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From: Bruno Marchal <marchal.domain.name.hidden>

Date: Tue, 24 Sep 2002 11:05:19 +0200

At 21:36 -0400 21/09/2002, Vikee1.domain.name.hidden wrote:

*>For those of you who are familiar with Max Tegmark's TOE, could someone tell
*

*>me whether Georg Cantor's " Absolute Infinity, Absolute Maximum or Absolute
*

*>Infinite Collections" represent "mathematical structures" and, therefore have
*

*>"physical existence".
*

Hi Dave,

Cantor was aware that his "absolute infinity" was strictly speaking

inconsistent. I also deduce from letters Cantor wrote to bishops that

his absolute infinity was some sort of un-nameable "god". The class of all

sets (or of all mathematical structures) can play that role in axiomatic

set theory, but keep in mind that in those context the class of all set is

not a set, nor is the class of all mathematical structure a mathematical

structure. Formalization of this "impossibility" has lead to the "reflection

principle", the fact that if you find a nameable property of such universal

class, then you get a set (a mathematical structure) having that

property, and thus approximating the universal class in your universe

(= model of set theory).

Please read Rudy Rucker "infinity and the mind" which is the best and quasi

unique popular explanation of the reflection principle.

Now "physical existence" is another matter. With the comp hyp in the

cognitive science, physical existence is mathematical existence seen

from inside arithmetics. I agree with Tim and Hal Finney that mathematical

existence is more, and different, from the existence of formal description

of mathematical object. For example, arithmetical truth cannot be unified

in a sound and complete theory, and if comp is true, arithmetical truth

escape all possible consistent set theories even with very large cardinal

axioms. The "seen from inside", that is the 1-person/3-person" distinction

is the key ingredient missed by Schmidhuber and Tegmark (although Tegmark

is apparantly aware of the distinction in his interpretation of QM).

See also Rossler's papers or Svozil's one, for works by physicist who are

aware of that distinction (under the labels exo/endo-physics).

Bruno

Received on Tue Sep 24 2002 - 02:08:38 PDT

Date: Tue, 24 Sep 2002 11:05:19 +0200

At 21:36 -0400 21/09/2002, Vikee1.domain.name.hidden wrote:

Hi Dave,

Cantor was aware that his "absolute infinity" was strictly speaking

inconsistent. I also deduce from letters Cantor wrote to bishops that

his absolute infinity was some sort of un-nameable "god". The class of all

sets (or of all mathematical structures) can play that role in axiomatic

set theory, but keep in mind that in those context the class of all set is

not a set, nor is the class of all mathematical structure a mathematical

structure. Formalization of this "impossibility" has lead to the "reflection

principle", the fact that if you find a nameable property of such universal

class, then you get a set (a mathematical structure) having that

property, and thus approximating the universal class in your universe

(= model of set theory).

Please read Rudy Rucker "infinity and the mind" which is the best and quasi

unique popular explanation of the reflection principle.

Now "physical existence" is another matter. With the comp hyp in the

cognitive science, physical existence is mathematical existence seen

from inside arithmetics. I agree with Tim and Hal Finney that mathematical

existence is more, and different, from the existence of formal description

of mathematical object. For example, arithmetical truth cannot be unified

in a sound and complete theory, and if comp is true, arithmetical truth

escape all possible consistent set theories even with very large cardinal

axioms. The "seen from inside", that is the 1-person/3-person" distinction

is the key ingredient missed by Schmidhuber and Tegmark (although Tegmark

is apparantly aware of the distinction in his interpretation of QM).

See also Rossler's papers or Svozil's one, for works by physicist who are

aware of that distinction (under the labels exo/endo-physics).

Bruno

Received on Tue Sep 24 2002 - 02:08:38 PDT

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