Re: D correctly simulated by H cannot possibly halt --- templates and infinite sets

On 31/05/24 10:51, Mikko wrote:

On 2024-05-30 11:33:03 +0000, Richard Damon said:
 

On 5/30/24 6:11 AM, immibis wrote:

On 30/05/24 05:48, olcott wrote:

On 5/29/2024 9:55 PM, Richard Damon wrote:

On 5/29/24 10:36 PM, olcott wrote:

On 5/29/2024 9:25 PM, Richard Damon wrote:

On 5/29/24 9:55 PM, olcott wrote:

When the category is examined all at once then there is no need
to look at each individual element.

 So, which one or ones gave the correct answer for their input?
 

 *Formalizing the Linz Proof structure*
∃H  ∈ Turing_Machines
∀x  ∈ *Turing_Machines_Descriptions*
∀y  ∈ Finite_Strings
such that H(x,y) = Halts(x,y)
 When we formalize it that way then some simulating halt deciders
get the correct answer.
 *Everyone else implicitly assumes this incorrect formalization*
∃H  ∈ Turing_Machines
∀x  ∈ *Turing_Machines*
∀y  ∈ Finite_Strings
such that H(x,y) = Halts(x,y)
 

 Nope.
 You just don't understand the meaning of a "Description" in the problem.
 

 A deciders compute the mapping FROM ITS INPUTS
to it own accept or reject state
Deciders cannot take ACTUAL TURING MACHINES AS INPUTS
Deciders can only take FINITE STRINGS AS INPUTS
 

 If you want to be pedantic, you made the mistake.
It's actually H(DescriptionOf(x),y) = Halts(x,y)
DescriptionOf is an injective function that converts Turing machines into finite strings.

 OR in his formal language:
  ∃H  ∈ Turing_Machines, and for
∀M  ∈ Turing_Machines, which have a description Wm, and
∀w  ∈ Finite_Strings
such that H(Wm,w) = Halts(M,w)
 The problem is that there doesn't exist a FUNCTION "Description" as one machine can have an potentially infinite number of descriptions.

 At this point we may assume the axiom of choice.
 

It is more that there is an inverse function, Described where we could say that M = Described(Wm)

 

also the function "Description" doesn't have to 'know' all of the descriptions - one per machine is enough.