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Is the cardinality of the reals 0 or $\aleph _0$?

It is easy to talk about a formal system plus an uncountable number of axioms that state the existence of all reals. Each real number can be defined as an infinite sequence of digits. We cannot write the entire sequence but we have a sense of what an arbitrary real is. In this way mathematicians talk about the true set theory that includes all reals.

The human mathematical mind is the product of biological evolution. There is no evidence of a special facility that transcends the finite. On the contrary all the evidence suggests the opposite. The current `theological' approach to mathematical truth flies in the face of the evidence. We believe the cardinality of the real numbers is 0. We do not think any real number as a completed infinite sequence exists. Of course the integers and rational numbers are also considered reals so one could argue that the cardinality of the reals is $\aleph _0$ or the same as $\omega$.

It makes sense to consider all possible paths that a nondeterministic computer simulation (or for that matter biological evolution) can take. Mathematical theorems about searching all possible paths have a form of absolute meaning because their truth is determined by a recursively enumerable set of events. They are a complex statement about a well defined set of events all of which can occur in a potentially infinite universe. This does not require that the set of all reals or for that matter a single real exists as some objective reality.


Completed second draft of this book

PDF version of this book
next up previous contents
Next: A philosophy of mathematical Up: Creative mathematics Previous: Extending mathematics   Contents


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