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"Marsden's data stores contain a fragmented catalogue of mathematical variants. All founded on the postulates of arithmetic, but differing in their resolution of undecidable hypothesis."
"Undecidability. You're talking about the incompleteness theorems,"
"Right. No logical system rich enough to contain the axioms of simple arithmetic can ever be made complete. It is always possible to construct statements that can be neither disproved nor proved by deduction from the axioms; instead the logical system must be enriched by incorporating the truth or falsehood of such statements as additional axioms..."
The Continuum Hypothesis was an example.
There were several orders of infinity. There were 'more' real numbers, scattered like dust in the interval between zero and one, that there were integers. Was there an order of infinity between the reals and the integers. This was undecidable, within logically simpler systems like set theory; additional assumptions had to be made.
"So one can generate many versions of mathematics, by adding these true-false axioms."
"And then searching on, seeking out statements which are undecidable in the new system. Yes. Because of incompleteness, there is an infinite number of such mathematical variants, spreading like the branches of a tree...."
While only briefly mentioned, there are living logical entities made of mathematical postulates that this section is talking about. Is this evidence of something?
"Undecidability. You're talking about the incompleteness theorems,"
"Right. No logical system rich enough to contain the axioms of simple arithmetic can ever be made complete. It is always possible to construct statements that can be neither disproved nor proved by deduction from the axioms; instead the logical system must be enriched by incorporating the truth or falsehood of such statements as additional axioms..."
The Continuum Hypothesis was an example.
There were several orders of infinity. There were 'more' real numbers, scattered like dust in the interval between zero and one, that there were integers. Was there an order of infinity between the reals and the integers. This was undecidable, within logically simpler systems like set theory; additional assumptions had to be made.
"So one can generate many versions of mathematics, by adding these true-false axioms."
"And then searching on, seeking out statements which are undecidable in the new system. Yes. Because of incompleteness, there is an infinite number of such mathematical variants, spreading like the branches of a tree...."
While only briefly mentioned, there are living logical entities made of mathematical postulates that this section is talking about. Is this evidence of something?
