First-order logic

Main article: First-order logic

Whereas universal algebra provides the semantics for a signature, logic provides the syntax. With terms, identities and quasi-identities, even universal algebra has some limited syntactic tools; first-order logic is the result of making quantification explicit and adding negation into the picture.

A first-order formula is built...

view entire post

First-order logic

Main article: First-order logic

Whereas universal algebra provides the semantics for a signature, logic provides the syntax. With terms, identities and quasi-identities, even universal algebra has some limited syntactic tools; first-order logic is the result of making quantification explicit and adding negation into the picture.

A first-order formula is built out of atomic formulas such as R(f(x,y),z) or y = x + 1 by means of the Boolean connectives {\displaystyle \neg ,\land ,\lor ,\rightarrow } and prefixing of quantifiers {\displaystyle \forall v}or {\displaystyle \exists v}. A sentence is a formula in which each occurrence of a variable is in the scope of a corresponding quantifier. Examples for formulas are φ (or φ(x) to mark the fact that at most x is an unbound variable in φ) and ψ defined as follows:

1. (φ ∨ ψ), (φ ∧ ψ), (¬φ), (φ → ψ) 2. (∃xφ), (∀xφ)

(Note that the equality symbol has a double meaning here.) It is intuitively clear how to translate such formulas into mathematical meaning. In the σsmr-structure {\displaystyle {\mathcal {N}}}of the natural numbers, for example, an element n satisfies the formula φ if and only if n is a prime number. The formula ψ similarly defines irreducibility. Tarski gave a rigorous definition, sometimes called "Tarski's definition of truth", for the satisfaction relation {\displaystyle \models }, so that one easily proves:

Thus ∀y (rxy) ∨ ∃y (rxy) means (∀y (rxy)) ∨ (∃y (rxy)) .

A set T of sentences is called a (first-order) theory. A theory is satisfiable if it has a model M |= T, i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set T. Consistency of a theory is usually defined in a syntactical way, but in first-order logic by the completeness theorem there is no need to distinguish between satisfiability and consistency. Therefore, model theorists often use "consistent" as a synonym for "satisfiable".

A theory is called categorical if it determines a structure up to isomorphism, but it turns out that this definition is not useful, due to serious restrictions in the expressivity of first-order logic. The Löwenheim–Skolem theorem implies that for every theory T[2] which has an infinite model and for every infinite cardinal numberκ, there is a model {\displaystyle {\mathcal {M}}\models T} such that the number of elements of {\displaystyle {\mathcal {M}}} is exactly κ. Therefore, only finitary structures can be described by a categorical theory.

Lack of expressivity (when compared to higher logics such as second-order logic) has its advantages, though. For model theorists, the Löwenheim–Skolem theorem is an important practical tool rather than the source of Skolem's paradox. In a certain sense made precise by Lindström's theorem, first-order logic is the most expressive logic for which both the Löwenheim–Skolem theorem and the compactness theorem hold.

As a corollary (i.e., its contrapositive), the compactness theorem says that every unsatisfiable first-order theory has a finite unsatisfiable subset. This theorem is of central importance in infinite model theory, where the words "by compactness" are commonplace. One way to prove it is by means of ultraproducts. An alternative proof uses the completeness theorem, which is otherwise reduced to a marginal role in most of modern model theory.

From: Model Theory, Wikipedia

Also see, Introduction to the CTMU:

That this structure is completely self-distributed implies that it is locally indistinguishable for subsystems s; it could only be discerned against its absence, and it is nowhere absent in S. Spacetime is thus transparent from within, its syntactic structure invisible to its contents on the classical (macroscopic) level. Localized systems generally express and utilize only a part of this syntax on any given scale, as determined by their specific structures. I.e., where there exists a hological incoversion endomorphism D:Sà{rÎS} carrying the whole structure of S into every internal point and region of S, objects (quantum-geometrodynamically) embedded in S take their recognition and state-transformation syntaxes directly from the ambient spatiotemporal background up to isomorphism. Objects thus utilize only those aspects of D(S) of which they are structural and functional representations.

From: http://www.ctmu.org/Articles/IntroCTMU.htm

Here we see a relationship between Quantum Mechanics and Logic.

Logic is Quantum. Quantum, logic. Recall that reality is logic(al). Hence, so to is Quantum reality. First order logic formulas are built by atomic formulas. Which themselves combine to form first order theories. A first order theory that is satisfiable has a model M |= T. We derive Quantum logic from this and the CTMU derives a reality that is Quantum-geometrodynamically embedded. Where spatiotemporal containment defines the location of objects within time and space. As the objects move through time and space, they take their state-recognition and state-transformation syntaxes directly from this ambient spatiotemporal background. Now, what does this have to do with first order logic? Well, if you read the very first paragraph of what I posted from the Wikipedia article, it says that Universal Algebra provides the semantics for the signature of a formal language, whereas logic provides the syntax. So we have reality being logically Quantum-geometrodynamically embedded within itself using syntax.

