The entropy of a system of finite volume is a finite number if the energy is finite. Even if you allow for the energy to be arbitrarily high, the number of possible states in the volume is bounded via the holographic principle.
This is true if and only if the phase space is finite-dimensional, and so it does not include systems with an infinite number of degrees of freedom, such as (dynamic) continuous spacetime. Therefore the argument is circular.The entropy of a system of finite volume is a finite number if the energy is finite.
This is true if and only if the phase space is finite-dimensional, and so it does not include systems with an infinite number of degrees of freedom, such as (dynamic) continuous spacetime. Therefore the argument is circular.
Why can't nature be both formally describable and be a continuum?formally describable and hence not a real continuum.
The proof for LST relies on the fact that there are at most countably many descriptions of anything, viz. names, sentences, paragraphs, books... There are at most countably many strings of symbols (when the strings are finite in length). This fact is easily proved by arithmetizing the alphabet of our language of description. Each wff then becomes a natural number. Since there are only countably many natural numbers, there are at most countably many wffs to do the describing.
One countable model that is always available for inspection, if only to demystify LST a bit, is the interpretation in which the terms of the language are assigned to their own tokens, or to the typographic strings which express them. We've seen that by arithmetization there are at most countably many such strings. Hence, even if the intended interpretation of the marks on paper refers to the uncountably many real numbers, one obvious alternate interpretation refers only to the countably many marks on paper that comprise the system.
As the typographic interpretation of S shows, the interpretations with merely countable models will be (or may as well be) non-standard. The "meaning" of the marks changes at the same time as the domain. The theorem which in the intended interpretation made some assertion about uncountable reals speaks of something entirely different in the countable LST models.
Remember, a formal system "about the reals" is really a system of wffs of some formal language. The language is inherently uninterpreted. We might give its symbols some interpretation, and on that intended interpretation we may say that the system is "about" the reals. LST asserts that every consistent first-order theory can be intepreted as being "about" some set of things no more numerous than the natural numbers, even if we thought it was --indeed, even if under another interpretation it is-- "about" the uncountable reals.
LST does not deny the possibility that some system S might have uncountable models alongside the critical countable model. In that sense, S might "succeed". LST qualifies this success by giving S other models whether it wanted them or not. How does this detract from S's success? In only one way: S is thwarted if it aspired to capture its intended domain unambiguously or uniquely.
Then follows several sections involving "Puzzles in the foundation of mathematics" sets,classes, Godels Theorem which seems inconclusive and introduces further analyses in later chapters.There are some...who would prefer a universe..that is finite in extent..only finitely divisible..so that a fundamental discreteness might begin to emerge at the tinest levels...(it's distinctly unconventional..but not inherently inconsistent. In the early days of quantum mechanics.... a hope was not realized by future developments...that the theory was leading to a picture of discreteness at the tinest levels. In the successful theories of our present day we take spacetime as a continum even when quantum concepts are involved and ideas that involve small scale discreteness must be regarded as 'unconventional'....It appears, for the time being at least, we have to take the use of the infinite seriously.
A reason for hoping (Maldacena's) ADS/CFT is true appears to be that it might provide a handle on what a a string theory could be like, without resorting to the usual pertebative methods with all the severe limitations such methods have.
No, there is no problem with defining the continuum with a finite number of axioms. The continuum hypothesis does not pertain to this at all.I'll defer to logicians to make rigorous statements on this topic, but my understanding is that the reason why things like the continuum hypothesis are undecidable is precisely because with only finite number of axioms there is no way you could precisely define the uncountable continuum.
First of all, countable != discrete. The rational numbers Q are countable, but in fact every point has infinitely many arbitrarily close neighbors!The Löwenheim-Skolem Theorem
Here you go again assuming the world can be simulated by a finite digital computer. What is the basis for this assumption? If the world were a continuum, then in principle I could encode all the information on the internet into a scratch on a rod.Suppose that using a huge computer you simulate a planet with mathematicians and physicsits living on it. The computer computes everything down to the atomic level, so it would be able to reproduce most of the mathematical and physics results right until the 19th century.
Mathematics is much more than symbolic manipulation, and a comuter which contains only the finite-bit strings of various mathematical papers without containing the semantic meaning of these has not captured all of the information.So, what is really ging on is that people can represent phenomena in their world using some abstract rules which involves manipulating finite bistrings.
You can't prove anything about the physical world, ever. But you saying we can't prove continuum physics is a world a part from saying that continuum physics cannot possibly be the case because of some information perspective.The only way you could prove that the continuum really exists is by constructing a machine that produces results that are not formally describable, e.g. the so-called "rapidly accelerating computer" I wrote about earlier in this thread.
According to Chaitin it does.No, there is no problem with defining the continuum with a finite number of axioms. The continuum hypothesis does not pertain to this at all.
The real numbers are the unique complete totally ordered field; students are exposed to the rigorous construction as senior undergraduates or beginning graduates.
His work is considered to be controversial, it is of a philosophical nature, and it is not widely accepted by mathematicians. Fortunately we do not need to argue about the existence of uncountable sets, since the rationals are countable, and the set of analytic functions from Q to Q is countable, and everything we do with the continuum in physics could be translated to smooth rational functions. (Discrete Or Uncountable) is a false dichotomy, the rationals are a counterexample.According to Chaitin it does.
I disagree, you keep coming back to the same circular assumption. To the contrary, as far as we know the universe is best described by quantum field theory. Please tell me what theory captures the world more accurately than QFT and suggests that the "physical world is computable."and as far as we know the physical world is computable.
A few physicists persisted in developing semiclassical models in which electromagnetic radiation is not quantized, but matter appears to obey the laws of quantum mechanics. Although the evidence for photons from chemical and physical experiments was overwhelming by the 1970s, this evidence could not be considered as absolutely definitive; since it relied on the interaction of light with matter, a sufficiently complicated theory of matter could in principle account for the evidence. Nevertheless, all semiclassical theories were refuted definitively in the 1970s and 1980s by photon-correlation experiments.[Notes 2] Hence, Einstein's hypothesis that quantization is a property of light itself is considered to be proven.
No, that is known for sure. The continuum hypothesis is that there are no cardinalities between that of the naturals and that of the reals.The continuum hypothesis is the claim that the cardinality of the real numbers is equal to the cardanality of the power set (set of all subsets of) the natural numbers.
I can't make heads nor tails of what you're trying to say here.Take for example category theory which breaks the mathematical world into the dichotomy of objects and morphisms (the discrete object/the continuous morphism). After many years floundering around with set theory and null sets, maths gave up either/or to embrace a fundamental duality (in interaction).
Just making the point that everywhere you turn when people are trying to make deep distinctions, you find dichotomies emerging. As in category theory. Definitions by mutal exclusion, followed by the interaction of what has been created.I can't make heads nor tails of what you're trying to say here.