On Diagonal Quantifiers; 6 of 9

6.    Diagonal Quantifiers in Higher Math

Here are diagonal-quantifier versions of mathematical induction:

                    All(n)( P(n) iff P(n+1) )       =      Con(n)( P(n) )

                   Var(n)( P(n) )                        =      Some(n)( P(n) xor P(n+1) )

          On the integers, the iffs and xors of ollerno and sumbunol need only be between elements separated by adding one. The integers are deductively linked by succession.

          In nonstandard analysis, where there are infinitesimal quantities, you can express the intermediate value theorem in Wilsonian terms:

          If f(x) is continuous on [a,b], and i is any infinitesimal, then

          Con(x)( f(x)>0 )           =       All(x) ( f(x)>0   iff  f(x+i)>0 )

          f’s sign is constant if it is constant under any infinitesimal change.

          Var(x)( f(x)>0 )           =       Some(x) ( f(x)>0   xor  f(x+i)>0 )

          f’s sign varies if it varies under some infinitesimal change.

          Conditional equations are variable, singular equations are constant:

          Var(x)(x=1)

          Con(x)(x=x+1)

          Con(x)(x=x)

          If a real number R equals 0. R₁ R₂ R₃ R₄ … in base 2, then

          R is dyadic                   =       Conv(n)(R_n=0)