To Infinity and Beyond. Buzz Lightyear to all economists.

To Infinity and Beyond.

Buzz Lightyear to all economists.

http://journalofcosmology.com/Cosmology8.html

The concept of “red shift” is based on the Doppler effect; i.e. wave lengths of light contract or expand as they approach and then speed toward or away from Earth. Hence, for red shifts to have any meaning, the Earth becomes the center of the universe; which, of course, is absurd.

http://en.wikipedia.org/wiki/Georg_Cantor

Continuum hypothesis

Main article: Continuum hypothesis

Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is exactly aleph-one, rather than just at least aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety.[9]

The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by Gödel and a 1963 one by Paul Cohen together imply that the continuum hypothesis can neither be proved nor disproved using standard Zermelo–Fraenkel set theory plus the axiom of choice (the combination referred to as “ZFC”).[47]

http://en.wikipedia.org/wiki/Ludwig_Boltzmann

The Boltzmann equation

Boltzmann’s bust in the courtyard arcade of the main building, University of Vienna.

Main article: Boltzmann equation

The Boltzmann equation was developed to describe the dynamics of an ideal gas.

$\frac{\partial f}{\partial t}+ v \frac{\partial f}{\partial x}+ \frac{F}{m} \frac{\partial f}{\partial v} = \frac{\partial f}{\partial t}\left.{\!\!\frac{}{}}\right|_\mathrm{collision}$

where ƒ represents the distribution function of single-particle position and momentum at a given time (see the Maxwell–Boltzmann distribution), F is a force, m is the mass of a particle, t is the time and v is an average velocity of particles.

This equation describes the temporal and spatial variation of the probability distribution for the position and momentum of a density distribution of a cloud of points in single-particle phase space. (See Hamiltonian mechanics.) The first term on the left-hand side represents the explicit time variation of the distribution function, while the second term gives the spatial variation, and the third term describes the effect of any force acting on the particles. The right-hand side of the equation represents the effect of collisions.

Boltzmann’s grave in theZentralfriedhof, Vienna, with bust and entropy formula.

In principle, the above equation completely describes the dynamics of an ensemble of gas particles, given appropriate boundary conditions. This first-order differential equation has a deceptively simple appearance, since ƒ can represent an arbitrary single-particle distribution function. Also, the forceacting on the particles depends directly on the velocity distribution function ƒ. The Boltzmann equation is notoriously difficult to integrate. David Hilbertspent years trying to solve it without any real success.

The form of the collision term assumed by Boltzmann was approximate. However for an ideal gas the standard Chapman–Enskog solution of the Boltzmann equation is highly accurate. It is expected to lead to incorrect results for an ideal gas only under shock wave conditions.

Boltzmann tried for many years to “prove” the second law of thermodynamics using his gas-dynamical equation — his famous H-theorem. However the key assumption he made in formulating the collision term was “molecular chaos“, an assumption which breaks time-reversal symmetry as is necessary for anything which could imply the second law. It was from the probabilistic assumption alone that Boltzmann’s apparent success emanated, so his long dispute with Loschmidt and others over Loschmidt’s paradox ultimately ended in his failure.

Finally, in the 1970s E.G.D. Cohen and J.R. Dorfman proved that a systematic (power series) extension of the Boltzmann equation to high densities is mathematically impossible. Consequentlynonequilibrium statistical mechanics for dense gases and liquids focuses on the Green–Kubo relations, the fluctuation theorem, and other approaches instead.