Does this make sense?

Nicholas I. Hosein, 2016.

view post as summary

These BH are fascinating.We know so few about them.We know how our electromagnetism and informations act near them.The works of Hawking about the event horizons are relevant.We see their compostments at these walls.I beleive strongly that these BH have rules, properties....and that their mechanics are different.They probably,for the supermassive BH central to galaxies,recycle the bosonic...

view entire post

These BH are fascinating.We know so few about them.We know how our electromagnetism and informations act near them.The works of Hawking about the event horizons are relevant.We see their compostments at these walls.I beleive strongly that these BH have rules, properties....and that their mechanics are different.They probably,for the supermassive BH central to galaxies,recycle the bosonic informations and fermionics.This matter that said seems not baryonic like dark matter.The relevance is to link with the serie of supermega BH towards thecentral BH of our universe.If the primordial informations come from this center.So the others are like adaptators, transfromators.They are essential for a galaxy these BH.I beleive really that dark matter is correlated.They produce this matter in logic but we cannot see it still due to our technological limits.If we don't analyse differently these BH, we shall not find the good method it seems to me humbly.The aether time of the photonic sphere and the aether time of gravitation seems different.Mr Agnew I am asking me really if the real age of our universe is correct.It seems that no considering this gravitation, BH and dark matter.The destiny of all informations is to be encoded, recycled, produced,imrpoved....But the puzzle is so complex.Informations are never lost in this line of reasoning.But how to find the good road to analyse this other universal newtonian mechanic.You can do it with the team at MIT ,Harvard....The correct mathematical method exists showing us this mechanic not relativistic.But frankly I search this method and I don't find it.I am limited in my Tools.It seems really that this gravitation is the main chiel orchestra connected with the agravitational primordial aether coming from this central BH of the universe.So bosonic informations and fermions ,stndard model,special and general relativity,heat and thermo are just a system Inside in y opinion.The stars and photons ....all these créations were necessary for this infinite entropy to create matters in evolution by encodings of particles produced by cosmological sphères.Heat ,thermo and photons are a tool for the créations and dynamics of evolutive minerals, animals and vegetals,but it is not all, these systems seems encircled really by BH and dark matter.Even our mass must be rethought like our universal age it seems to me.If my equation is correct E=mc²+ml² it could be interesting to develop it .Best Regards

view post as summary

Once you have a universal quantum force like aethertime, it is a lot easier to deal with the event horizon of a black hole. In fact, with quantum gravity there are incipient black holes at the center of any matter body larger than about six Jupiters. So all stars have a boson matter center that is a black hole.

Black holes are not really the voracious singularities of spacetime but rather...

view entire post

Once you have a universal quantum force like aethertime, it is a lot easier to deal with the event horizon of a black hole. In fact, with quantum gravity there are incipient black holes at the center of any matter body larger than about six Jupiters. So all stars have a boson matter center that is a black hole.

Black holes are not really the voracious singularities of spacetime but rather just the inevitable boson matter destiny for all fermion matter in the shrinking aethertime universe. With a quantum gravity like aethertime, the mechanics of relativity in spacetime are just a convenient representation for the simpler underlying reality of matter and action. For black hole event horizons and at the Planck limit, you need quantum gravity to predict action.

So black holes do not really need a supernova to form...black holes just need lots of mass. That is why the mass of a galaxy stabilizes a supermassive black hole. The spin of a black hole does have a very important role in quantum gravity since black hole spin stores all of the information of accretion.

Dark matter and dark energy are simply manifestations of exchange of quantum gravity. Since stars emit large amounts of mass as light and neutrinos, that star matter decay couples the motions of inner and outer stars of a galaxy. Inner stars slow down while outer stars speed up and a galaxy therefore rotates at a constant velocity without the need to invoke a cold dark matter halo to keep outer stars bound.

It would appear that just about any kind of quantum gravity would show this kind of quantum exchange, but there would need to be decoherence for quantum gravity. Dark energy has a similar explanation but at a much greater scale of galaxy clusters and superclusters. The only event horizon is that that defines the limits of the universe and of what we can know.

Thus in billions more years, people will still look up in the sky and see the same CMB light that represents our common creation. Of course, the universe will be a very different place in a few billion years from what it is today. The Andromeda galaxy will likely have merged and that will be quite a spectacle that humanity will miss.

view post as summary