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> The thing that I don't like about this sort of construction is that it also implies all sorts of nonsense. For example, I can ask the question "Is 2 a member

> The thing that I don't like about this sort of construction is that it also implies all sorts of nonsense. For example, I can ask the question "Is 2 a member of 4", which is clearly nonsensical, but will get the answer "yes" from this model. Why? Any set of 4 things also contains 2 things. Where's the nonsense? Within this definition, that's consistent. > Type theory and category theory give us a much better way of constructing these sorts of objects without having to resort to creating constructions with all sorts of side effects. Out of curiosity, what is the category theoretic construction that avoids Russell's paradox? I don't know that there isn't one, but I can't think of it off the top of my head. I know there are category theoretic constructions in general (I responded to one someone else posted in this thread).

andrewla 8y ago
The definition of the Peano numerals has two constructors, Zero: N and Succ: N -> N. Russell's paradox (assuming that we're talking about the idea of "set of all sets that are not members of themselves") is avoided simply because the objects produced are not sets, and sets have no exalted position within the mechanics of category theory. Talking about concepts like "the category of categories that don't contain themselves" is kind of navel-gazey; and ends up falling apart in most constructive variants just because you can't give a comprehensive construction of elements of this category. Admittedly I'm throwing some concepts from intuitionalism and type theory in the mix here; if I took the time I could make these statements more precise.
- throwawaymath 8y ago
  I see what you're getting at. My (similarly hand wavey) perspective is that constructions using category theory end up defining natural numbers as categories of sets with cardinality n. What I was really getting at is, how do you define your relation for the category in such a way that isn't pathological ("n is the category of sets with cardinality n except all such sets containing n")? Alternatively, what are you selecting as the objects for your category if not sets of a given cardinality? To be clear, I consider a lot of discussion about the foundations of set theory (and paradoxes thereof) to be pretty navel-gazey.
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 ago
  It'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 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 ago
    I 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.

andrewla 8y ago

The definition of the Peano numerals has two constructors, Zero: N and Succ: N -> N. Russell's paradox (assuming that we're talking about the idea of "set of all sets that are not members of themselves") is avoided simply because the objects produced are not sets, and sets have no exalted position within the mechanics of category theory. Talking about concepts like "the category of categories that don't contain themselves" is kind of navel-gazey; and ends up falling apart in most constructive variants just because you can't give a comprehensive construction of elements of this category. Admittedly I'm throwing some concepts from intuitionalism and type theory in the mix here; if I took the time I could make these statements more precise.

throwawaymath 8y ago
I see what you're getting at. My (similarly hand wavey) perspective is that constructions using category theory end up defining natural numbers as categories of sets with cardinality n. What I was really getting at is, how do you define your relation for the category in such a way that isn't pathological ("n is the category of sets with cardinality n except all such sets containing n")? Alternatively, what are you selecting as the objects for your category if not sets of a given cardinality? To be clear, I consider a lot of discussion about the foundations of set theory (and paradoxes thereof) to be pretty navel-gazey.

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 ago
It'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 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 ago
  I 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.