Proofs Of Infinity Research Articles

HOW DID FREGE FALL INTO THE CONTRADICTION?

AbstractQuine made it conventional to portray the contradiction that destroyed Frege's logicism as some kind of act of God, a thunderbolt that descended from a clear blue sky. This portrayal suited the moral Quine was antecedently inclined to draw, that intuition is bankrupt, and that reliance on it must therefore be replaced by a pragmatic methodology. But the portrayal is grossly misleading, and Quine's moral simply false. In the person of others – Cantor, Dedekind, and Zermelo – intuition was working pretty well. It was in Frege that it suffered a local and temporary blindness. The question to ask, then, is not how Frege was overtaken by the contradiction, but how it is that he didn't see it coming. The paper offers one kind of answer to that question. Starting from the very close similarity between Frege's proof of infinity and the reasoning that leads to the contradiction, it asks: given his understanding of the first, why did Frege did not notice the second? The reason is traced, first, to a faulty generalization Frege made from the case of directions and parallel lines; and, through that, to Frege's having retained, and attempted incoherently to combine with his own, aspects of a pre‐Fregean understanding of the generality of logical principles.

Final value theorem in number sequences

By definition in sequence number theory, the most primitive entity is the natural number sequence. Numbers are always viewed as members of global sequences. Number sequences can be classified into two primitivity classes, namely the primary sequences and the secondary sequences. Many unsolved number theoretic problems are connected with the search for rigorous proofs of finity or infinity of number sequences. Infinities in most sequences in the primary class have already been proved but unsolved problems occur in the secondary class. With the development of generating functions in the z-domain, the possibilities of applying the final value theorem on number sequences look promising but precaustions must be taken in its applications as there are pitfalls. Breakthroughs in proofs of infinities in Mersenne number and Fermat number sequences are found. However, there are sequences in the secondary category which still do not yield to this method of proof. This note explains why.