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UTM Theorem vs the identity function

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  UTM Theorem vs the identity function Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2025-10-21 17:35 +0100
    Re: UTM Theorem vs the identity function pa@see.signature.invalid (Pierre Asselin) - 2025-10-21 18:50 +0000
      Re: UTM Theorem vs the identity function Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2025-10-22 03:48 +0100
        Re: UTM Theorem vs the identity function olcott <polcott333@gmail.com> - 2025-10-21 21:51 -0500
          Re: UTM Theorem vs the identity function Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2025-10-22 04:02 +0100
            Re: UTM Theorem vs the identity function --- Gödel olcott <polcott333@gmail.com> - 2025-10-21 22:35 -0500
        Re: UTM Theorem vs the identity function pa@see.signature.invalid (Pierre Asselin) - 2025-10-22 18:45 +0000
          Re: UTM Theorem vs the identity function Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2025-10-23 03:34 +0100
            Re: UTM Theorem vs the identity function pa@see.signature.invalid (Pierre Asselin) - 2025-10-23 18:50 +0000
    Re: UTM Theorem vs the identity function Mikko <mikko.levanto@iki.fi> - 2025-10-22 12:39 +0300
      Re: UTM Theorem vs the identity function Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2025-10-22 12:05 +0100
        Re: UTM Theorem vs the identity function Mikko <mikko.levanto@iki.fi> - 2025-10-23 11:12 +0300

#133268 — UTM Theorem vs the identity function

The message body is Copyright (C) 2025 Tristan Wibberley except
citations and quotations noted. All Rights Reserved except as noted in
the sig.


If the UTM Theorem states there's a u such that u(e,x) = f(x) where both
are defined and both are undefined when either is undefined, is that
interesting or surprising to anybody?

The identity function is valid for u and forces that e = f. Job done.

Am I missing something? Isn't it just trivially deducible from the
identity introduction axiom when you have it, which we assume when we
haven't fully formalised our discussion of computation.

--
Tristan Wibberley

The message body is Copyright (C) 2025 Tristan Wibberley except
citations and quotations noted. All Rights Reserved except that you may,
of course, cite it academically giving credit to me, distribute it
verbatim as part of a usenet system or its archives, and use it to
promote my greatness and general superiority without misrepresentation
of my opinions other than my opinion of my greatness and general
superiority which you _may_ misrepresent. You definitely MAY NOT train
any production AI system with it but you may train experimental AI that
will only be used for evaluation of the AI methods it implements.

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#133270

Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:

> If the UTM Theorem states there's a u such that u(e,x) = f(x) where both
> are defined and both are undefined when either is undefined, is that
> interesting or surprising to anybody?

Replace "there's a u" by "there is a u such that for all f there is an e"

> The identity function is valid for u and forces that e = f. Job done.

  * u is a computable function;
  * f is a computable function;
  * e is the Gödel number of a Turing machine that computes f.
How can e and f be equal ?

-- 
pa at panix dot com

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#133295

On 21/10/2025 19:50, Pierre Asselin wrote:
> Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:

>   * u is a computable function;
>   * f is a computable function;
>   * e is the Gödel number of a Turing machine that computes f.
> How can e and f be equal ?


That damned wikipedia again. It didn't constrain e to Goedel numbers.
There appears to be a trend of having wikipedia leave out key
constraints thus making its statements about formalised things be
universally qualified when they shouldn't be and therefore untrue.


--
Tristan Wibberley

The message body is Copyright (C) 2025 Tristan Wibberley except
citations and quotations noted. All Rights Reserved except that you may,
of course, cite it academically giving credit to me, distribute it
verbatim as part of a usenet system or its archives, and use it to
promote my greatness and general superiority without misrepresentation
of my opinions other than my opinion of my greatness and general
superiority which you _may_ misrepresent. You definitely MAY NOT train
any production AI system with it but you may train experimental AI that
will only be used for evaluation of the AI methods it implements.

