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.

Comments

  1. 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...

What is Mythopoetics?

  The narrative grew in the process of being told, as myths always do. The Blog has become more labyrinthine over the years. It contains my Mythopoetics book and a few other things. For those who access these texts without knowing anything about Mythopoetics, I am going to post the introduction of the first part, so you can decide if you want to spend your precious time thinking about the identity narratives that we humans have developed over the years. throughout our eventful existence as a species. "Mythological narratives are the only intellectual activity that has been continuously practiced by human beings, a fact that makes them a unique tool for thinking synthetically our evolution as homo-sapiens. In this sense, they are the first valuation settings that humans have made about themselves and their environment, and as such, they have conditioned the ones that have come afterwards, both in form and content. Their communicative function places them at the basis o...

An Epistemological Perspective of Individuation

For the ancient Romans, "Terminus" was the god of boundaries, represented as large stones used to divide and delimit fields. Festivals were held, called Terminalia, in which the stones that "generated" human space were sanctified. Our word "term" is the heir of that god, or better, it is that god incorporated into an everyday space, in our Lebenswelt or world of life. A philosophical term, whatever its semantic content, is the conceptual mark that we make by establishing a referential sign, it is the action of determining, of generating a reference in a mental space, a reference with which we make a sign correspond, or if we deal with a physical space, the correspondence with an object, be it a milestone, a stone, or an indicator sign. Since its beginnings, philosophy has used binary semantic terms as thinking tools, something that analytical psychology has also made good use of. One of the longest-running binary semantic terms for psychology...