Re: Cantor Confusion


William Hughes schrieb:


It cannot have a cardinal number at all. We have no cardinals in
potential infinity. Cantor knew that.

Piffle.

Extending the concept of bijection from sets to potentially
infinite sets is trivial.

May be if you apply your personal definition of potentially infinity,
but not if you apply the generally accepted definition.


Recall, you want to do more than just state that A does
not have a cardinal number. You want to show that assuming
that A has a cardinal number leads to a contradiction.

The problem is although the term "finite" is used in the
definition of A, A does not have many of the properties
that are usually associated with finite sets. In particular
it is not possible to produce A by induction.

A potentially infinite set like N can be produced by induction. An
actually infinite set cannot be produced by induction nor can it be
produced by other means because it is simply nonsense. The only way of
getting it is to postulate its existence by an axiom, although the
degree of nonsense is not reduced by this method.

Please do not mix up potential and actual infinity. Actual infinity is
required, e.g., for the real numbers.


Piffle.

The rational numbers form a potentially infinite set A.

Try to learn the commonly accepted meaning of "potentially infinite". I
am not wiling to discuss your personal definitions.

Let B and C be two potentially infinite sets such that:


For any element a, that can be shown to exist in A,
it is possible to show that a exists in exactly one
of B and C.

for any element, b, that can be shown to exist in B,
for any element, c, that can be shown to exist in C
and b < c.

for any element b_1 that can be shown to exist in B, there
is an element b_2 that can be shown to exist in B, and
b_1 < b_2


The potentially infinite set of pairs (B,C) is the real numbers.

No. Your definition already fails with the meaning of "that can be
shown to exist". In fact only the existence of a finite number of
numbers can be shown to exist. So your set of real numbers is finite?

Regards, WM

.

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