Set Theory and Logic

The Non-uniqueness of the Set of Natural Numbers

Authors: Hongyi Li
Comments: 18 Pages.

Some counterexamples to the uniqueness of the set of natural numbers were given and the non-uniqueness of the set of natural numbers was proved on the basis of strict definition of the number of elements of any set. It was proved that the number of elements in any infinite set is more than that in its proper subset and any one-to-one correspondence cannot be established between two sets with different number of elements according to the definition of bijection. As a result, an infinite set cannot correspond one-to-one to its proper subset. So, there is no infinite hotel paradox and Galilean paradox. The set of natural numbers that corresponds one-to-one to the rational numbers set Q is not the proper set of Q, but another set of natural numbers. The part does not equal the whole. The number of digits after the decimal point for infinite decimals may also be different for infinite decimals. When the number is the same, there is no one-to-one correspondence between different dimensional spaces and there is no one-to-one correspondence between different length segments. The real numbers are countable. There is no uncountable set. Almost all of Cantor's counterintuitive theories are wrong in their logic, so the conflict between intuition and logic is not normal and intuitionism and logicism must be united. It is possible to reach limit but the infinite process can never be completed. There is no Zeno paradox. The fact that there are so many errors in the most rigorous mathematics and that they remain uncorrected for so long shows that human logical thinking ability needs to be greatly improved and no one can be sure that he is correct. Therefore, it is foolish to impose one's own views, including ideology and values, on others, even by force, to push mankind towards self-destruction in nuclear and biological wars.
Category: Set Theory and Logic