Just out of curiosity, can we make illegitimate operations with $0$, say, division by $0$ legitimate simply by imposing additional axioms? If so, then what consequences may follow? If the answer is negative, what are the differences between making operations with $+\infty, -\infty$ legitimate in the extended reals?
A number of efforts have been made as listed below. Patrick Suppes provides an excellent overview and succinct survey of the issue of division by zero in his Introduction to Logic, Chapter 8.5, The Problem of Division by Zero, and Chapter 8.7, Five Approaches to Division by Zero. http://stuff.mit.edu/afs/athena.... Suppes notes that the fourth of the five approaches he identifies has been the one most commonly used in mathematics. Essentially, he is referring to item 1 listed below. As far as I know, prior to the 1950s when Suppes wrote, the extended complex plane was the only** arithmetic defining division by 0 that existed. Items 2 and 3 descend from 1. Meadows is an example of one of the other approaches Suppes covers. Item 4, Carlstrom's Wheels seems to me yet another, sixth, approach although it owes a debt to the fourth.
1) The extended complex plane** http://mathworld.wolfram.com/DivisionbyZero.html
2) The floating point arithmetic known as exact real arithmetic. http://www.doc.ic.ac.uk/exact-co...
3) Wheels, Anton Setzer http://www.cs.swan.ac.uk/~csetzer/articles/wheel.pdf (unpublished paper 1997)
4) Wheels, Jesper Carlstrom, http://www2.math.su.se/reports/2... (published 2001)
5) Meadows - models of “an equational specification ENA (elementary number algebra) which specifies a super class of the class of zero-totalized fields." http://staff.science.uva.nl/~janb/FAM/topFAM.html
6) My own work simply replaces 0 using several ideas, one of which is a set different from the empty set based on a change to predicate logic. The algebra is here while the paper Replacing 0 - A NonEuclidean Arithmetic is more in depth.
7) Also, it may be of interest, the software package Mathematica returns 0 upon entering 1/Infinity Although it cannot be said to be mathematically correct, this is consistent with a fairly common practice in physics (e.g, when interpreting the mathematics of singularities).
footnote ** Related to this are the affinely extended real number system, the Real projective line, and the extended non-negative Real line. Just as the complex numbers are extended by a point at infinity, all these arithmetics use a point or points at infinity to extend some or all of the Real numbers. Some of these may well have existed before Suppes' book as well.
In some contexts, things like $5/0$ should be taken to be $\infty$, where that is neither $+\infty$ nor $-\infty$, but a single $\infty$ at both ends of the real line or at the extreme of the complex plane. This makes all trigonometric functions and rational functions continuous.
This is really an example of one of the points at infinity in projective geometry.
Sure. Lots of math programming libraries essentially do this by adding NAN (not a number) as a possible value of functions. Things don't break because NAN is not the domain of any function/operator/etc.
However, saying anything divided by zero equals 42 would result in an inconsistent theory, which makes everything true. We see lots of examples of this in "proofs" that 12 equals 14 or whatever.