Skip to main content

Undecidability

 Let us construct a symbolic formal system with the following elements.

    1. An arbitrary axiomatic system which contains Gödel’s axiomatic system together with its rules of inference (Ga)

    2. The functions and relations of the system are recursively defined and free from contradiction.

    3. We construct an isomorphic representation of the subsystem of non-numerical symbols by a system of positive integers, ascribing natural numbers to the symbols. Therefore, we can express any formula in numerical terms (particularly as a sequence of primes), and proofs as sequences of positive integers.

    4. We construct a set of formulas F which are directly deducible within the system and which represent common expressions of our calculus.

For every formula fi ∈ F, there is a numerical formula pi ∈ P, for P⊂ F, such that Ga ⊢ pi. 

Construct a fj which expresses “this formula is not deducible”, a valid and meaningful expression of our calculus. Therefore, there is a pj numerical formula that corresponds to fj.

The undecidability theorem says that pj is undecidable. Suppose pj is true. Then pj is not deducible, but pj ∈ F, set of directly deducible formulas, so there is a contradiction. Suppose pj is false. Then ¬ pj is true, id est, pj is deducible, but pj says that it is not deducible, so there is a contradiction.

Ga is not complete.

Labels:

Comments

  1. UnknownOctober 18, 2021 at 3:44 PM

    Is this is similar to the following informal famous logic proof: S="This statement S is False". S cannot be true because it contradicts its definition which is "S is false." S cannot be false because if S was false then, it contradicts Non S "This statement is not false"?

    ReplyDelete

Post a Comment

Please write here your comments

Popular posts from this blog

Limen et Continuum

  Existence is Encounter. Meeting at the limen. In the limen, the masks disappear, that is, the basic intuitions of identities, such as the identity that I feel and think in relation to the tree that I see in front of me. The identity of the tree is a projection of mine: the unity of my process of perceiving the tree generates a mask in me, the ghost of a limited unity separated from everything else. The simplest form of intuitive understanding of masks and limen is given to us by numbers. Numbers intuitively express the liminal tension that is Existence. A little etymological note. Rythmos in Greek means flow. Arythmos (number) is what does not flow, what remains solidified. Numbers express the liminoid, and flow, rhythm, expresses the liminal. A rhythm becomes liminoid when we can trace patterns in it, that is, when we can construct masks of identities. Mathematics has spoken of flow using the Latin word “continuum”, the continuous. All modern science, since Leibni...
Post a Comment
Read more

Metalanguages are formal metaphors

  In a logic class, the professor tells his students: "Yesterday, while talking with my Sufi gardener about happiness, we ended up talking about metalanguages, because he said that orchids are 'chambers where light plays between amorous encounters.' I told him: 'You have to be a poet to talk about poetry.' He replied: 'You just have to be human.'" In what way can we say that my gardener is proposing that every metalanguage is a formalized metaphor for its object language and what would be the metaphor for arithmetical addition? Furthermore” -he asks-how does this little narrative show that Kurt Gödel was a Platonist? One student answers: “The gardener uses orchids as a metaphor for biological reproduction, and from this he makes a second-order metaphor at the human level, calling reproduction a loving encounter. The gardener is a Sufi; in Sufi ontology, the word 'encounter' is used as equivalent to 'existence,' a double meaning (Wujud)....
Post a Comment
Read more

Rhapsodies of Anima Mundi: Fear of Death

In the nascent dawn of consciousness, when the human spirit still danced in rhythmic harmony with the grand, elemental pulse of nature, the enigma of cessation—that profound silence we name death—arose as the most formidable of shadows. Yet, it was not then perceived as an absolute, terminal end in the stark, isolated sense we often conceive today. For those early societies, intimately imbricated in the vast and primordial canvas of the Anima Mundi, death was seamlessly woven into the very ur-tapestry of existence as a continuity, a fluid dissolution into the great soul of the world, or a joyous return to a collective paradise, utterly devoid of the strict, solitary individuation that modernity has, unwittingly, imposed upon us. This is not merely the clinical apprehension of biological cessation, but rather a primordial panic before the void, a visceral anguish in the face of the "I's" dissolution and the potential loss of all that imbues life with meaning. Confronted...
Post a Comment
Read more