On Thursday, 1 May 2014 22:41:27 UTC+2, Dan Christensen wrote: > On Tuesday, April 29, 2014 2:50:51 AM UTC-4, muec...@rz.fh-augsburg.de wrote: > > > Couldn't just the seemingly so fruitful hypothesis of the infinite have straightly inserted contradictions into mathematics and have fundamentally distroyed the basic nature of this science which is so proud on its consistency? > > > > > > > > > > > > Find the resolution tomorrow in > > > > > > > Whoever said it has obviously not tried to formalize number theory on a finite set of any size (one with a beginning and an end).
No, he (Zermelo) has not. But his fear has come true. Set theory shows the uncountability of all countably many finite expressions.
> You might be able to construct an add function, but, from my experience, simple properties like associativity and commutativity seem to be impossible to establish. Addition, multiplication and exponentiation must be closed on the natural numbers.
Of course. That is the problem pointing to neither finite nor actually infinite mathematics. Potential infinity appears to be the only resolution unless one is willing to consider MatheRealism. But that cannot be recommended as a general basis.