
All of Mathematics is inconsistent. Turing machines are consistent - ukj
There is a conflation between <i></i>identity<i></i> and <i></i>equality<i></i> in Classical logic, but a distinction between these concepts on a Turing machine.<p>IDENTITY means unique memory address.
EQUALITY means contents-of-memory address.<p>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.<p>In the real world A = A is allowed to be false (when interpreted as identity) because the two &quot;A&quot;s exist at different space-time coordinates. And so what does it mean for <i></i>TWO<i></i> <i></i>individual<i></i> things to be &quot;equal&quot; are they entangled or what?<p>Classical logic overloads &quot;=&quot; to mean both identity and equality. That&#x27;s why it&#x27;s inconsistent. Classical logic doesn&#x27;t have UUIDs - computers do. Memory addresses.<p><pre><code>   for all x: x = x  =&gt; Undefined, Complexity: O(1) to O(∞)
   for all x: id(x) = id(x) =&gt; True, Complexity: O(1)   
   for all x: id(x) = x =&gt; False, Complexity: O(1)
</code></pre>
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.<p>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)<p>Solution to Symbol-grounding problem. ( https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Symbol_grounding_problem)<p>Q.E.D<p>λ-calculus ⇔ λ-calculus ⇔ λ-calculus ⇔ λ-calculus ⇔ λ-calculus ⊇ Mathematics<p>Feedback welcome.
======
throway88989898
[https://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_t...](https://en.wikipedia.org/wiki/Set_theory#Objections_to_set_theory_as_a_foundation_for_mathematics)

~~~
ukj
Also, I didn't start with set theory to get here. I started with Type theory
as foundational.

~~~
throway88989898
It's outside my knowledge space. What are the implications to computability or
mathematics?

~~~
ukj
The 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 !

------
jasonhansel
It'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)

~~~
ukj
I 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.

------
ukj
The principle of explosion is hiding in x = x itself!

The complexity of the task is O(1) to O(∞) !!!

