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## Logic, Physics, Set Theory, Mathematics, Philosophy,

## and other things, by Charles Michael Fox

## Home- This Site's Contents

## Introduction

This site is for presenting some ideas that I have had about logic, physics, set theory, mathematics, philosophy, and possibly other things, including music, literature, and politics, if people seem interested (I have written papers about three of these, described below, which have been rejected for what I consider to be very poor reasons, when reasons were given, by what I thought were appropriate journals.), questions I have about some of these subjects, and comments from those who email me comments, to which I may reply, that I decide to post here. (My email for this site is charlesmfoxweb@gmail.com.)

## Present Site Contents

This site contains, among other things, copies to two papers of mine:

A. A set theory paper, "Derivation of the Axiom of Choice from the axioms of Zermelo-Fraenkel set theory", earlier versions of which I submitted to the Journal of Symbolic Logic, and later to its companion, the Bulletin of Symbolic Logic, and were rejected by both for what I consider ridiculous reasons, instead of their being read with understanding and rejected, if they were, for the part that I realized conceivably was defective

There have been, reputedly, many papers incorrectly purporting to prove AC in ZF. That AC is true, in the standard interpretation of the terms of ZF, is by most people considered to be obvious. The question of whether it can, however, be proven in ZF was supposedly settled in the negative by a two-part paper published by Paul Cohen in 1963, for which he won the Fields Medal. To (supposedly) prove that AC cannot be proved in ZF, and also that the Continuum Hypothesis cannot be proved in ZF+AC = ZFC (I have no idea whether the CH is true, or whether Cohen's supposed proof that it cannot be proved in ZFC, which may be somewhat independent of his supposed proof that AC cannot be proven in ZF, is correct), these papers use a technique, forcing (like implication, but weaker), said by Cohen to be a delicate one, which he introduced and which I have not yet studied in detail, although I may later. It, or its use, may be the culprit in Cohen's defective, I think, claimed proof. Instead of directly criticizing Cohen's supposed proof, my paper indirectly does so by giving a very simple 8 line (in its latest version) proof (derivation) of AC that, as far as I can tell, rigorously follows the rules of ZF together with its first-order logic.

B. A physics paper, "A Possible Severe Conflict between Quantum Mechanics and Special Relativity", its current title, earlier versions of which I submitted to and were rejected by seven! (possibly a world record for rejections) journals, either for no stated reason, or for what I consider to be reasons as ridiculous (well, almost) as those given for rejecting the Axiom of Choice paper.

One may usually have a moral obligation to keep his/her communications from journals private unless one has gotten permission from them to make the communications public. However, I think the journals have a moral obligation to review one's submitted papers with some care, and reasonable competency in the subject matter of the journal. This was not done, for the two papers described above, and to some extent for the Gödel paper described below, by the journals I submitted them to, so I consider my obligation to keep private their communications with me about the papers to be now nonexistent. The total communications between me and the journals for A and B above occupy about 5 MB, so I haven't put them on this site, but if anyone wants to see how badly the journals botched their evaluations of A and B, I will (probably) email the files to him/her if he/she requests me to by email. The journals to which I submitted B were: Physical Review A, Nature Physics, Science, Proceedings of the National Academy of Sciences, Philosophy of Science, Foundations of Physics, and International Journal of Physics.

In addition to the above, there are currently, in Entries for items 1-10, 11-20, and 21-30, linked at the top, full entries for numbers 1, 2, 3, 5, 6, 14, 19, and 22 briefly described below.

## Future Site Contents

If I ever get finished with revising my paper "Meaninglessness of the Gödel string 'G' and consequent invalidity of Gödel's supposed proof of his First Incompleteness Theorem, and related invalidity of supposed proofs of four other claimed limitative theorems", also rejected by the JSL, this time not for technical reasons, but, as far as I can tell from the comments of Mr. Welch of the JSL, for being too philosophical, I will put a copy of the revised version on this site. I have no quarrel with Kurt Gödel's First Incompleteness Theorem (G1IT) itself, which Theorem may or may not be true, but only with Gödel's supposed proof of it, which he based on his (in)famous claimed sentence "G": "This sentence is not provable." (There are, in fact, known valid proofs of incompleteness theorems which are similar to, although not identical with, G1IT.) I am far from being the first person to claim that "G" is meaningless (because of the circularity of its supposed definition, which directly follows its form). I found on the Internet some time ago an article making that claim; however, I have been unable to locate the article since then. Actually, there are two different "G"s in Gödel's paper. The second (in the order in which they occur in Gödel's paper) one, which I call "G2" in my paper, is actually the one Gödel used in his formal supposed proof of G1IT, in section 2 of his famous 1931 paper. "G2" is formally equivalent (given a plausible assumption about exactly what Gödel intended in one slightly unclear step in his paper) to the first "G", which I call "G1", which is trivially formally equivalent to the quoted infamous sentence above, but showing that "G2" is formally equivalent to "G1", and that its supposed definition is circular, and so doesn't define anything, both of which I show in my paper, has not, to my knowledge, been done elsewhere.

