>>11241052First off, the notation "lim" describes an algorithm which produces a number. I use the standard epsilon-delta definition which is implicit in the "lim" notation. The definition of the number that is output is separate from the definition of the algorithm implied by the "lim" notation.
Second, the reason it seems like my R is a superset of everyone else's R is because everyone else was considering a tiny subset of R to be all of R. I have proven that numbers less than any natural number are only a small subset of R, and the only axiom have taken to do so is that R is a connected open interval of the form
R = ( -inf , inf ).
Also, your statement
>reals defined with sequences... usuallyis highly tenuous. You are not wrong, persay, because "usually" can mean "usually right now," but if you take it to mean "usually in the history of mathematics" then you are totally wrong. Usually in the history of mathematics, and even usually in 1859 when Riemann published his hypothesis, R was defined as per Euclid as a cut in an infinite line.
Since the main purpose of modern mathematics is algebra rather than geometry, I agree that it is good to have an algebraic definition of R. However, my conjecture is that Cauchy's and Dedekind's attempts to do so are suboptimal because they contradict the geometric definition that R is a connected open interval of the form
R = ( -inf , inf ).
My opinion is that any algebraic definition of R needs to preserve the axiom that R = ( -inf , inf ), and it is a cold fact that the Cauchy and Dedekind definitions do not do this. The extension, however, is quite easy to make, and I have done so in the case of the Cauchy definition, pic related.
