I reject Cantor's infinite set theory as inconsistent (see The Countable-Infinity Contradiction Preprint The Countable-Infinity Contradiction
). Is mathematics more than the unbounded set of consistent definitions formed from the concept of 1?
Dear Ed Darnell ,
The concept of {1} is essential to evolve the thoughts of sets and numbers up to any basis. But this is not enough to find an algebraic structure. Operations over sets are a must; this will lead to constructing N, Z, Q, R, C, quaternions, etc. Besides, we need the idea of space and dimension as a first step to surround sets with imaginations and the geometry of figures. Best regards
Agreed. We must define consistent operations. Interestingly it is the inverse operators which tend to generate new sets. So we use + to generate the natural numbers but - to generate all integers including 0.* generates nothing new but / generates the rationale. Powers generate nothing new but roots generate the algebraic irrationals and also the complex numbers. Coordinates and vectors generate from , giving the concept of dimensions.
In mathematics are some concepts that haven't definition. For example concept of point. I think that the concept of point is one of this concept that not build from 1.
Interesting. I would argue that a point is an infinite concept (infinitely small so of zero size). So it is clearly defined by mathematics but strictly lies outside as infinity must if we are to preserve consistency. It is an analytic concept based on considering the limit.
I mentioned in my previous answer the need of the dimension concept to build the geometry; this includes the points which are defined as objects that have zero dimension. To locate the position of the point, we need the algebraic system of numbers. For example, p = p(x, y, z) ∈ R3 where x, y, and z are numbers represents the length, width, and height.
Best regards
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