RZC shares a journey into an estranged class.

Bertrand Russell's old paradox of logic is solved.

In here is a powerful new formula for the Leibniz derivative, an opening fundamental.

The following author's commentary may help bodies decide whether this is the book for them.

For further enticement, click here for a proof of Euclid's fifth axiom, using the Leibniz derivative as defined in the book.
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Round Trip (vacancy unfilled in signposting)

Russell Z Christensen (Of Self)

In relation to complex things that are supposed to work, as a kid I was a gurdjieff. The superheterodyne radio receiver made from discrete components at the tender age of eleven, boasting seven transistors in all, actually worked. I started with only a circuit diagram, bought the components individually, designed and etched the circuit board myself. One day I was called to help another boy in the neighbourhood who had decided to make a radio receiver. But I could not help, the wiring undecipherable in tangle, the boy's mind a blank in several areas I understood but could not really voice.

Over the years I have learnt better how to express technical matters. I have also learnt that some people have minds totally closed off to the sensitivities pertaining. As a community, to liken ourselves to a rugby team, we do need specialists; we do need wings as well as the forward pack, to tackle the challenges of the times. My new book is one for specialists, but the group is broadly conceived - calculus students, mathematics teachers and other persons interested in the fundamental ideas.

I could not have written the book without having succeeded with excellence at the levels requisite for a bachelor's degree in mathematics. But I aim to entertain with the book, as well as expound in the mathematical sense.

Objective Framework

From the time of Leonardo da Vinci and King Richard III, western governments have progressively turned to an objective framework for their decision making in relation to microbes and inventions. This involves respectable on-going research about our enemies among microbes and inventions. It involves a delicate balancing act which is exemplified in the case of CCA timber treatment in New Zealand. Copper chromium arsenic (CCA) is a toxic substance, but one which has been the making of New Zealand. It enabled a nation with very limited stocks of hardwood and little brick to make wonderfully long-lasting houses out of quick growing pinus radiata.

Mankind needs to deal with toxic substances and with technologies that may do harm. Being objective means knowing what we are doing. We have not really failed to know what we are doing when dealing with CCA, except possibly in isolated, relatively insignificant cases. On the other hand, we emphatically do not know we are doing with the conversion of the energy of tidal motion into electricity. The lack of a clean energy-balance accounting for this conversion technology is widespread and endemic. This is a far more serious black mark against a technology than exists against any other on the books. Even with nuclear fission, we are aware of the problems, can take measures to protect the innocents.

What is the problem? Gravity has the form of a conservative force, so all analyses concerning energy if gravity is the obliging force from which Man benefits, must needs refer to a cycle. In the case of the hydro-electric turbine, the water cycle is the relevant one. In the case of the tidal turbine, it is the tidal cycle instead. It is impossible to cheat the negative work of gravity in the water cycle as the Sun's work in evaporating water cannot be undone or prevented. By contrast it is inevitable that tidal turbines will cheat the negative work of gravity. In the tidal cycle, the only source of energy for the negative work is the energy provided in the alternate phase of the cycle by the positive work of gravity. Nature has been designed for the positive work to balance the negative. Man only began thwarting this design of Nature on an industrial scale in the mid-1960s, just prior to the well-noted and now seemingly determined upward swing in the surface temperature of the Earth. It has been particularly bad in a certain large area of the Northern Hemisphere.

On the Trail of Loose Assumptions

After discerning the solution to Russell's Paradox, I was beset by thoughts that space, as also space-time, could be described as an improper class of places. One of the fruits of that investigation is the proof of Euclid's fifth axiom, available as PDF from the hyperlink above. Extending from the two dimensions of the proof into the four dimensions of space-time allowed me some confidence in looking at Einstein's derivation for the Total Energy law. Eventually I was able to contrive a pretty alternative derivation, introducing the concept of the permebasin. If the tides' soft landings imply the existence of a gravitational energy reserve, then why not the permebasin? It is all the more interesting that, if each heavenly body has its own permebasin, then theoretically the combined permebasin energies are all the energy the gravitational system really needs. It can even still conform to Einstein's tensor equation, to a sufficient extent at least that this will remain as undisputed law.

The new picture is tight and satisfying. Of course I am always trying to test it out. It has so far proven robust but the investigation has introduced me to various lacks and weaknesses in popular arguments.

