By Professor Hao Wang (auth.)
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Extra resources for Computation, Logic, Philosophy: A Collection of Essays
The matter of constructing an exact theory of (say) probability contains an additional factor. Since ordinary language is not exact, new words are coined or ordinary words are given technical usage. In order to evaluate the theory, you have first to understand it. In order to understand it, you have first to learn a new language. Since it is usually impossible to explain clearly and exactly even the technical usages, a formal or 10 Computation, Logic, Philosophy exact theory can almost always be defended against charges that it does not conform to fact.
Zil in HB or Heyting's system), since the functions which are provably general recursive in this system can be enumerated and the diagonal argument can be applied to get a new general recursive function. , on p. 82 and p. 101 of Kleene 1952). Second, if one looks at ordinary informal proofs, one can distinguish two classes. There are many proofs ("purely arithmetical" ones), where effective majorizing functions f3(m) can be extracted by the substitution method of HE. For example, by Ackermann 1940, if (m) (En) R (m,n), R primitive recursive, is provable in ZIl' then a transfinite recursive function f3(m) of order 80 (compare below) can be extracted from the proof such that (m) (En) [n < f3(m)& R (m,n)] holds.
It is also known that theoretically we can rearrange the natural numbers to get a well-ordered set of any ordinal type of the second number class. For example, if we put all even numbers after all odd numbers, we get a set whose ordinal number is w2. Or again, we can rearrange natural numbers to get a set with the first c-number as ordinal type in the following manner. Let Ai be the class of all positive integers divisible by 2i but not by 2 i + 1 (i = 1,2,3, ... ), VI (aj,az) be 2a1 -1 (2az + 1), and vi + I (a'" a· ) be 2v ;(a" ...