Thus the fundamental ideas of geometry (e.g. those of points and of straight lines) are not ideas of determinate entities, but of any entities for which the axioms are true. And a set of formal geometrical axioms cannot in themselves be true or false, since they are not determinate propositions, in that they do not refer to a determinate subject matter. The axioms are propositional functions.[25] When a set of axioms is given, we can ask (1) whether they are consistent, (2) whether their “existence theorem” is proved, (3) whether they are independent. Axioms are consistent when the contradictory of any axiom cannot be deduced from the remaining axioms. Their existence theorem is the proof that they are true when the fundamental ideas are considered as denoting some determinate subject matter, so that the axioms are developed into determinate propositions. It follows from the logical law of contradiction that the proof of the existence theorem proves also the consistency of the axioms. This is the only method of proof of consistency. The axioms of a set are independent of each other when no axiom can be deduced from the remaining axioms of the set.

Gödel's incompleteness theorems Gödel's incompleteness theorems is the name given to two theorems (true mathematical statements), proved by Kurt Gödel in 1931. They are theorems in mathematical logic. Mathematicians once thought that everything that is true has a mathematical proof. A system that has this property is called complete; one that does not is called incomplete. Also, mathematical ideas should not have contradictions. This means that they should not be true and false at the same time. A system that does not include contradictions is called consistent. A system is a collection of theorems (logical consequences) based on axioms (basic assumptions). Axioms are statements that are accepted as true, and need no proof. Gödel said that every non-trivial formal system (consistent and axiomatic system with theorems listable by following an algorithm) is incomplete and not provably consistent:[1][2] - There will always be questions that cannot be answered, using a certain set of axioms; there are truths that cannot be proved using the axioms of the system. - You cannot prove that a system of axioms is consistent according to the axioms of the system.

The fish farming industry is expanding, and to achieve economic and ecological sustainability, new fish feeds are being developed. When developing new feeds, it can be useful to first simulate on a computer how the biological network will react. This can be done with metabolic models. Metabolic models consist of the reactions and metabolites arranged into a stoichiometric matrix. Constraints on the network are imposed in the form of stoichiometric coefficients and bounds on reaction rates. To trust the results from a simulation, it is important that the model is well annotated and internally consistent, i.e. of high quality. The software Memote tests the model for a set of quality criteria and presents the score in a report. This thesis will discuss the application, development and validation of Memote’s tests. This is done by implementing three possible improvements iteratively to a model and testing with Memote for each iteration, eventually composing a Memote history report to inspect the change in score for the different model versions. There is an overall emphasis on annotations to databases in the tests; a wide range of annotations for genes, metabolites and reactions will increase the score. Including physiologically important reactions, such as secretion of CO2 will also increase the score. Memote is a good tool to show what the model contains, the scope and notify you if a feature you think was added to the model, was in fact not added. To more thoroughly review the