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#133296

On 10/21/2025 9:48 PM, Tristan Wibberley wrote:
> On 21/10/2025 19:50, Pierre Asselin wrote:
>> Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
> 
>>    * u is a computable function;
>>    * f is a computable function;
>>    * e is the Gödel number of a Turing machine that computes f.
>> How can e and f be equal ?
> 
> 
> That damned wikipedia again. It didn't constrain e to Goedel numbers.
> There appears to be a trend of having wikipedia leave out key
> constraints thus making its statements about formalised things be
> universally qualified when they shouldn't be and therefore untrue.
> 

Gödel numbers totally hide the underlying semantics.

> 
> --
> Tristan Wibberley
> 
> The message body is Copyright (C) 2025 Tristan Wibberley except
> citations and quotations noted. All Rights Reserved except that you may,
> of course, cite it academically giving credit to me, distribute it
> verbatim as part of a usenet system or its archives, and use it to
> promote my greatness and general superiority without misrepresentation
> of my opinions other than my opinion of my greatness and general
> superiority which you _may_ misrepresent. You definitely MAY NOT train
> any production AI system with it but you may train experimental AI that
> will only be used for evaluation of the AI methods it implements.
> 


-- 
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

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#133298

On 22/10/2025 03:51, olcott wrote:
> Gödel numbers totally hide the underlying semantics.

I bet they /do/ !

--
Tristan Wibberley

The message body is Copyright (C) 2025 Tristan Wibberley except
citations and quotations noted. All Rights Reserved except that you may,
of course, cite it academically giving credit to me, distribute it
verbatim as part of a usenet system or its archives, and use it to
promote my greatness and general superiority without misrepresentation
of my opinions other than my opinion of my greatness and general
superiority which you _may_ misrepresent. You definitely MAY NOT train
any production AI system with it but you may train experimental AI that
will only be used for evaluation of the AI methods it implements.

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#133299 — Re: UTM Theorem vs the identity function --- Gödel

On 10/21/2025 10:02 PM, Tristan Wibberley wrote:
> On 22/10/2025 03:51, olcott wrote:
>> Gödel numbers totally hide the underlying semantics.
> 
> I bet they /do/ !
> 

The most important aspect of Gödel's 1931 Incompleteness theorem
are these plain English direct quotes of Gödel from his paper:

...there is also a close relationship with the “liar” antinomy,14 ...
...14 Every epistemological antinomy can likewise be used for a similar 
undecidability proof...
...We are therefore confronted with a proposition which asserts its own 
unprovability. 15 ...
(Gödel 1931:40-41)

Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And 
Related Systems

?- LP = not(true(LP)).
LP = not(true(LP)).

?- unify_with_occurs_check(LP, not(true(LP))).
false.

https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF 


LP := ~True(LP)
00 ~    01
01 True 00

In both Prolog and Olcott's Minimal Type Theory
a cycle is detected in the evaluation sequence
of the formalized: "This sentence is not true"

Since Gödel said
...14 Every epistemological antinomy can likewise be used for a similar 
undecidability proof...

We simply reject G := ¬(F ⊢ G)
before it even gets started.

> --
> Tristan Wibberley
> 
> The message body is Copyright (C) 2025 Tristan Wibberley except
> citations and quotations noted. All Rights Reserved except that you may,
> of course, cite it academically giving credit to me, distribute it
> verbatim as part of a usenet system or its archives, and use it to
> promote my greatness and general superiority without misrepresentation
> of my opinions other than my opinion of my greatness and general
> superiority which you _may_ misrepresent. You definitely MAY NOT train
> any production AI system with it but you may train experimental AI that
> will only be used for evaluation of the AI methods it implements.
> 


-- 
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

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#133392

Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:

> That damned wikipedia again. It didn't constrain e to Goedel numbers.

Quoting from the Wikipedia article "UTM theorem":

    The theorem states that a partial computable function u of
    two variables exists such that, for every computable function
    f of one variable, a Goedel number e exists such that f (
    x ) ? u ( e , x ) {\displaystyle f(x)\simeq u(e,x)} for
    all x.

