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universally been assumed that with the continued development of mathematics any of these questions could in the end be answered.

However, what Gödel's Theorem shows is that there must always exist some questions that cannot ever be answered using the normal axioms of arithmetic. Yet the fact that the few known explicit examples have been extremely complicated has made this seem somehow fundamentally irrelevant for the actual practice of mathematics.

But from the discoveries in this book it now seems quite certain that vastly simpler examples also exist. And it is my strong suspicion that in fact of all the current unsolved problems seriously studied in number theory a fair fraction will in the end turn out to be questions that cannot ever be answered using the normal axioms of arithmetic.

If one looks at recent work in number theory, most of it tends to be based on rather sophisticated methods that do not obviously depend only on the normal axioms of arithmetic. And for example the elaborate proof of Fermat's Last Theorem that has been developed may make at least some use of axioms that come from fields like set theory and go beyond the normal ones for arithmetic.

But so long as one stays within, say, the standard axiom systems of mathematics on pages 773 and 774, and does not in effect just end up implicitly adding as an axiom whatever result one is trying to prove, my strong suspicion is that one will ultimately never be able to go much further than one can purely with the normal axioms of arithmetic.

And indeed from the Principle of Computational Equivalence I strongly believe that in general undecidability and unprovability will start to occur in practically any area of mathematics almost as soon as one goes beyond the level of questions that are always easy to answer.

But if this is so, why then has mathematics managed to get as far as it has? Certainly there are problems in mathematics that have remained unsolved for long periods of time. And I suspect that many of these will in fact in the end turn out to involve undecidability and

Captions on this page:

Smallest solutions for various sequences of integer (or so-called Diophantine) equations. indicates that it can be proved that no solution exists. A blank indicates that I know only that no solution exists below a billion. Methods for resolving some of the equations in the first column were known in antiquity; all had been resolved by the 1800s. Practical methods for resolving the so-called elliptic curve equations in the second column were developed only in the 1980s. No general methods are yet known for most of the other equations given—and some classes of them may in fact show undecidability.

From Stephen Wolfram: A New Kind of Science [citation]