3 ms·
> Why? Any set of 4 things also contains 2 things. Where's the nonsense? Within this definition, that's consistent. That's the problem -- it's only within this
by andrewla 8y ago
> Why? Any set of 4 things also contains 2 things. Where's the nonsense? Within this definition, that's consistent.
That's the problem -- it's only within this definition. In other definitions, notable the nesting example that I gave, that theorem is false. And the theorem makes no sense in and of itself, because we're talking about numbers, so it is unexpected that the membership operator would apply at all.
Whereas I can take two definitions of the natural numbers
PN =
Peano_One: PN
Peano_Succ: PN -> PN
BN =
Binary_One: BN
Binary_2x: BN -> BN
Binary_2xp1: BN -> BN
and I can define a Plus: PN x PN -> PN and a Plus: BN x BN -> BN, and so on, and once I can define Binary_Succ: BN -> BN and Peano_2x and Peano_2xp1 I can prove that these are isometric types, so all theorems derivable from PN apply to BN and vice versa, not just the convenient theorems that don't use any syntax from the meta-language (e.g. set theory).
- xamuel 8y agoIt's similar to the way in C you can malloc memory for two data structures, and then check which pointer is smaller. Say, one pointer points at the data for Mario, and the other pointer points at the data for Luigi. You look at the raw pointers and notice that the Mario pointer is a smaller 64-bit number than the Luigi pointer. That seems absurd or arbitrary because why should "Mario" be "smaller" than "Luigi"? But it doesn't really matter: the game still runs fine and nothing breaks as a result of the arbitrariness.
- throwawaymath 8y agoWhat 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.