On Nodal Structure of Cofinallity of Cardinals and Dimensionality (part 1)

NOTICE

• DO NOT TRUST THE FOLLOWING

"Je le vois, mais je ne le crois pas"

• In 1877, Georg Cantor found bijection between $\mathbb{R}$ and $\mathbb{R^2}$
• He was surprised that dimensionality of space does not matter for the number of points in space, or cardinality
• In reply to him, Richard Dedekind pointed out that $\mathbb{R^2} \to \mathbb{R}$ is not a continuous mapping
• Other than cardinality, what represents dimensionality in set theory?

Singular Cardinal with Countable Cofinality

• What are $\aleph_\omega$ and $\aleph_{\omega 4}$, which appear in a Saharon Shelah's well-known result of PCF theory?
• On the way of cardinal construction, $\aleph_0, \aleph_1, .. \aleph_\omega, \aleph_{\omega + 1}, .. \aleph_{\omega + \omega} = \aleph_{\omega 2} .. \aleph_{\omega 3} .. \aleph_{\omega 4} .. \aleph_{\omega \omega} = \aleph_{\omega^2} .. \aleph_{\omega^\omega} .. \aleph_{\epsilon_0} .. \aleph_{\Gamma_0} .. \aleph_{\omega_1} .. \aleph_{\omega_1+1} .. \aleph_{\omega_1 + \omega_1} = \aleph_{\omega_1 2} ..$
• For their cofinallity, singular cardinals form "nodes", or minima in cofinality graph: $cf(\aleph_\omega) = cf(\aleph_{\omega n}) = \omega = \aleph_0$

"But we all know the world is nonlinear"

• Halord Hotelling once questioned validity of linear programming in a presentation by George Dantzig (Dantzig 1991)
• John von Neumann adovocated for Dantzig, saying "If you have an application that satisfies the axioms, well use it. If it does not, then don't"
• To believe the validity of linear spaces, we need to ask the consistency of axioms of orthogonal linear domain
• So we need to prove existence of a model of othogonal n-dimension additive space and its geometry