Gaps in Solutions of Polynomial Equations
„Mathematics is not settled. Even concerning the basic objects of study, like numbers and geometric figures, our ignorance is much greater then our knowledge” (Jordan Ellenberg)1
The consequences of the modification of the mathematical notion of infinity are manifold, and I have already mentioned several of them, though I have not explained them all in detail. An important property of this notion of infinity is that it is closely related to Cantor's continuum hypothesis (CH) and with it to the axiomatic foundation of set theory, since with a quantified version of CH and its alternatives2 we can "produce" three different set theories, similar to the way we arrive at different geometries in geometry depending on the definition of the sums of angles of triangles. A quantified version of CH can be used to unify probability calculus, and three different versions can be axiomatized.3 If all this is not enough, it is necessary to mention that the most elementary relations describing our real spaces can be described by the numerical models of these CH-variations, the two-element numbers, since the multiplication by the parabolic unit vectors models the Galilean transformation, and the multiplication by the hyperbolic unit vector models the Lorentz transformation.4
But I have not written about the need for changes to elementary algebra, which also have far-reaching consequences. One of these is the subject of this article.
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1 Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking
2 See for example article titled „Linking Hilbert’s 1st and 6th Problems”:
https://www.infinitemath.hu/en/archieve/others/373-linking-hilbert-s-1st-and-6th-problems
3 See article titled „Towards a Universal Probability Theory, Part III (Final part)”;
https://www.infinitemath.hu/en/archieve/mathematics/375-towards-a-universal-probability-theory-part-iii-final-part
4 See article titled „The Galilean Transformation and the Parabolic Numbers”;
https://www.infinitemath.hu/en/mathematics/413-the-galilean-transformation-and-the-parabolic-numbers
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