Section 5-3b - Austin Mohr
... Items marked with an asterisk (∗) require you to submit a written explanation or solution. Please form your explanations with complete sentences and in your own words (as though you are explaining the concept to another student in the course). If the problem requires computation, make sure that it i ...
... Items marked with an asterisk (∗) require you to submit a written explanation or solution. Please form your explanations with complete sentences and in your own words (as though you are explaining the concept to another student in the course). If the problem requires computation, make sure that it i ...
Incompleteness Result
... working mathematicians to continue their pursuit to prove or disprove those historically well-known mathematical conjectures. However, the bad news is that Godel also gives us the incompleteness proofs (1931), which in effect apply a self-undermining “Godel sentence” which says that “I am not provab ...
... working mathematicians to continue their pursuit to prove or disprove those historically well-known mathematical conjectures. However, the bad news is that Godel also gives us the incompleteness proofs (1931), which in effect apply a self-undermining “Godel sentence” which says that “I am not provab ...
On integers with many small prime factors
... much to get as good results as possible, but it should be noted that it is possible to improve upon our results by using not yet published results of Stark and others, or by improving upon the auxiliary results. This is possible because of additional information available here. For the proofs of The ...
... much to get as good results as possible, but it should be noted that it is possible to improve upon our results by using not yet published results of Stark and others, or by improving upon the auxiliary results. This is possible because of additional information available here. For the proofs of The ...
On a Symposium on the Foundations of Mathematics (1971) Paul
... axioms, as was done simultaneously by A.A. Fraenkel and Thoralf Skolem. The systems presented by W.V. Quine mediated, as it were, between type theory and the systems of axiomatic set theory. It turned out, however, by the results of Kurt Gödel and Skolem, that all such strictly formal frameworks fo ...
... axioms, as was done simultaneously by A.A. Fraenkel and Thoralf Skolem. The systems presented by W.V. Quine mediated, as it were, between type theory and the systems of axiomatic set theory. It turned out, however, by the results of Kurt Gödel and Skolem, that all such strictly formal frameworks fo ...
Document
... Note: the construction need not be easy or obvious (most of the time it isn't). Stare at the problem, think about it, draw pictures (if possible), doodle... eventually something may happen (no guarantee). The only way you get reasonably proficient at concocting proofs of (simple) theorems is t ...
... Note: the construction need not be easy or obvious (most of the time it isn't). Stare at the problem, think about it, draw pictures (if possible), doodle... eventually something may happen (no guarantee). The only way you get reasonably proficient at concocting proofs of (simple) theorems is t ...
The First Incompleteness Theorem
... The terminological situation is a bit messy. Some labels we’ll be using are entirely standard; some are used in importantly different ways by different authors; while some natural ideas seem to have no commonly used labels at all. It might be helpful, then, if I star my own non-standard terminology ...
... The terminological situation is a bit messy. Some labels we’ll be using are entirely standard; some are used in importantly different ways by different authors; while some natural ideas seem to have no commonly used labels at all. It might be helpful, then, if I star my own non-standard terminology ...
DOC - John Woods
... A big question is, “Why do we bother with proof theory?” After all, its principal concepts – axiom, theorem, deduction, proof – have no intuitive meaning there. What’s the point? Suppose we could show that for each of these uninterpreted properties of CPL’s proof theory theory is a unique counterpar ...
... A big question is, “Why do we bother with proof theory?” After all, its principal concepts – axiom, theorem, deduction, proof – have no intuitive meaning there. What’s the point? Suppose we could show that for each of these uninterpreted properties of CPL’s proof theory theory is a unique counterpar ...
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. Such arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being ""trivial"", or ""difficult"", or ""deep"", or even ""beautiful"". These subjective judgments vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.