Trying to learn mathematics. Comments and corrections welcome. Piece of a work in progress.
All consistent axiomatic formulations of number theory include undecideable propositions.
The single, circular ‘loop’ of the statement above is the assertion that, although the statement is true, it cannot be proven to be true. This parallels the way Principia Mathematica contained mathematical statements of truth that could not be proven through the text itself.
Gödel showed that probability is a weaker notion than truth. His sentence G showed that no fixed system could adequately represent the complexity of whole numbers-no matter how complicated or elegant. No connective set of principles could explain ‘a whole’. ‘A whole’ is a sum greater than its constituent parts.
See Paul’s first letter to the Corinthians which makes the point that although sun, water, and soil are required and can explain growth, they are not growth itself. Growth results in death (sic. cell senescence) which returns as birth/reproduction.
The main paradox in math is trying to enliven our intuitive realizations with formalized, axiomatic explanations?