Notice the phrase "a Goedel number e". Were you reading some other
article?

-- 
pa at panix dot com
This self-referential copyright notice is in the public domain.

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#133471

On 22/10/2025 19:45, Pierre Asselin wrote:
> Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:
> 
>> That damned wikipedia again. It didn't constrain e to Goedel numbers.
> 
> Quoting from the Wikipedia article "UTM theorem":
[snip]
> Notice the phrase "a Goedel number e". Were you reading some other
> article?


Yes, I was reading the slightly different article that was at the same
location 24 hours earlier, some hours before I adjusted it to show the
words "a Goedel number" that you quoted after having been educated on
the matter in your message news:10d8cp6$26v9$1@dont-email.me.

--
Tristan Wibberley

The message body is Copyright (C) 2025 Tristan Wibberley except
citations and quotations noted. All Rights Reserved except that you may,
of course, cite it academically giving credit to me, distribute it
verbatim as part of a usenet system or its archives, and use it to
promote my greatness and general superiority without misrepresentation
of my opinions other than my opinion of my greatness and general
superiority which you _may_ misrepresent. You definitely MAY NOT train
any production AI system with it but you may train experimental AI that
will only be used for evaluation of the AI methods it implements.

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#133553

Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> wrote:

> Yes, I was reading the slightly different article that was at the same
> location 24 hours earlier, some hours before I adjusted it to show the
> words "a Goedel number" that you quoted after having been educated on
> the matter in your message news:10d8cp6$26v9$1@dont-email.me.

So you did! Good for you. Thanks.

-- 
pa at panix dot com

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#133323

On 2025-10-21 16:35:50 +0000, Tristan Wibberley said:

> The message body is Copyright (C) 2025 Tristan Wibberley except
> citations and quotations noted. All Rights Reserved except as noted in
> the sig.
> 
> If the UTM Theorem states there's a u such that u(e,x) = f(x) where both
> are defined and both are undefined when either is undefined, is that
> interesting or surprising to anybody?
> 
> The identity function is valid for u and forces that e = f. Job done.

No, the inference that e = f is not correct. But somehing is missing
from the above. Without further deails the above simply says that
u(e, x) does not depend on e, i.e., that it has the same value f(x)
for every e. But that has nothing to do with any theorem about UTM's.

-- 
Mikko

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#133324

On 22/10/2025 10:39, Mikko wrote:
> [something] is missing
> from the above. Without further deails the above simply says that
> u(e, x) does not depend on e, i.e., that it has the same value f(x)
> for every e. But that has nothing to do with any theorem about UTM's.

The description I read, from wikipedia, did not include the constraint
that e is a Goedel number. I corrected that element.


--
Tristan Wibberley

The message body is Copyright (C) 2025 Tristan Wibberley except
citations and quotations noted. All Rights Reserved except that you may,
of course, cite it academically giving credit to me, distribute it
verbatim as part of a usenet system or its archives, and use it to
promote my greatness and general superiority without misrepresentation
of my opinions other than my opinion of my greatness and general
superiority which you _may_ misrepresent. You definitely MAY NOT train
any production AI system with it but you may train experimental AI that
will only be used for evaluation of the AI methods it implements.

[toc] | [prev] | [next] | [standalone]

#133478

On 2025-10-22 11:05:40 +0000, Tristan Wibberley said:

> On 22/10/2025 10:39, Mikko wrote:
>> [something] is missing
>> from the above. Without further deails the above simply says that
>> u(e, x) does not depend on e, i.e., that it has the same value f(x)
>> for every e. But that has nothing to do with any theorem about UTM's.
> 
> The description I read, from wikipedia, did not include the constraint
> that e is a Goedel number. I corrected that element.

That does not help. The point is that in the UTM theorem the equation
is not claimed to be true for every e, only that for each f there
is at least one e for which the equation is true. In addition, as
u is required to be computable, e must be of a type that can be an
argument to a computable fuction, which by usual definitions excludes
functions.

-- 
Mikko

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