Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Counterexamples to Cantorian Set Theory   This paper provides some counterexamples to Cantor's contributions to the
foundations of Set Theory. The first counterexample forces Cantor's Diagonal
Method (DM) to yield one of the numbers in the target list. To study this
anomaly, and given that for the DM to work the list of numbers have to be
written down, the set of numbers that can be represented using positional
fractional notation, $\mathbb{W}$, is properly characterized. It is then shown
that $\mathbb{W}$ is not isomorphic to $\mathbb{R}$, meaning that results
obtained from the application of the DM to $\mathbb{W}$ in order to derive
properties of $\mathbb{R}$ are not valid. It is then shown that Cantor's DM for
a generic list of reals can be forced to yield one of the numbers of the list,
thus invalidating Cantor's result that infers the non-denumerability of
$\mathbb{R}$ from the application of the DM to $\mathbb{W}$. Cantor's Theorem
about the different cardinalities of a set and its power set is then
questioned, and by means of another counterexample we show that the theorem
does not actually hold for infinite sets. After analyzing all these results, it
is shown that the current notion of cardinality for infinite sets does not
depend on the size of the sets, but rather on the representation chosen for
them. Following this line of thought, the concept of model as a framework for
the construction of the representation of a set is introduced, and a theorem
showing that an infinite set can be well-ordered if there is a proper model for
it is proven. To reiterate that the cardinality of a set does not determine
whether the set can be well-ordered, a set of cardinality
$\aleph_{0}^{\aleph_{0}}$ is proven to be equipollent to the set of natural
numbers $\mathbb{N}$. The paper concludes with an analysis of the cardinality
of the ordinal numbers, for which a representation of cardinality
$\aleph_{0}^{\aleph_{0}}$ is proposed.

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The categories of this paper are: econ
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The categories of this paper are: adap-org
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The categories of this paper are: comp-gas
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
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The categories of this paper are: eess
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The categories of this paper are: stat
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: astro-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ex
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The categories of this paper are: q-fin