**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. R

_{1}R_{2}R_{3}R_{4}… in base 2, then**R is dyadic = Conv(n)(R**

_{n}=0)
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