232 A History ofMathematics
solvability of all problemsas formulated by Hilbert in 1900 is equivalent to the logical Principle of
the Excluded Middle; therefore, since there are no sufficient grounds for this axiom and since logic
is based on mathematics—and not vice versa—the use of the Principle of the Excluded Middle
isnot permissibleas part of a mathematical proof. The Principle of the Excluded Middle has only
scholastic and heuristic value, so that theorems that in their proof cannot avoid the use of this
principle lack all mathematical content.Appendix C. Hilbert’s programme
(Hilbert, ‘The New Grounding of Mathematics’, in Mancosu 1998, p. 204)But we can achieve an analogous point of view if we move to a higher level of contemplation, from
which the axioms, formulae, and proofs of the mathematical theory are themselves the objects of
a contentual investigation. But for this purpose the usual contentual ideas of the mathematical
theory must be replaced by formulae and rules, and imitated by formalisms. In other words, we
need to have a strict formalization of the entire mathematical theory, inclusive of its proofs, so
that—following the example of the logical calculus—the mathematical inferences and definitions
become a formal part of the edifice of mathematics. The axioms, formulae, and proofs that make
up this formal edifice are precisely what the number-signs were in the construction of elementary
number theory...and with them alone, as with the number-signs in number-theory, contentual
thought takes place—that is, only with them is actual thought practiced. In this way the contentual
thoughts (which of course we can never wholly do without) are removed elsewhere—to a higher
plane, as it were; and at the same time it becomes possible to draw a sharp and systematic distinction
in mathematics between the formulae and formal proofs on the one hand, and the contentual ideas
on the other.
In the present paper my task is to show how this basic task can be carried out in a rigorous and
unobjectionable manner, and to show that our problem of proving the consistency of the axioms
of arithmetic and analysis is thereby solved.Solutions to exercises
- (a) We have already done this—the definition of ‘
√
2’ works for the square root of any number.
(That is, except when the number is already a square; then of course one defines (for example)√
4 to be 2, which is already rational, and not a problem.)
(b) First we must define the product of two positive real numbers. We use the upper classes,
otherwise we keep having to make special provision for the negative numbers. LetUandU′
(sets of rational numbers) be the upper classes belonging to the positive real numbersaanda′.
Then all numbers in each of these are positive. DefineVto be the set of allzsuch thatz≥uu′
for someu∈U,u′∈U′. (This is complicated, but ensures thatVobeys the rules for an upper
class; that is, ifz∈Vandz′>z, thenz′∈V.) By definition,aa′is the real number whose upper
class isV.
[What happens whenaa′is rational—for example,πand 2/π?]
Now letVbe the upper class for√
2.
√
3, andV′the upper class for√
6; we have to show they
are the same. Ifz∈V,z≥aa′, wherea^2 >2 and(a′)^2 >3. Soz^2 >a^2 (a′)^2 >6, and clearly