fastest way to print a lot of pdf files 7140

Peter T. Breuer

This wasn't your objection earlier; your objection earlier was that there wasn't enough time to enumerate all the possibilities. That indicates you don't know what cardinality means, because cardinality has nothing to do with an actual enumeration of all possible instances of a set in some finite amount of time. Apparently now that you've been brushed out of that corner, you're now talking about bijections. So you're shifting the goalposts.

It's trivially easy to create a bijection between an arbitrary length array of a finite set of symbols and the integers. So the set of all possible (arbitrary length) PDF files is clearly denumerable.

Now you're talking about the axiom of choice, which only applies to infinite sets here, so if you're next choice of goalpost shifts is to say that you're talking about countable and not denumerable sets, then that won't work either.

You're right, that constructivist versions of mathematics exist and are perfectly consistent with all the mathematics that have come before. But that also means that they can't possibly invalidate what's come before, either. Countable and denumerable still have the same definitions that they had before, and your interest in a constructivist interpretation doesn't change that.

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