[FOM] Is CH vague?

Arnon Avron aa at tau.ac.il
Thu Jan 31 06:32:02 EST 2008

> 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

Actually, all of the four notions are not altogether clear,
but with different degrees of unclarity.

  Historically, it was very unclear what are "real numbers". 
The intuition about these  creatures (that may well be legendary)
come from Geometry. However, the official definition I was
taught at my first year at the university was that of a Dedckind
cut. As such  the concept of a real number relies on the concept
of an *arbitrary* set of rationals - and here is the problem.
Since I do not believe in God or any other superhuman entities,
I could never fully understand and fully trust the notion
of an "arbitrary set of X" where X is an infinite collection.
I can well understand sets of X when they are defined by some
definite properties of elements of X (and this is, I believe,
the original notion of a "set"), but I could never understand
in what sense an "arbitrary set of X" exists (or where it exists
or how it exists). Unfortunately, "God Mind" is not a 
satisfactory answer for me.

  It should be obvious that the concept of "an arbitrary set of reals" is 
even (perhapse much) more  doubtful than that of a real number.
An arbitrary set of arbitrary sets of rational numbers???
I can only envy those who feel that they fully grasp
such creatures. I humbly admit that my own personal
finite and very limitted  brain is incapable of doing so.

Now the notion of an arbitrary function whose domain and range are sets
of real numbers goes even further, but at this point I do not
really feel worse about it than I feel about the previous one.
Committing a first crime is very difficult. 
Comitting the second - still problematic, though easier.
But after that comitting more and more primes cause 
no real further mental difficulties.
> 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.

They have also been cheerfully working for many decades
without caring a bit about FOM - and with no difficulties. So what?
Should we forget about FOM because of this? 

 And in ancient Rome people cheerfully work for centuries
with their sets of gods - and they strongly believe 
that difficulties face only those who ignore their gods.
What does this prove about the existence of Jupiter and Venus? 

 When I read this type of arguments I again and again realize
that something was very wrong in the mathematical education
I got. I was taught that mathematics is characterized by
truth and rigor.  However, it seems that my teachers forgot to 
mention to me that cheerful work is an important criteria for
both either.

> 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.

Again it is obvious that something is very wrong with my way
of thinking. I always thought that the confidence in the
validity of a mathematical proof can never be greater than 
that of the assumptions on which that proof is based. But it seems
that now the logic goes in the opposite direction. Now it is
taken as *given* and beyond doubt that some proof is correct,
and from this one infers the truth of its 
underlying assumptions! I feel depressed for being unable
to accept this new approach to mathematical rigor and logic.

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