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To quote Planck, “The logarithmic connection between entropy and probability was first stated by L. Boltzmann in his kinetic theory of gases”.[6] This famous formula for entropy S is[7] [8]

$S = k \log_e W \,$

where k = 1.3806505(24) × 10−23 J K−1 is Boltzmann’s constant, and the logarithm is taken to the natural base e. W is the Wahrscheinlichkeit, the frequency of occurrence of a macrostate [9] or, more precisely, the number of possible microstates corresponding to the macroscopic state of a system — number of (unobservable) “ways” in the (observable) thermodynamic state of a system can be realized by assigning different positions and momenta to the various molecules. Boltzmann’s paradigm was an ideal gas of N identical particles, of which Ni are in the ith microscopic condition (range) of position and momentum. W can be counted using the formula for permutations

$W = \frac{N!}{\prod_i N_i!}$

http://en.wikipedia.org/wiki/Kurt_G%C3%B6del

The Incompleteness Theorem

In 1931 and while still in Vienna, Gödel published his incompleteness theorems in “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme” (called in English “On Formally Undecidable Propositions of Principia Mathematica and Related Systems“). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g. the Peano axioms or Zermelo–Fraenkel set theory with the axiom of choice), that:

If the system is consistent, it cannot be complete.
The consistency of the axioms cannot be proven within the system.

These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in Principia Mathematica and Hilbert’s formalism, to find a set of axioms sufficient for all mathematics. The incompleteness theorems also imply that not all mathematical questions are computable.

In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false, which contradicts the idea that in a consistent system, provable statements are always true. Thus there will always be at least one true but unprovable statement. That is, for any computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that obtains in arithmetic, but which is not provable in that system. To make this precise, however, Gödel needed to produce a method to encode statements, proofs, and the concept of provability as natural numbers. He did this using a process known as Gödel numbering.

In his two-page paper “Zum intuitionistischen Aussagenkalkül” (1932) Gödel refuted the finite-valuedness of intuitionistic logic. In the proof he implicitly used what has later become known as Gödel–Dummett intermediate logic (or Gödel fuzzy logic).

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In 1951, Gödel demonstrated the existence of paradoxical solutions to Albert Einstein’s field equations in general relativity. He gave this elaboration to Einstein as a present for his 70th birthday.[15] These “rotating universes” would allow time travel and caused Einstein to have doubts about his own theory. His solutions are known as the Gödel metric.

During his many years at the Institute, Gödel’s interests turned to philosophy and physics. He studied and admired the works of Gottfried Leibniz, but came to believe that a hostile conspiracy had caused some of Leibniz’s works to be suppressed.[16] To a lesser extent he studied Immanuel Kant and Edmund Husserl. In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz’s version of Anselm of Canterbury‘s ontological proof of God‘s existence. This is now known as Gödel’s ontological proof. Gödel was awarded (with Julian Schwinger) the first Albert Einstein Award in 1951, and was also awarded the National Medal of Science, in 1974.

http://en.wikipedia.org/wiki/Alan_Turing

After Sherborne, Turing went to study at King’s College, Cambridge. He was an undergraduate there from 1931 to 1934, graduating with first-class honours in Mathematics, and in 1935 was elected afellow at King’s on the strength of a dissertation on the central limit theorem.[17]

In 1928, German mathematician, David Hilbert had called attention to the entscheidungsproblem (decision problem). In his momentous paper “On Computable Numbers, with an Application to theEntscheidungsproblem“,[18] Turing reformulated Kurt Gödel‘s 1931 results on the limits of proof and computation, replacing Gödel’s universal arithmetic-based formal language with what became known as Turing machines, formal and simple devices. He proved that some such machine would be capable of performing any conceivable mathematical computation if it were representable as analgorithm. He went on to prove that there was no solution to the Entscheidungsproblem by first showing that the halting problem for Turing machines is undecidable: it is not possible to decide, in general, algorithmically whether a given Turing machine will ever halt. While his proof was published subsequent to Alonzo Church’s equivalent proof in respect to his lambda calculus, Turing was unaware of Church’s work at the time.

In his memoirs Turing wrote that he was disappointed about the reception of this 1936 paper and that only two people had reacted – these being Heinrich Scholz and Richard Bevan Braithwaite.

Turing’s approach is considerably more accessible and intuitive. It was also novel in its notion of a ‘Universal (Turing) Machine’, the idea that such a machine could perform the tasks of any other machine. Or in other words, is provably capable of computing anything that is computable. Turing machines are to this day a central object of study in theory of computation, the simplest examplebeing a 2 state 3 symbol Turing machine discovered by Stephen Wolfram.[19]

The paper also introduces the notion of definable numbers.

