Re: Ordinals

From	Mild Shock <janburse@fastmail.fm>
Newsgroups	sci.math
Subject	Re: Ordinals
Date	2024-03-01 20:38 +0100
Message-ID	<urtarb$qese$1@solani.org> (permalink)
References	(8 earlier) <b8dMn7_7dL9kYr-1qK_KxCDw7gU@jntp> <urptma$abti$2@i2pn2.org> <2NRiNAMtOqLljuoKQ0Z4UbqiHbE@jntp> <urrh33$cbpp$7@i2pn2.org> <thicnWRrTbih0Hz4nZ2dnZfqn_udnZ2d@giganews.com>

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Somehow I am quite new to the ordinal analysis
that Alan Turing started. Nice find Gödel introduces
a nice notion in his incompletness paper,

namely he says a function is primitive recursive,
to degree N, if the primitive recursion schema is
applied N times. So I guess this definition of

factorial would then have degree 3, since 'R' is used 3 times:

/* The R combinator */
natrec(_, 0, X, X) :- !.    % attention, not steadfast
natrec(F, N, X, Z) :- M is N-1, natrec(F, M, X, Y), call(F, Y, Z).

plus(X, Y, Z) :- natrec(succ, X, Y, Z).

mult(X, Y, Z) :- natrec(plus(X), Y, 0, Z).

step((X,Y),(Z,T)) :- succ(X,Z), mult(Z,Y,T).

factorial(X,Y) :- natrec(step,X,(0,1),(_,Y)).

Works fine, although eats quite some computing resources,
i.e. 9_303_219 inferences, since it must form
3268800 successors:

?- factorial(10,X).
X = 3628800.

?- time(factorial(10,X)).
% 9,303,219 inferences, 0.578 CPU in 0.617 seconds
(94% CPU, 16092054 Lips)
X = 3628800.

Ross Finlayson schrieb:
> ... fourier my ass, what has it to do with ordinals ...
>

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Re: Ordinals "markus...@gmail.com" <markusklyver@gmail.com> - 2024-02-20 11:35 -0800
  Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-20 20:54 +0100
    Re: Ordinals "mitchr...@gmail.com" <mitchrae3323@gmail.com> - 2024-02-20 11:59 -0800
  Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-20 12:15 -0800
    Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-20 12:27 -0800
    Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-20 17:02 -0500
      Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-20 19:36 -0800
        Re: Ordinals Mild Shock <bursejan@gmail.com> - 2024-02-21 05:47 -0800
          Re: Ordinals Mild Shock <bursejan@gmail.com> - 2024-02-21 05:50 -0800
            Re: Ordinals Mild Shock <bursejan@gmail.com> - 2024-02-21 06:16 -0800
        Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-21 10:24 -0800
          Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 09:03 +0100
            Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 09:11 +0100
            Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-22 10:16 -0800
              Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 19:20 +0100
                Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 19:22 +0100
                Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 19:40 +0100
                Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 20:00 +0100
                Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-22 10:55 -0800
                Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 20:08 +0100
                Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 20:13 +0100
                Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-22 11:01 -0800
                Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 20:11 +0100
                Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 20:13 +0100
            Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-22 14:34 -0500
              Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-22 11:39 -0800
              Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-23 00:41 +0100
      Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-21 08:33 +0000
        Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-21 12:59 -0500
          Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-22 13:00 +0000
            Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-22 11:13 -0500
              Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-23 08:47 +0000
                Semanticists candy (Was: Ordinals) Mild Shock <janburse@fastmail.fm> - 2024-02-26 12:58 +0100
                Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-27 14:25 -0500
                Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-27 11:59 -0800
                Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-27 20:05 +0000
                Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-27 17:24 -0500
                Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-28 09:48 +0000
                Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-28 06:52 -0500
                Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-28 17:24 +0000
                Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-02-28 17:07 -0500
                Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-29 08:23 +0000
                Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-02-29 07:35 -0500
                Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-29 19:25 +0000
                Re: Ordinals "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-02-29 13:48 -0800
                Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-02-29 22:12 -0500
                Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-29 19:36 -0800
                Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 20:38 +0100
                Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 20:52 +0100
                Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 21:08 +0100
                Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-03-01 15:56 -0500
                Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 22:53 +0100
                Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 23:11 +0100
                Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 23:33 +0100
                Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-03-01 19:41 -0500
                Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-03-01 20:15 -0800
                Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-03-02 10:49 -0800
                Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-02 20:40 +0100
                Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-03-01 14:45 -0800
                Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-03-01 08:47 +0000
                Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-03-01 09:44 -0500
                Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-03-01 18:29 +0000
                Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-03-01 14:00 -0500
                Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-03-02 12:36 +0000
                Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-03-02 09:22 -0500
                Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-03-02 15:25 +0000
  Re: Ordinals "mitchr...@gmail.com" <mitchrae3323@gmail.com> - 2024-02-20 17:21 -0800
    RSemanticists candy (Re: Ordinals) [Addendum] Mild Shock <janburse@fastmail.fm> - 2024-02-26 13:01 +0100

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