Complete Theories with Finitely Many Countable Models. I | SpringerLink
Editorial team. Abraham Robinson. North-Holland Areas of Mathematics in Philosophy of Mathematics. Edit this record. Mark as duplicate. Find it on Scholar. Request removal from index. Revision history. From the Publisher via CrossRef no proxy Setup an account with your affiliations in order to access resources via your University's proxy server Configure custom proxy use this if your affiliation does not provide a proxy.
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Hourya Benis-Sinaceur - - Dialogue 27 4 Middelburg - - Journal of Applied Logic Introduction to Model Theory and to the Metamathematics of Algebra. Abraham Robinson - - North-Holland. Elementary Topics in Mathematical Logic. Alonzo Church - - Brooklyn, N. George Boole - - Dover Constable. Esko Turunen - - Mathematical Logic Quarterly 41 2 Volker Peckhaus - - Bulletin of Symbolic Logic 5 4 T he combination of quantum mechanics and relativity implies an immediate scaling problem. There is an inherent uncertainty in energy and momenta that can never be reduced.
When this fact is combined with special relativity, the conclusion is that you cannot actually even constrain the number of particles present in a small volume for short times. One striking effect of this is that when we measure the force between electrons, say, the actual measured charge on the electron—the thing that determines how strong the electric force is—depends on what scale you measure it at. Since positive virtual particles are attracted to the electron, the deeper you penetrate into the cloud, the less of the positive cloud and more of the negative charge on the electron you see.
Then, when you set out to calculate the force between two particles, you need to include the effects of all possible virtual particles that could pop out of empty space during the period of measuring the force. This includes particles with arbitrarily large amounts of mass and energy, appearing for arbitrarily small amounts of time. When you include such effects, the calculated force is infinite. We know of no theory that both makes contact with the empirical world, and is absolutely and always true. Richard Feynman shared the Nobel Prize for arriving at a method to consistently calculate a finite residual force after extracting a variety of otherwise ambiguous infinities.
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As a result, we can now compute, from fundamental principles, quantities such as the magnetic moment of the electron to 10 significant figures, comparing it with experiments at a level unachievable in any other area of science. Now, though, we understand things differently. The problem was not with the theory, but with trying to push the theory beyond the scales where it provides the correct description of nature. That is impractical at best, and impossible at worst.
Thus, theories that make sense must be insensitive, at the scales we can measure in the laboratory, to the effects of possible new physics at much smaller distance scales or less likely, on much bigger scales. This is not just a practical workaround of a temporary problem, which we expect will go away as we move toward ever-better descriptions of nature. Since our empirical knowledge is likely to always be partially incomplete, the theories that work to explain that part of the universe we can probe will, by practical necessity, be insensitive to possible new physics at scales beyond our current reach.
It is a feature of our epistemology, and something we did not fully appreciate before we began to explore the extreme scales where quantum mechanics and relativity both become important. This applies even to the best physical theory we have in nature: quantum electrodynamics, which describes the quantum interactions between electrons and light.
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They correspond to extrapolating the theory to domains where it is probably no longer valid. Feynman was wrong to have been disappointed with his own success in maneuvering around these infinities—that is the best he could have done without understanding new physics at scales far smaller than could have been probed at the time. Even today, half a century later, the theory that takes over at the scales where quantum electrodynamics is no longer the correct description is itself expected to break down at still smaller scales.
T here is an alternative narrative to the story of scale in physical theory.
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Rather than legitimately separating theories into their individual domains, outside of which they are ineffective, scaling arguments have revealed hidden connections between theories, and pointed the way to new unified theories that encompass the original theories and themselves apply at a broader range of scale.
For example, all of the hoopla over the past several years associated with the discovery of the Higgs particle was due to the fact that it was the last missing link in a theory that unifies quantum electrodynamics with another force, called the weak interaction. These are two of the four known forces in nature, and on the surface they appear very different. But we now understand that on very small scales, and very high energies, the two forces can be understood as different manifestations of the same underlying force, called the electroweak force.
If we expect our theories to be complete, that means that before we can have a theory of anything, we would first have to have a theory of everything. Asymptotic Freedom causes the strong force between quarks to get weaker as the quarks are brought closer together. If the strong force becomes weaker at small distances, it presumably can be strong enough at large distances to ensure that no free quarks ever escape their partners.
The discovery that the strong force gets weaker at small distances, while electromagnetism, which gets united with the weak force, gets stronger at small distances, led theorists in the s to propose that at sufficiently small scales, perhaps 15 orders of magnitude smaller than the size of a proton, all three forces strong, weak, and electromagnetic get unified together as a single force in what has become known as a Grand Unified Theory.
Over the past 40 years we have been searching for direct evidence of this—in fact the Large Hadron Collider is just now searching for a whole set of new elementary particles that appear to be necessary for the scaling of the three forces to be just right. But while there is indirect evidence, no direct smoking gun has yet been found.
Naturally, efforts to unify three of the four known forces led to further efforts to incorporate the fourth force, gravity, into the mix. In order to do this, proposals have been made that gravity itself is merely an effective theory and at sufficiently small scales it gets merged with the other forces, but only if there are a host of extra spatial dimensions in nature that we do not observe.
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This theory, often called superstring theory, produced a great deal of excitement among theorists in the s and s, but to date there is not any evidence that it actually describes the universe we live in.