"The classic example of an is that of plane geometry formulated by Euclid... It forms the model of all rigorous mathematical schemes. The axioms are the initial assumptions... From them, logical deductions can proceed under stipulated rules of reasoning... analogous to the scientists' laws of Nature, whilst the axioms play the role of s. We are not free to pick any axioms... They must be logically consistent... Euclid and most other pre-nineteenth-century mathematicians... were also strongly biased towards picking axioms which mirrored the way the world was observed to work... Later mathematicians did not feel so encumbered and have required only consistency from their lists of axioms. ...It remains to be seen whether the initial conditions appropriate to the deepest physical problems, like the cosmological problem... will have initial conditions which are directly related to visualizable physical things, or whether they will be abstract mathematical or logical notions that enforce only self-consistency. ...one can quantify the amount of information that is contained in a collection of axioms. None of the possible deductions... can possess more information than was contained in the axioms. ...this is the reason for the famous limits to the power of logical deduction expressed by Gödel's incompleteness theorem. ...however, ...an axiomatic system ...not as large as the whole of arithmetic does not suffer... incompleteness."