3 ms·
I don't understand what you mean. Higher-order logic does have classical negation. First-order PA tries to approximate the (second-order) induction axiom by rep
by cubefox 3y ago
I don't understand what you mean. Higher-order logic does have classical negation. First-order PA tries to approximate the (second-order) induction axiom by replacing it with an infinite axiom schema, but that doesn't rule out non-standard numbers.
- nyrikki 3y agoCantor's proofs showing that Z and Q are countable and R is uncountable. Cantor's "diagonalization proof" showed that. Turing extended to the computable numbers K, which can be conceptualized as a number where you can write a f(n) that returns the nth digit in a number. The reals numbers are un-computable almost everywhere, this property holds for all real numbers in a set except a subset of measure zero, the computable reals K which is Aleph Zero, a countable infinity. The set of computable reals is only as big as N, and can be mapped to N. It is not 'non-standard numbers' that are inaccessible, it is most of the real line is inaccessible to any algorithm. Note the following section for the first part. "Non-Absoluteness of Truth in Second-Order Logic" https://plato.stanford.edu/entries/logic-higher-order/#NonAbsoTrutSecoOrdeLogi https://plato.stanford.edu/entries/logic-higher-order/#NonAb...
- nyrikki 3y agoWell-founded relations is probably a good lens on why the above is important.