One more thing: The omega notation is only used for ordinal numbers. When discussing the size of sets we are dealing with cardinal numbers. Thus the reasoning is invalid.
I do not think you have to know the unique prime factorisation theorem. It is enough to know that every non-prime has a smallest divisor which is a prime.
The flaw is that you assume that Cantor's diagonalisation theorem can only be applied to one ordering of the image of the integers. By reordering them and appllying the theorem you can show that there are many different real numbers that are not in the image of the mapping. Most of these will in fact be members of ]0,1[.
It is not correct to say that c = infinity^2. However, it could be said that c = 2^infinity. This is in anology with sets of finite cardinality, since the power set of a set with the finite cardinality n has 2^n members. Still, saying that c = infinity^2 is obviously false.
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One more thing: The omega notation is only used for ordinal numbers. When discussing the size of sets we are dealing with cardinal numbers. Thus the reasoning is invalid.
I do not think you have to know the unique prime factorisation theorem. It is enough to know that every non-prime has a smallest divisor which is a prime.
The flaw is that you assume that Cantor's diagonalisation theorem can only be applied to one ordering of the image of the integers. By reordering them and appllying the theorem you can show that there are many different real numbers that are not in the image of the mapping. Most of these will in fact be members of ]0,1[.
It is not correct to say that c = infinity^2. However, it could be said that c = 2^infinity. This is in anology with sets of finite cardinality, since the power set of a set with the finite cardinality n has 2^n members. Still, saying that c = infinity^2 is obviously false.