Why should I switch to Python? - Infinity of Primes

Moshe Zadka moshez at math.huji.ac.il
Wed Apr 5 01:55:24 EDT 2000

On 4 Apr 2000, Cameron Laird wrote:

> To what extent are constructive proofs supplanting *... ad
> absurdum* as standards?

I don't know, but the given proof is easily transformed into a
constructive proof:

Here's a way to find one more prime:

Take any n-primes, take the product, add 1, via brute force find a prime
factor. A constructive way to find in a bounded time (though long!) an
infinite sequence of primes,

Note that a similar transformation shows that the diagonal argument is
usually just as constructive.

There are some reductio ad absurdum which cannot be trnsformed thus: this
is usually because of a use of the axiom of choice. So, in a very real
sense, this al boils down to: do you believe the axiom of choice (or some
weaker variant, like the ultra-filter axiom, which is almost always
enough, or the countable axiom of choice, which is enough in many cases)?

Thus, the question was taken out of mathematics and into philosophy, and
hence cannot be answered, so the only answer I can give is my own: I
refuse to use the axiom of choice, but I do use the ultra-filter axiom.
This relates in a deep way to how I view the future of mathematics.

was-that-offtopic-enough-ly y'rs, Z.

Moshe Zadka <mzadka at geocities.com>. 
http://www.linux.org.il -- we put the penguin in .com

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