#05 A third new number always between any two numbers (density) is

Now another fake consequence in Old Math since they never had a border between finite and infinity, is this idea of density of numbers or density of points in geometry. In Old Math, such as the Hilbert Axioms they proved a theorem that basically says given points A and B on a line, there always exists a new point C between A and B. In Algebra, this idea becomes that of between any two Rationals, exists a third new Rational number. Also, between any two Reals is a third new Real. For example, between the two Rationals of 1/2 and 2/3, one can subtract them 4/6 - 3/6 = 1/6 and divide by 2 giving 1/6/2 = 1/12 and add 1/12 to 1/2 to find a third new Rational of 1/12 + 6/12 = 7/12. Same process for Reals only more messy. In True Math, that process no longer is true because if the length is microinfinity, there is no new third number.

In True Geometry, we see such a theorem is false because if the distance or length of A to B is microinfinity empty space, then no new finite point exists in that region.

So, at least one of Hilbert's axioms is a phony axiom and must be deleted. As I wrote earlier that Hilbert had 3 primitive notions and 3 primitive relations and 20 axioms which really is 26 axioms altogether, that at least one of those 26 must be deleted so that such a theorem of density can no longer be possible.
In True Geometry, the microinfinity imposes a barrier or limit to how close points can get to one another on a line. And that barrier prevents a theorem from saying between any two points A and B is a new third point. For if the two points A and B are microinfinity apart such as .1 and .2 in 10-Grid then there is no new third point. If A is .1 and B is .4 then there are new points of .2 and .3 in between A and B.

In Old Math they had the Rationals and Reals as dense sets of having a new third number between A and B. In True Math, density stops at microinfinity distance.

Now be careful because in New Math, sometimes a line can stretch from 0 to infinity and contain only two points of A and B and no other point between them, other than empty-space between them, as I show for the Sawtooth function in Calculus inside a cell. But that takes me far ahead. Here I just want to dwell on the outright consequences of True Geometry that has empty space between successive points.

So in True Geometry, density is no longer continuity or continuum of points.. Density in True Math is where we have discrete points in space separated uniformly by empty-space between those discrete points. Space has a point then a hole then another point, then a hole.

In Old Math, they gave the illusion or delusion that they could get points so dense on a line that the line would be continuous as a continuum and have squeezed out all holes or empty space. Which is again, the return to the contradiction at the heart of Old Math, where you have lines as having length composed of only points which have no length-- contradiction.

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Drexel's Math Forum has done an excellent search engine for author posts as seen he
http://mathforum.org/kb/profile.jspa?userID=499986

Now, the only decent search for AP posts on Google Newsgroups, is a search for for it brings up posts that are mostly authored by me and it brings up only about 250 posts. Whereas Drexel brings up nearly 8,000 AP posts. Old Google under Advanced Search
could bring up 20,000 of my authored posts but Google is deteriorating in search of old posts to the science newsgroups.

So the only search engine today doing author searches is Drexel.

All the other types of Google searches of AP are just top heavy in hate-spam posts due to search-engine-bombing practices by thousands of hatemongers who have nothing constructive to do in their lives but attack other people.

Funny, and I have to laugh here, for in the recent news of spying on Europe by USA, such a big fuss, whereas I want that sort of full recording of all of my posts to Usenet. I am hoping the spy agencies of the USA has recorded every last one of the posts that AP has posted to the Usenet and please keep up that spy activity. I have kept a personal archive and figure that in these 20 years of posting of say 5 posts per day makes 365 x 5 = 1825 times 20 = 36500. So roughly I have about 36,500 posts to Usenet in these past 20 years. I hope many computers and servers have all those 36,500 posts safely stored and please feel free to display them. Whereas Google is working in the exact opposite, by trying to hide or destroy them, or bias the listing of those author posts.

Archimedes Plutonium