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June 7th, 2014, 01:30 PM   #11
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OK, but are there many interesting facts that require CH? Even if this is the case, CH then serves as as an axiom whose independence from other axioms and whose importance was realized in the 20th century. So we have AC and CH. Are there many more? I was saying that hunting for new axioms just to be able to justify interesting results is not a common occurrence. But even if it were, this does not change the concept of a formal proof (in the narrow sense). Or do you want to say that this concept of mathematics is deficient? For example, that we determine what's true intuitively rather than using formal derivations? Then how would you define a formal proof?
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June 7th, 2014, 01:49 PM   #12
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Originally Posted by Evgeny.Makarov View Post
OK, but are there many interesting facts that require CH? Even if this is the case, CH then serves as as an axiom whose independence from other axioms and whose importance was realized in the 20th century. So we have AC and CH. Are there many more? I was saying that hunting for new axioms just to be able to justify interesting results is not a common occurrence. But even if it were, this does not change the concept of a formal proof (in the narrow sense). Or do you want to say that this concept of mathematics is deficient? For example, that we determine what's true intuitively rather than using formal derivations? Then how would you define a formal proof?
The problem I have with CH as an axiom is that it's not obvious. It's not an obvious statement, which we clearly want for axioms to be. I place intuition above logic, yes. I just don't know enough about all this to answer. Maybe a formal proof is simply a way from one truth to another.
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