[FOM] Re: The Search for Mathematical Roots

Schaefer, Marcus MSchaefer at cti.depaul.edu
Mon Mar 8 22:45:58 EST 2004

Let me retract the following quote:
> > "He also proved here the existence of irrational numbers by
> a lovely
> > reductio argument that has never gained the attention that it
> > deserves: assume that the equation t^2 = Du^2 in integers (D not a 
> > square) has a solution, and let u be the smallest integer involved; 
> > then exhibit a smaller integer also to satisfy the equation, a 
> > contradiction which establishes sqrt(D) as irrational."

When reading this, I got the impression the author was
not aware that the result was more than 2000 years old 
and covered in every other elementary textbook. However, 
in all fairness, the paragraph is probably about 
Dedekind's proof (which is online at
rather than the result. So my reading says more about my trust in 
the author than anything else, and I apologize for my mistake.

However, let me add (as a replacement) a new
quote I just came across (Section 5.2.3). In full
view of Peano's induction axiom, transcribed here into ASCII:

  9. (k in K and 1 in k and (x in N and x in k =>_x  x+1 in k)) => N
contained in k

the author states

"Secondly, the induction axiom ... was stated in first-order form with
no quantification over K; and the universal quantification over x
characterizes it as of the strong form (in modern terminology) in
involving all integers preceding x."

In explanation of the notation in the axiom (and the transcription): 
K is the class of all classes (so the author must have meant
over k"); N denotes the natural numbers, and =>_x is Peano's (and later
Russell/Whitehead's) notation for universal quantification over x. We 
would now write (for all x: if x in N and x in k => x+1 in k).


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