5 ms·
All of Mathematics is inconsistent. Turing machines are consistent
There is a conflation between identity and equality in Classical logic, but a distinction between these concepts on a Turing machine.
IDENTITY means unique memory address.
EQUALITY means contents-of-memory address.
The error in Mathematics is precisely the conflation of identity and value. Or in terms of a physics conception. Mistaking the space-time coordinates with that which occupies them.
In the real world A = A is allowed to be false (when interpreted as identity) because the two "A"s exist at different space-time coordinates. And so what does it mean for TWO individual things to be "equal" are they entangled or what?
Classical logic overloads "=" to mean both identity and equality. That's why it's inconsistent. Classical logic doesn't have UUIDs - computers do. Memory addresses.
for all x: x = x => Undefined, Complexity: O(1) to O(∞)
for all x: id(x) = id(x) => True, Complexity: O(1)
for all x: id(x) = x => False, Complexity: O(1)
Further. ALL operators are supposed to do actual, physical work.
The energy spent on deriving the correct result of x = x is the proof-of-work.
If x is an infinite-byte object then comparing it to itself should take infinite time e.g machine will not halt, whereas determining its identity is O(1)
Solution to Symbol-grounding problem. ( https://en.wikipedia.org/wiki/Symbol_grounding_problem)
Q.E.D
λ-calculus ⇔ λ-calculus ⇔ λ-calculus ⇔ λ-calculus ⇔ λ-calculus ⊇ Mathematics
Feedback welcome.
- deleted 8y ago[deleted]
- throway88989898 8y agohttps://en.wikipedia.org/wiki/Halting_problem https://en.wikipedia.org/wiki/Halting_problem https://en.wikipedia.org/wiki/Proof_of_impossibility https://en.wikipedia.org/wiki/Proof_of_impossibility https://en.wikipedia.org/wiki/Set_theory#Objections_to_set_theory_as_a_foundation_for_mathematics https://en.wikipedia.org/wiki/Set_theory#Objections_to_set_t...
- ukj 8y agoPrecisely. The point is that ALL operators do actual, physical work. Work == Energy. Energy == Proof-of-work. If X is an infinite-byte object then comparing it to itself should take infinite time, whereas determining its identity is O(1)
- ukj 8y agoAlso, I didn't start with set theory to get here. I started with Type theory as foundational.
- throway88989898 8y agoIt's outside my knowledge space. What are the implications to computability or mathematics?
- ukj 8y agoThe law of identity is a blunder. It's the Principle of explosion in disguise. If one x = x can be trivial to determine But another x = x is infinitely complex then in one single law you have a triviality and infinity. That's the principle of explosion !
- deleted 8y ago[deleted]
- jasonhansel 8y agoIt's not an inconsistency, it's an axiom: https://en.m.wikipedia.org/wiki/Axiom_of_extensionality https://en.m.wikipedia.org/wiki/Axiom_of_extensionality
- throway88989898 8y agoI think they meant it's inconsistent with the 'memory value recorded vs memory address value' distinction in a Turing machine (as in, it's not consistent with its physical instantiation: "the real world"), not inconsistent with axioms of formal logic. I'm getting confused because I think these are two or more different research camps (computability, philosophy, logic, etc) and I don't know how much jargon is shared. OP's other post's comment is related https://news.ycombinator.com/item?id=19265632 https://news.ycombinator.com/item?id=19265632
- ukj 8y agoI have tossed set theory out entirely. I consider Lambda foundational. I am not even talking about comparing sets. I am talking about comparing infinite-precision floats to themselves. Any system that does it in O(1) is broken.
- deleted 8y ago[deleted]
- ukj 8y agoThe principle of explosion is hiding in x = x itself! The complexity of the task is O(1) to O(∞) !!!