[FOM] Is CH vague?

Martin Davis martin at eipye.com
Mon Jan 28 22:01:10 EST 2008

Charles Silver says that Sol Feferman has convinced him that CH is 
vague and asks: "what's the argument that it's not?"

Here is one of the many equivalent form of CH:

For every uncountable set of real numbers, there is a function that 
maps it *onto* the set of all real numbers.

If this statement is vague it must be because there is something 
unclear about one of the concepts with which it deals. These concepts are:
1. real number
2. set of real numbers
3. uncountable set
4. function whose domain and range are sets of real numbers
Mathematicians (Weyl, Borel) famously did worry about these concepts 
in the early years of the 20th century. But analysts have been 
cheerfully working with them for many decades with no dificulties.

Now there are constructivists and predicavists (we hear from some of 
them on FOM) who demand in the one case that abstract objects be 
given constructively, and in the other, that they at least not 
require impredicative definitions. From my point of view (and I 
believe that of the great majority of working mathematicians) either 
view imposes an unnatural straitjacket on mathematics.

In the famous article by Feferman to which a number of FOMers have 
referred in which he expresses his view that CH is indeed inherently 
vague, he also points to the recent success with Fermat's Last 
Theorem as an indication that "hard work" rather than new axioms are 
what is needed for mathematical progress. I therefore find it ironic 
that the infinitary notions in terms of which the proof of FLT is 
cast go far beyond the concepts 1-4 above.


                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
                          (Add 1 and get 0)

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