By Klaus Metsch

A well-known theorem within the thought of linear areas states that each finite linear house has at the very least as many traces as issues. This results of De Bruijn and Erd|s resulted in the conjecture that each linear house with "few strains" canbe got from a projective aircraft through altering just a small a part of itsstructure. Many effects with regards to this conjecture were proved within the final two decades. This monograph surveys the topic and provides a number of new effects, equivalent to the new facts of the Dowling-Wilsonconjecture. ordinary tools utilized in combinatorics are built in order that the textual content could be understood with out an excessive amount of historical past. therefore the publication might be of curiosity to anyone doing combinatorics and will additionally support different readers to benefit the suggestions utilized in this actual field.

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**Example text**

Thus we should have two different numbers with the same successor. This failure of the third axiom cannot arise, however, if the number of individuals in the world is not finite. 2 Assuming that the number of individuals in the universe is not finite, we have now succeeded not only in defining Peano’s [page 25] three primitive ideas, but in seeing how to prove his five primitive propositions, by means of primitive ideas and propositions belonging to logic. It follows that all pure mathematics, in so far as it is deducible from the theory of the natural numbers, is only a prolongation of logic.

In the case of an assigned number, such as 30,000, the proof that we can reach it by proceeding step by step in this fashion may be made, if we have the patience, by actual experiment: we can go on until we actually arrive at 30,000. e. by proceeding from 0 step by step from each number to its successor. Is there any other way by which this can be proved? Let us consider the question the other way round. What are the numbers that can be reached, given the terms “0” and [page 21] “successor”? Is there any way by which we can define the whole class of such numbers?

We have thus reduced Peano’s three primitive ideas to ideas of logic: we have given definitions of them which make them definite, no longer capable of an infinity of different meanings, as they were when they were only determinate to the extent of obeying Peano’s five axioms. We have removed them from the fundamental apparatus of terms that must be merely apprehended, and have thus increased the deductive articulation of mathematics. ” How stands it with the remaining three? It is very easy to prove that 0 is not the successor of any number, and that the successor of any number is a number.