instance I think Lindenbaum algebras are mentioned (possibly in the exercises) in Chang and Keisler's _Model Theory_. This question is really a bit off-topic for sci.math.symbolic; you'd be better off posting any follow-up questions on Lindenbaum algebras, logic, and topology in sci.logic or sci.math. Robert E. Beaudoin claude govaerts wrote: > > I am interested in Topology, I found a book of Steven Vickers : Topology via > Logic > Ed. Cambridge University Press that has as prerequisite some knowledge of > LINDENBAUM Algebra. So far I couldn't find what this is all about. > Can anyone help? > Thanks and regards, > > Claude. ============================================================================== From: Dave Rusin Subject: Re: LINDENBAUM Algebra Date: Fri, 12 Nov 1999 03:00:27 -0600 (CST) Newsgroups: sci.math.symbolic To: claude.govaerts@iping.be [bad address] In article <80efno$bes$3@news3.Belgium.EU.net> you write: >I am interested in Topology, I found a book of Steven Vickers : Topology via >Logic >Ed. Cambridge University Press that has as prerequisite some knowledge of >LINDENBAUM Algebra. So far I couldn’t find what this is all about. I never heard of it before, but a search for "Anywhere = LINDENBAUM Algebra" found hundreds of matches in MathSciNet (Math Reviews). This is definitely a part of logic; its connections with mainstream topology are small. Here is one of the references: 84g:01045 Surma, Stanis\l aw J. On the origin and subsequent applications of the concept of the Lindenbaum algebra. Logic, methodology and philosophy of science, VI (Hannover, 1979), pp. 719--734, Stud. Logic Foundations Math., 104, North-Holland, Amsterdam-New York, 1982. (Reviewer: Siegfried Gottwald) 01A60 (01A70 03-03 03G05) This might also be helpful: 20 #6353 02.00 Rasiowa, H.; Sikorski, R. On the isomorphism of Lindenbaum algebras with fields of sets. Colloq. Math. 5 1958 143--158. If $S$ is a logical system based on the first order predicate calculus, then a choice of a set $A$ of certain well-formed formulas to be axioms yields an elementary theory $S(A)$. If $C\sb n(A)$ denotes the theorems of the theory, one can define an equivalence relation of the set $W$ of well-formed formulas of $S\colon a\sim b$ if and only if $a\rightarrow b$ and $b\rightarrow a$ are both in $C\sb n(A)$ for all $a,b\in W$. The Lindenbaum algebra $L(A)$ of $S(A)$ is the algebra obtained by regarding $W$ as an abstract algebra and forming the equivalence classes relative to the equivalence just described. This paper describes conditions under which $L(A)$ is representable as a field of sets. A theorem due to Rieger [Fund. Math. 38 (1951), 35--52; MR 14, 347] is generalized to the case card $W>\boldsymbol\aleph\sb 0$. dave