From September 1936 to July 1938 he spent most of his time at the Institute for Advanced Study, Princeton, New Jersey, studying under Alonzo Church. As well as his pure mathematical work, he studied cryptology and also built three of four stages of an electro-mechanical binary multiplier.[20] In June 1938 he obtained his Ph.D. from Princeton; his dissertation introduced the notion of relative computing, where Turing machines are augmented with so-called oracles, allowing a study of problems that cannot be solved by a Turing machine.

Back in Cambridge, he attended lectures by Ludwig Wittgenstein about the foundations of mathematics.[21] The two argued and disagreed, with Turing defending formalism and Wittgenstein arguing that mathematics does not discover any absolute truths but rather invents them.[22] He also started to work part-time with the Government Code and Cypher School (GCCS).

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http://www.bbc.co.uk/news/science-environment-13545453

This was on the BBC news site yesterday the shape of an electron of course has everything to do with the

whole Pythagoran Concept of the perfect ratio and the notion of the infinitely large and the ifinitely small.

Of Course Keynes is always king, in the long run we are all dead ( our bodies at least entropy takes care of that).

The most accurate measurement yet of the shape of the electron has shown it to be almost perfectly spherical.

Electrons are negatively-charged elementary particles which orbit the nuclei of atoms.

The discovery is important because it may make some of the emerging theories of particle physics – such as supersymmetry – less likely.

The research, by a team at Imperial College London, is published in the latest edition of Nature journal.

In their scientific paper, the researchers say the electron differs from being perfectly round by a minuscule amount.

“Conventionally, people think that the electron is round like a little ball. But some advanced theories of physics speculate that it’s not round, and so what we’ve done is designed an experiment to check with a very, very high degree of precision,” said lead author Jony Hudson, from Imperial.

The current best theory to explain the interactions of sub-atomic particles is known as the Standard Model. According to this framework, the electron should be close to perfectly spherical.

But the Standard Model is incomplete. It does not explain how gravity works and fails to explain other phenomena observed in the Universe.

Egg off the menu

So physicists have tried to build on this model. One framework to explain physics beyond the Standard Model is known as supersymmetry.

However, this theory predicts that the electron has a more distorted shape than that suggested by the Standard Model. According to this idea, the particle could be egg-shaped.

Experimental set-up used to measure electron

The researchers used lasers to measure the shape of the electron

Researchers stress that the new observation does not rule out super-symmetry. But it does not support the theory, according to Dr Hudson.

He hopes to improve the accuracy of his measurements four-fold within five years. By then, he said, his team might be able to make a definitive statement about supersymmetry and some other theories to explain physics beyond the Standard Model.

“We’d then be in a position to say supersymmetry is right because we have seen a distorted electron or supersymmetry has got to be wrong because we haven’t,” he told BBC News.

Dr Hudson’s measurement is twice as precise as the previous efforts to elucidate the shape of the electron.

Future prospects

That in itself does not alter scientists’ understanding of sub-atomic physics, according to Professor Aaron Leanhardt of Michigan University in the US.

But the prospect of improved measurements and the potential to shed light on current theories of particle physics has made the research community “sit up and take notice”.

The Large Hadron Collider is searching for signs of supersymmetry

“A factor of two doesn’t change the physics community’s general opinion of what’s going on,” he told BBC News.

But he added that improved measurements could start “constraining the possible theories, and what could be discovered at the Large Hadron Collider at Cern and what you might expect in cosmological observations.”

Current theories also suggest that if the electron is more or less round, then there ought to be equal amounts of matter and anti-matter – which, as its name suggests, is the opposite of matter.

Instead, astronomers have observed a Universe made up largely of matter. But that is an observation that could be explained if the electron were found to be more egg-shaped than the Standard Model predicts.

Although the shape of the electron could have an important bearing on the future theories of particle physics, Dr Hudson’s main motivation is simply curiosity.

“We really should know what the shape of the electron is,” he said.

“It’s one of the basic building blocks of matter and if this isn’t what physicists do I don’t know what we should do”.