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September 1st, 2017, 11:56 AM   #51
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That's certainly a good point to bring up - while you can deduce any true or false statement from a false one, you can't deduce a meaningless statement from one. However, I'm not seeing why the statement that Maschke is the pope is meaningless (or that it depends on who is saying it) - surely it's just false?
Masche's conclusion was 'I am the Pope' and I think he meant it as a reductio ad absurdum to support his claim that what I was saying was absurd , but the conclusion is not absurd if the actual Pope draws the same conclusion.
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September 1st, 2017, 12:13 PM   #52
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But I know that 5 isn't divisible by 2 that's why I said "Unfortunately 5 is prime"

The tone of your postings seems hostile to me.
I am sorry you feel that way. I do not believe I have made any ad hominem comments at all.

My position has been quite consistent although my very first post was poorly worded. If we assume that 5 is not prime, we are dealing with a different arithmetic (or a different definition of prime). Without further specification of what is intended, we do not know whether your original statement that ALL primes would be of the form 6n + 1 if 5 were not prime can be validated. It obviously cannot be validated for any arithmetic in which 5 is not prime because you yourself gave a counter-example.

I agree with mashcke on two points. First, from a false premise, anything can be deduced. Second, an argument that both denies and relies on the rules of normal arithmetic is not interesting. Where I deviated from him was that I actually was interested in seeing where the supposition of an arithmetic where 5 was composite might go. Your mod 11 arithmetic seems to fit the bill, but, alas, not all primes in that arithmetic are of the form 6n + 1.
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September 1st, 2017, 12:17 PM   #53
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Your mod 11 arithmetic seems to fit the bill, but, alas, not all primes in that arithmetic are of the form 6n + 1.
There are no primes in the ring of integers mod 11, so it's vacuously true that all primes in that ring are of the form 6n + 1.
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September 1st, 2017, 12:40 PM   #54
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There are no primes in the ring of integers mod 11, so it's vacuously true that all primes in that ring are of the form 6n + 1.
By definition but is there a good reason for the definition or is it a convention?
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September 1st, 2017, 01:13 PM   #55
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By definition but is there a good reason for the definition or is it a convention?
Just as we exclude $1$ from being a rational prime and $1,-1,i,-i$ from being primes in the ring of Gaussian integers, we exclude units from being primes in integral domains in general. Ultimately, it is just convention, but it does have advantages (and no significant drawbacks). This way, a lot of results are much easier to state, such as unique factorisation (which only holds in certain rings!) and various things relating prime ideals and prime elements. Note that fields are just non-trivial commutative rings in which all non-zero elements are units

Last edited by cjem; September 1st, 2017 at 01:50 PM.
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