On Sun, Dec 30, 2012 at 8:27 PM, Joe Niederberger <firstname.lastname@example.org> wrote: > Paul Tanner III says: >>This asymmetry in the non-dual distributive property that "connects" the two binary operations of this type of ringoid fully explains the asymmetry between the binary operations we see in such examples of such a ringoid as the real numbers and its subsets like the integers. > > I've mentioned before Tarski's axioms for real numbers: they say nothing outright about either multiplication (existence and properties thereof are proved as theorems later), or (obviously) distributivity. > > So, under Tarskian real numbers, Paul, you are wrong, again. > > Cheers, > Joe N
No. I am not the one wrong here. Those who hold to the idea that "addition" is inherently fundamental to "multiplication" because of some asymmetry in their behavior in some ringoid are the ones who are wrong. This notion that there exists a ringoid in which one of the operations of the ringoid is fundamental to the other is a mirage caused by the defining properties of the ringoid. Just because Tarski derived one operation from assuming the other does not mean that one of the operations is fundamental to the other. Read on to see why.
Fact: A ringoid is simply a set under two binary operations such that one of them distributes over the other. This distributivity can go in one direction only, in which case we can call it non-dual distributive property, or it can go in both directions, in which case we can call it a dual distributive property. In the non-dual case, if one wants to define the operation being distributed over as being fundamental to the other one simply because it is the one being distributed over, then that's one thing, but it's another thing to say that the one being distributed over is inherently fundamental to the other one. (Notice that I did not name these operations. This is because the operations are named as they are only by convention. There is nothing magical about the word "addition" so as to make the operation of a ringoid named in such a way fundamental to the other one named "multiplication".)
Fact: Tarski's axioms assume addition among things and yield a ringoid under a non-dual distributive property, this particular yielded ringoid under the non-dual distributive property being the set of real numbers. (See http://en.wikipedia.org/wiki/Tarski's_axiomatization_of_the_reals for more.) That is, they yield a Dedekind-complete ordered field, which is isomorphic to the set of areal numbers. (See http://en.wikipedia.org/wiki/Ordered_field for more.) What Tarski did was made possible by the defining properties of the target object to be derived, a ringoid under a non-dual distributive property, and the non-dual nature of this property is the key as to why there we have the mirage in question. (Can Tarski's axioms yield a ringoid under the dual distributive property?)
Fact: For every ringoid that is under the dual distributive property, because of the symmetry of the dual distributive property, the asymmetries in question do not exist unless there some set of axioms for some such ringoid that defines an asymmetry or from which we can derive an asymmetry. (The reason they do not exist is because we can perform the derivations in question for the non-dual distributive property context in both directions, not in just one direction. I give an example below.)
Fact: For every ringoid that is under the non-dual distributive property, because of the asymmetry of the non-dual distributive property, the asymmetries in question exist and we can derive the asymmetries in question using only algebraic properties.
in which I derive the behavior of multiplication as repeated addition for every ringoid under the barest minimum of algebraic properties, and adding more and more properties until we reach a field, that barest minimum starting with the ringoid having only a multiplicative identity, where, for all b in a specific subset of the set of all sums in the ringoid and for all a, I derive in that minimal context the equality
ab = a + a,
multiplication behaving as repeated addition.
In fact, even just the distributive property itself, which is the only algebraic property that is part of the definition of a ringoid, is a form of one operation as another operation.
For every ringoid that is under the dual distributive property and that contains (ever merely) an additive identity, we can derive
a + b = aa,
addition behaving as repeated multiplication, using the same method I used in that post above but with things appropriately reversed.
To sum up: For any ringoid (including the set of real numbers), there is an asymmetry in the behavior of the two operations if and only if the defining properties of the ringoid are such that they define some asymmetry in question or are such that we can derive some asymmetry in question. (The defining property most important in this regard is the defining property of the distributive property: If it is an asymmetrical non-dual distributive property, then we have the asymmetries in question, and if it the symmetrical dual distributive property, then we do not have the asymmetries in question unless there is some additional defining properties of the given ringoid that defines some asymmetry in question or from which can derive some asymmetry in question.)