Somewhere recently I read that one can create a chain of mathematical universes, each modelling the next, in which particular objects alternate between properties. (Was it countably vs uncountably infinite, or infinite/finite? If it matters to you, I'll try to trawl my history)
Another aspect of infinity: our (human) Turing Machines have an infinite tape, but only a finite repertoire of symbols. One might imagine angelic Turing Machines which allowed an infinite variety of symbols (on a finite tape?) but I currently believe that to actually compute these could be no more powerful than ours, for to model lambda calculus requires continuity à la Scott. As below, so above?
Edit: Note that the difficulties Cantor faces here with orthodoxy and Spinoza rhyme with Einstein's declaration that his beliefs were in the "God of Spinoza".
Another aspect of infinity: our (human) Turing Machines have an infinite tape, but only a finite repertoire of symbols. One might imagine angelic Turing Machines which allowed an infinite variety of symbols (on a finite tape?) but I currently believe that to actually compute these could be no more powerful than ours, for to model lambda calculus requires continuity à la Scott. As below, so above?
Edit: Note that the difficulties Cantor faces here with orthodoxy and Spinoza rhyme with Einstein's declaration that his beliefs were in the "God of Spinoza".