I have been thinking very hard about the title question and its
implications, re: Goedel's constructions and Goedel numbering/coding. I
refer the the following papers:
We can find in psychology cases of Dysmorphia, such as
http://www.anred.com/musdys.html), where a distortion of the self-image
occurs. The fact that distortions themselves are possible seems, IMHO, to
require us to consider whether or not there is an assumption implicit in
discussions of Self-reference modeled in terms of Goedel's (and Church,
Turing, et al) treatment of the question of completeness and provability of
formal systems what should be examined more closely.
For example, we have the following quotes:
> "Self-reference in formal arithmetic can be compared to a formal way of
> self-recognition in the mirror. The basis for
> this comparison is the role of the Gödel code in arithmetical
> self-reference. The comparison is developed in a series of diagrams which
> show the stages of construction of a self-referential sentence and relate
> them to the irreflexivity of vision and ways of overcoming it. The aim of
> the comparison is to turn arithmetical slf-reference into an idealized
> formal model of self-recognition and the conception(s) of self based on this
> The incompleteness of a formal system does not make it unsuitable for
> building models of the mind, but, on the contrary, as Hofstadter's
> comparison suggests, it is precisely its incompleteness which makes a formal
> system suitable for modeling the mind.
> Furthermore, it can be said that a mirror-like device cannot be used to
> overcome incompleteness because it has already been used in producing it.
> Namely, the self-reference of a sentence on which arithmetical
> incompleteness depends is indirect, by way of a numerical code: the sentence
> refers to a certain number which belongs to that sentence itself under some
> system of numeric coding of sentences as numbers. This Gödel code functions
> as a numerical mirror in which a sentence can refer to itself, "see itself".
> To re-use Hofstadter's comparison, just as I cannot see my face, in
> particular my eye(s), except in a mirror, so sentences of arithmetic cannot
> refer to, "see" themselves, except in a numerical mirror. More generally,
> this is how a sentence of arithmetic can refer to another formula: by
> refering to its numerical image, Gödel number."
> "An Hierarchy of Levels
> So we evolved the idea that the formal system serving to extend a given
> basic system (or "ground level") could be
> structured by making use of special functions enabling references to levels.
> In effect, if we used references indicating a
> higher level then it would become permissible to "overview" the formulae and
> procedures of a lower level.
> And the basic idea, at least at the beginning of the construction of an
> hierarchy of extension levels, is quite like Turing's concept in that the
> first extension level, the first level above the ground level, incorporates
> the possibility to prove as a
> theorem the Goedel assertion for the logical system of the "ground level".
> The first higher level, corresponding in our approach to any acceptable
> definition of the first ordinal number, would be > such that the Goedel
> assertion from the ground level would now become provable if that Goedel
> assertion had been simply added as an axiom usable on the higher level. Or
> instead it would be effective to introduce as an axiom an assertion of
> formal consistency of the ground level logic. This type of assertion can be
> seen to imply the assertion of Goedel's type. Formal consistency is the
> assertion to the effect that "on the ground level nothing false can be
> The verbal argument for the truth of Goedel's formula depends in effect
> on an overview of the specific formal system in which it is stated and on
> the basis of the axioms, etc. of which it was constructed. Also this
> argument depends on the assumption that that formal system is actually
> perfectly consistent. Our idea of extension and of "hierarchical
> introspective logics" is that there is to be considered an hierarchy of
> levels of logic such that higher levels have an "overview" of the
> proceedings and results obtainable on lower levels.
> Thus the totality is "introspective" because it is looking inward to
> study itself but this self-study is not unrestricted.
> And effectively a higher level is supposed to be able to see the truth of a
> Goedel assertion deriving from a lower level by a process analogous to the
> verbal argument originally known for the truth of the Goedel assertion in a
> formal system although there is no proof of that assertion in that original
> formal system.
> A logical system cannot effectively state its own consistency; this
> relates to the incompleteness originally found by Goedel. But one logical
> system CAN easily state the formal consistency of another system. We use
> this idea in creating our hierarchy of levels so that on a higher level it
> is possible to, in effect, assume the validity and the consistency of
> proceedings possible on lower levels."
My question is whether or not it is unproblematic to assume that the
numbering via a mapping between the "levels" to the set of ordinals and the
isomorphism between Goedel numbers and Z (the integers). Is the well
ordering of the ordinals and integers prior to any instantiation thereof?
My question is further compounded by David Deutsch's paper: Machines, Logic
and Quantum Physics (http://xxx.lanl.gov/abs/math.HO/9911150):
> "Though the truths of logic and pure mathematics are objective and
> independent of any contingent facts or laws of nature, our knowledge of
> these truths depends entirely on our knowledge of the laws of physics.
> Recent progress in the quantum theory of computation has provided practical
> instances of this, and
> forces us to abandon the classical view that computation, and hence
> mathematical proof, are purely logical notions
> independent of that of computation as a physical process. Henceforward, a
> proof must be regarded not as an abstract object
> or process but as a physical process, a species of computation, whose scope
> and reliability depend on our knowledge of the physics of the computer
> concerned. "
Which leads to the related question: Why is it that there seems to be a
difference between physical systems (viz: objects of experimentation having
observable properties) and formal (and informal) representations of such
(viz: Goedelian codes, quantum algorithms, information structures, etc.)?