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

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.

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So, which one or ones gave the correct answer for their input?
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*Formalizing the Linz Proof structure*
∃H  ∈ Turing_Machines
∀x  ∈ *Turing_Machines_Descriptions*
∀y  ∈ Finite_Strings
such that H(x,y) = Halts(x,y)
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When we formalize it that way then some simulating halt deciders
get the correct answer.
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*Everyone else implicitly assumes this incorrect formalization*
∃H  ∈ Turing_Machines
∀x  ∈ *Turing_Machines*
∀y  ∈ Finite_Strings
such that H(x,y) = Halts(x,y)
>
>

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Nope.
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You just don't understand the meaning of a "Description" in the problem.
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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.
It is more that there is an inverse function, Described where we could say that M = Described(Wm)