NOTICE
- DO NOT TRUST THE FOLLOWING
Cardinal Geometry
- Desirable geometric view/interpretation for set-theoretic elementary embedding of universe into universe, not new math
- Infinite dimensional analogue of Riemannian embedding
- Ignoring finite sets, set-theoretic universe consists of ordinal "lines"
- Measurable cardinals or $\aleph_\omega$ (first singular cardinal) might work as "vanishing lines" or "horizon"
- But why ZFC? Categorical topology actually works and needs no additional concern about cardinality. Continuous geometry was invented for vaguely same objective, but
disappeared abandoned
So what?
Update: curse of dimensionality, integrability and chaos
- the very first result in dimension theory (Cantor(1877)): $card(\mathbb{R}^2) = \mathfrak{c}$
- existence of discontinuous mapping between 1-d real line and 2-d real area
- generally speaking, elementary embedding of universe into universe does not conserve information of dimensionality
- as change of topology, it changes continuity into singularity (making neighbors apart), it might also change singularity into continuity (making new neighborhood with aliens from distant galaxy)
- if you change order of enumeration, measure is not conserved (see summation of infinite series)
- curse of dimensionality: with requirement of conserving dimensional order information (for integrability) elementary embedding require further huge universe (if GCH holds it must be the power of current universe)
- continuum problem in numerical context: can you find measure-conserving chaotic (turbulent) circumstance characterized as having fractional dimensionality?
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