Application to the Environment Court

The situation with respect to converting the energy of tidal motion seemed to raise its head as somewhat urgent. Written-in to the Resource Management Act of New Zealand, there seemed to be provision for a judge to make orders against the Ministry for the Environment, in connection with the latter's support for the theory of the greenhouse effect. This theory is shabby and ultimately untenable. In trumpeting the greenhouse effect as cause for Earth's anthropogenic warming, the Ministry somewhat blocks the come-uppance of a more serious theory. I clutched at the straw and in due course it turned into an apparition. The judge's order denying jurisdiction was issued very swiftly and reactionary-like after my application had been received at the Court.

Revisiting the Years of U.S. President Richard Nixon

In world history, have there been any other monumental technology-stopping exercises, such as would be required in a shut-down of all the tidal turbines on Earth?

Part of my life has been spent as a resident in Pine Avenue, Niagara Falls, on the United States side of the border. While there I figured out that a widespread gratitude, to the American administration, for curtailing the toxic ion spills of sundry Great Lakes steel mills, was more particularly a gratitude felt towards the administration of President Richard Nixon. Before he was a quixotic figure through the Watergate upset, Nixon had been a bold upholder of the citizens' privilege set. One can refer, through the presidential archives, to the speech he gave upon the signing of an agreement between Canada and the United States. Here is someone touching upon the great themes of life, regulating the steel mills to ensure that downstream people may enjoy the great Niagara river.

The story goes that Richard Nixon, John Ehrlichmann and crew had to make a Berlin Wall in the Whitehouse in order to press forward to firm outcomes. In another way of describing it, they quarantined a philosophy. The reference to the Berlin Wall arises because before the fall of Wall, a certain communist philosophy was quarantined by Wall to the East side. If we can draw a lesson from the quarantining of a mischevious philosophy in the Whitehouse of President Nixon, then we may be thinking about quarantining the unqualified application of radiative transfer theory to the atmosphere. In May, 2016, I wrote an Open Letter to Our Childrens' Trust explaining in the language of a senior secondary-school student how this omits consideration of an important class of phenomena in the atmosphere. To commit public funds to remedial actions based on a theory resting on such an omission is certainly quite mischevious. It goes further than mere mischief if such philosophy prevents us from focussing on the likely real cause of anthropogenic climate change.

Possibly we can go further and quarantine the Zermelo-Fraenkel paradigm in mathematics. The two-birds-with-one-stone approach may offer the sweetner that some researchers no longer needed in certain areas may opt to learn and develop the improper class theory, as a way of keeping themselves useful at a university level. From a societal point of view, this should improve the delicate balancing act that is put upon us by the march of technology.

Solving the Paradox

The paradox calls us to consider the set of all sets which are not members of themselves. Once we have both mathematical set and improper class on board, the solution can be simply stated as follows: if one speaks of the class, it is not a member of itself, but if one speaks of the set, then the opposite applies. That this may apply in sets requires us to distinguish between a structural member (through its singleton being a subset) and an incidental member (through its possession of a generating property). Clearly the incidental membership cannot apply in this case but the structural membership can, if we are careful to contemplate a full range of mathematical entities when setting up for mathematics within a wider language context.

Of particular note, we need to allow that an unconstrained free variable can fall down two distinct holes. It can fall into a hole of non-existence, as is the case with tan(x) when x takes on the value of 90 degrees, but separately it can fall into a hole of indeterminancy. Assuming the axioms of the natural numbers do apply, x is indeterminate if it is equal both to 2 and to 3 at once. This is because in the system of the natural numbers 2 is not equal to 3. In dynamics, a free variable may be used for an object that is constituted by its context, rather than as this or that numerical value or functional result. The upshot is that for such a variable, being indeterminate is a possible state, a special complication if and where it occurs.


The dialogue from pages 11 thru 13 of Chapter One of Russell's Paradox - Two Sets is a slim distillation of the technical shortcomings I found in the Zermelo-Fraenkel paradigm for mathematics. Chapter One is available to read in PDF through a hyperlink on the right of this page above. From the heading Zermelo's Solution, through to the heading Intuitionism, the reading is fairly tame. In my efforts to understand how reductio-ad-absurdum proof could be cleared of the taint of Russell's Paradox, I discovered how New Mathematics had been born with its fundamentally webbed feet. It remained for me to recall some of the rules of nomenclature I had learnt in primary school before New Mathematics took hold.

Could it be that the language of improper classes has been around for many years? Possibly, yes. There is a strong intuitive appeal to the rules of nomenclature and once teased out they are logically sound. One can be inclined to speculate that the distinction between a mathematical set and an improper class is latent in many natural languages.

R Z Christensen 2016