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What you're saying isn't demonstrating any inconsistency. A thing is only true in mathematics if it follows from the definitions. If you change your definitions
by throwawaymath 8y ago
What you're saying isn't demonstrating any inconsistency. A thing is only true in mathematics if it follows from the definitions. If you change your definitions, you should expect that theorems built upon those definitions will no longer hold.
Stated another way, consistency is only a coherent mathematical concept from the perspective of a specific set of definitions. There's no problem here.
- andrewla 8y agoI don't claim that this makes it inconsistent. Just that it admits meaningless concepts. You can attribute this, if you like, to the fact that I am a computer scientist, and see things through that lens. If you give me a statement saying "For two sets, A and B, is 'A union intersection B empty", the statement would not be well-formed, so you could just say that it is not a valid proposition without asserting anything about its truthiness. Similarly, a good foundation of mathematics should be able to reject a statement like "2 is a member of 4" as poorly formed, and make no statement about its truthiness. I guess what I'm saying is that set theory is a bad basis for mathematics and should just be regarded as an interesting relic from previous attempts to formalize mathematics so that we can move on to more powerful systems.