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This was the answer I was looking for. You're reminding me that existence of infinity is an axiom, like parallel lines dont intersect. We can remove that axio
by nphardon 2mo ago
This was the answer I was looking for. You're reminding me that existence of infinity is an axiom, like parallel lines dont intersect. We can remove that axiom and get a new mathematical framework, but by their nature, axioms are not something to prove or disprove.
- amavect 2mo agoExactly. For example, John Mayberry wrote "The Foundations of Mathematics in the Theory of Sets" (2000). Half of the book consists of philosophical arguments for his "Euclidean set theory" contrasted against the big bad "Cantorian set theory". He takes inspiration from Euclid's common notion 5 "the whole is greater than the part". On page 277, formula 8.3.1, his Axiom of Euclidean Finiteness goes like this: any injective endofunction is also surjective, ∀f∀Y((f:Y→Y ∧ 1to1(f)) ⇒ onto(f)). I've come to believe that many related incompatible theories have interpretations between each other. For example, hyperbolic geometry has a Euclidean-like Poincare disk model, and Euclidean space exists locally in a hyperbolic space. Boolean logic contains intuitionistic logic (just add the law of excluded middle), but intuitionistic logic contains Boolean logic through the double negation translation. Similar might happen for finite set theories, infinite set theories, and neutral set theories. The fun includes finding the right translation so that we can all enjoy our different tastes in axioms.