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Count Iblis
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"[URL consistency of ordinary arithmetic has not yet been satifactorily settled[/URL]. What is the upper limit for N such that arithmetic modulo N is known to be consistent?
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torquil said:So I wonder why some "feel" that the problem is resolved
CRGreathouse said:There is a problem, phrased in natural language. There are proofs written in mathematical formalisms. Some feel that the problem described in natural language are solved by those proofs; some feel that the problem described in natural language is not addressed by the formal proofs.
The issue isn't the validity or existence of the proofs, but their applicability. It is fundamentally an issue of translation.
torquil said:Is it then just that the "problem described in natural language" has not been accurately and uniquely translated into mathematical terms? If so, it would be a case of people interpreting the concept of "consistency" in different ways?
Consistency of arithmetic Mod N refers to the property of arithmetic operations performed on numbers using the modulus N, where N is a positive integer. It means that the result of an arithmetic operation on two numbers will always be the same, regardless of the order in which the operations are performed.
Unlike regular arithmetic operations, where the order of operations can affect the result, consistency of arithmetic Mod N ensures that the order of operations does not change the final result. This is because Mod N operations only take into account the remainder after division by N, rather than the actual values of the numbers involved.
Consistency of arithmetic Mod N is important in mathematics because it allows for the simplification and generalization of mathematical concepts. It also helps in solving complex problems involving large numbers, as the modulus operation reduces the numbers to a smaller range, making them easier to work with.
Consistency of arithmetic Mod N has various practical applications, such as in cryptography, computer algorithms, and error-correction codes. It is also used in industries such as banking and finance, where large numbers are involved in calculations.
The consistency of arithmetic Mod N can be proved using various mathematical techniques, such as induction, direct proof, or proof by contradiction. It can also be demonstrated through examples and counterexamples to show that the property holds true for all numbers and operations.