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By Professor Hao Wang (auth.)

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They all can be considered as results of formalization or abstraction which serve as tools of thinking and research. As tools they help to economize our thought, as is often remarked. , all form groups; anything that we prove about groups in general, of course, applies also to the special groups which may differ from one another in many respects. Similarly, there are many different chairs which can all be employed to support buttocks. In this way formalization, closely tied up with abstraction, produces useful tools.

Xn) - o-[p(Xl> ... ,xn ,O), Xl> ... ,xn ,O]. 0- Here we have a typical case where, although we know, by (VI--c), that the process of computation will eventually terminate for any given argument values Xl, ••• X m we have no idea how soon that will happen. The condition (VI--c) in general gives us no information as to how big a number y could satisfy the equation p(Xl> ... ,x"y) = 0, for given Xl> ••• ,Xn. In other words, when the schema (VI) is involved in the definition of a certain function, the function is computable but not necessarily effectively computable.

Thus, there is Godel's famous theorem according to which, for any fairly rich 8 Computation, Logic, Philosophy system, we can find some property expressible in the system such that we can prove for each of the integers 1, 2, ... that it has the property, but we cannot prove the general statement that all positive integers have the property in question. , 1 has the property P), P(2), P(3), ... are all true, then it must be the case that all positive integers have the property P; yet in no fairly strong logistic system can we formalize adequately this intuition so as to guarantee the performability of such an inference for all the properties P expressible in the system.

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