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