Magari algebra




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A diagonizable or Magari algebra is a Boolean algebra together with a modal operator internalizing the diagonalization properties involved in Gödel's second incompleteness theorem.


A Magari algebra is a pair (M,)(M,\diamond) where MM is a Boolean algebra and :MM\diamond:M\to M is an operator satisfying:

  1. =\diamond\bot=\bot and (xy)=xy\diamond (x\vee y) =\diamond x\vee \diamond y .

  2. x=(x¬x)\diamond x=\diamond (x\wedge \neg\diamond x).


Suppose that TT is a ‘Gödelian’ theory containing enough arithmetics in order to admit a formula Cons\mathsf{Cons} internalizing consistency. Cons\mathsf{Cons} induces an operator T\diamond_T on the Lindenbaum-Tarski algebra of provable equivalence classes of formulas of TT by [φ][Cons(T+φ)][\varphi]\to [\mathsf{Cons}(T+\varphi)].

Then the condition M2 for T\diamond_T can be viewed as expressing the non-provability of the consistency of the theory T=T+φT'=T+\varphi since Cons(T)\mathsf{Cons}(T') says that TT' is consistent whereas Cons(T+¬Cons(T))\mathsf{Cons}(T'+\neg\mathsf{Cons}(T')) says that T+¬Cons(T)T'+\neg\mathsf{Cons}(T') is consistent i.e. not TCons(T)T'\vdash\mathsf{Cons}(T') (For more details see Beklemishev-Gabelaia 2012).


From M1 it follows that \diamond is monotone: xyx\leq y implies xy\diamond x\leq \diamond y.

Moreover, \diamond satisfies the following transitivity property:

xx,xM.\diamond\diamond x\leq\diamond x\quad ,\forall x\in M.

To see this, consider xxx=(xx)\diamond\diamond x\leq\diamond\diamond x\vee\diamond x=\diamond (x\vee \diamond x) where the last equation follows from the commutativity of \vee together with M1.

Set y=xxy=x\vee\diamond x whence sofar we have got : xy\diamond\diamond x\leq\diamond y. Now we want to show that yx\diamond y\leq\diamond x:

From M2 plus the definition of yy: y=(y¬y)=((xx)¬y)=((x¬y)(x¬y))\diamond y=\diamond(y\wedge\neg\diamond y)=\diamond((x\vee\diamond x)\wedge \neg\diamond y)=\diamond((x\wedge\neg\diamond y)\vee(\diamond x\wedge\neg\diamond y)). But x¬y=\diamond x\wedge\neg\diamond y=\bot since the expanded left side is a \wedge containing a factor x¬x\diamond x\wedge\neg\diamond x. Whence ((x¬y)(x¬y))=((x¬y))=(x¬y)\diamond((x\wedge\neg\diamond y)\vee(\diamond x\wedge\neg\diamond y))=\diamond((x\wedge\neg\diamond y)\vee\bot)=\diamond(x\wedge\neg\diamond y) but (x¬y)x\diamond(x\wedge\neg\diamond y)\leq\diamond x since x¬yxx\wedge\neg\diamond y\leq x and \diamond is monotone. Whence all in all we get yx\diamond y\leq\diamond x and we are done.



  • L. Beklemishev, D. Gabelaia, Topological interpretations of provability logic , arXiv:1210.7317 (2012). (abstract)

  • A. Macintyre, H. Simmons, Gödel’s diagonalization technique and related properties of theories , Coll. Math. 28 no.2 (1973) pp.165-180. (pdf)

Revised on July 16, 2017 13:56:58 by Refurio Anachro? (