After joking that there would always be prayer in schools as long as there are arithmetic tests, Mr. Gore said that "in some school settings the impression is given that it's wrong if a student wants to exercise or display his or her faith or engage in a truly voluntary prayer that the school employees have nothing to do with".
Then Kurt Gödel proved that there exist arithmetical statements that are undecidable in Peano Arithmetic (Gödel 1931).
What's the arithmetic there?
The first incompleteness theorem tells us that in any consistent formal theory whose theorems are recursively enumerable and which entails a certain (rather limited) amount of arithmetic, there will be an arithmetical sentence such that neither it nor its negation is provable.
What Gödel showed is that, for any consistent, recursively axiomatized formal system, F, strong enough for arithmetic, there are truths expressible in purely arithmetical language which are not provable in F.
Mr. Smith, of New Deal, offered this simple arithmetic: There are fewer than 7 million computers in American schools for a total of 51 million students.
There is an arithmetic for cardinal numbers based on natural definitions of addition, multiplication, and exponentiation (squaring, cubing, and so on), but this arithmetic deviates from that of the natural numbers when transfinite cardinals are involved.
A study of 2,300 Italian students by three European universities found that for simple arithmetic there was little difference between right- and left-handed performance but for difficult problem-solving left-handers won out.
In addition to this, there was additional arithmetic needed to understand the movement of the car and the algorithms used to find the angle and using that to move the car closer to its final destination.
Most mathematicians had little doubt of the eventual attainment of this goal until 1931, when the Austrian-born logician Kurt Gödel demonstrated the surprising result that there must exist arithmetic propositions (or questions) that cannot be proved or disproved.
