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Isn’t the order of a proposition included in its Gödel number? Each proposition is assigned to an increasing prime power, and the increasing list of primes has
by espadrine 5y ago
Isn’t the order of a proposition included in its Gödel number?
Each proposition is assigned to an increasing prime power, and the increasing list of primes has total order, such that swapping propositions yields a distinct Gödel number.
- ProfHewitt 5y ago[Gödel 1931] encoded the characters of a proposition in its Gödel number but did not include the order of the proposition.
- espadrine 5y agoCould you provide two statements with equal Gödel numbers but distinct proposition orders, and a detail of how you compute the Gödel numbers?
- ProfHewitt 5y agoIf two propositions have the same characters, then they have the same Gödel numbers.
- espadrine 5y agoCould you provide two statements with equal Gödel numbers and the same characters in different orders, and a detail of how you compute the Gödel numbers?
- ProfHewitt 5y agoIf two propositions have the same characters in the same order, then they have the same Gödel numbers. Unfortunately, Gödel number of a proposition leaves out the order of the proposition :-(
- espadrine 5y agoIf two statements have the same characters in the same order, how are they not the same statement? And why would that make Gödel’s statement invalid in Principia Mathematica?
- ProfHewitt 5y agoThere might be two propositions of different orders that have the same characters in the same order.
- tsimionescu 5y agoI think what ProfHewitt means here (based on other writing I've found, as he is frustratingly low on details in these conversations) is "order" in the sense of "first order logic", "second order logic"; not in the sense of "first proposition, second proposition" etc. His claim is that the proposition "This proposition is not provable", formalized as "P equivalent to P is not provable" is not well formed in a typed logic, as "P"'s type is first-order, while "P is not provable"'s type is second-order. Therefore, his claim is that the proposition is simply not well-typed and therefore not interesting. Godel's proofs were discussing an untyped logic, but according to Hewitt that is not an accurate representation of mathematics. I don't think anyone in the space agrees with him, though, as far as I could tell from some cursory reading.
- ProfHewitt 5y ago[Gödel 1931] was for a system for the foundations of mathematics with orders on propositions. The [Gödel 1931] proposition I'mUnprovable is certainly of historical interest and may perhaps be of philosophical interest. However, including the proposition I'mUnprovable in foundations makes the foundations inconsistent.