Logical difficulties in modern mathematics (2012)

Attempts at trying to define “real numbers” in the way applied mathematicians and physicists would prefer—as decimal expansions—run into the serious problems of how to define the basic operations, and prove the usual laws or arithmetic. The serious problems with the continuum are reflected by an attendant state of denial by our first year Calculus texts, which try to bluff their way through these difficulties by either pretending that the foundations have been laid out properly elsewhere, can be replaced by some suitable belief system dressed up using “axiomatics”, or can be glossed over by appeals to authority. A challenge to those pure mathematicians who object to these claims: can you show us some explicit first year examples of arithmetic with real numbers??

Source: njwildberger.com