I also will eventually put on this website, as I get them redacted, full entries for some of the following which are not mentioned above as being already on it:

(1) My simple derivation, based on special relativity and the Equivalence Principle, of the gravitational time dilation formula of general relativity (for not-too-complicated space-times)

(2) My calculation, based on the published LIGO gravitational wave interferometer waveforms of GW150914, the first gravitational wave detection event, of the combined masses of the two merging black holes involved in that GW's creation, using only basic Newtonian, special relativistic, and general relativistic formulas, without numerically solving the Einstein equation of general relativity for the merger, which combined mass value came surprisingly close to the final LIGO value, which was arrived at by LIGO through a much more complicated calculation

(3) A surprising conclusion of mine about the Ultraviolet Catastrophe, which supposedly was the problem in classical physics which Max Planck introduced his quantum hypothesis to solve, beginning quantum theory

(4) My criticism of an apparent error in common interpretations of plane-wave solutions of the interaction-free, time-dependent Schrodinger equation

Added some time after writing the above: The error was probably mine; the approximate group velocity (properly "speed") of a wave packet which is a near-plane-wave solution of the interaction-free TISE, which wave packet has a very small spread in its momentum probability distribution, is about twice the phase velocity of the waves making up the packet, yet is said to represent the true speed of the particle the wave packet represents (is a wave-function of). This seemed to me to be unlikely behavior for waves in free space (unlike, e.g., waves in wave guides, where the phase velocity of EM waves can be much higher than the group velocity. This, however, is almost just a geometric illusion, due to the peculiar definition of "phase velocity" in this situation.) I thought this apparent discrepancy was related to considering, following de Broglie, the kinetic energy of all quantum waves, even those particles moving much slower than light, to be equal to hf, the Planck formula for the energy of photons in terms of their frequency. However, some investigation showed that the group velocity of a wave packet which is a solution of the Schrodinger equation and is nearly a momentum eigenfunction is indeed about twice that of the phase velocity of the waves making up the packet, and that the represented particle's energy-momentum relationship seems to work out to be correct. That the energy of all particles, even those moving, in a reference frame, much slower than light speed, is given by hf even though their total energy, including their rest mass energy, may be much greater than their kinetic energy, whereas the total energy of photons with energy hf is entirely kinetic energy, still seems to me to be peculiar. I will think about this some more.

Still later: I thought about it, and realized that the approximate solution of the SE which is used in the (Wikipedia) argument I have seen which has a group velocity twice its phase velocity is a physically unrealistic non-normalizable solution, which has an infinite number of almost identical wave packets, and doesn't have defined for it what is standardly considered to be the speed of a wave packet- the speed of its position expectation value- since its position expectation value isn't defined. I have a computer program which may be capable of producing a numerical solution of the SE with a single, normalizable wave packet which may more nearly settle the question. If this program can't, I can probably write one myself, but to do so I will have to learn either the programming language Python or the Wolfram language, which however I am planning to do anyway, or write it in FORTRAN, the only major programming language I now know. FORTRAN is old but still used for some programming problems, but I don't have a version for my computer, and don't know whether one is conveniently available.

(5) My criticism of the common claim by Internet physics popularizers that there is no mass increase of a massive object with increasing relative speed of that object, and their related insistence that "m" in "E = mc^2" refers only to rest mass

(6) A question of mine about whether there has been any solution, which there probably hasn't been, of the problem of calculating wave-wave interactions in quantum mechanics, which really is necessary to properly do QM, and is closely related to the QM measurement problem

The full entry for (6), accessed from "Entries for items 1-10" above, consists of an email I sent to the well-known Internet physics popularizer and critic Sabine Hossenfelder (which she never replied to). It contains (1) a discussion of the wave-wave interaction problem, followed by a criticism of Hossenfelder's claim, popular on the Internet, that according to the standard model (involving quantum mechanics) an electron can be in 2 places at once, (2) what I think is a disproof of her dismissal of the causality-violation objection to faster-than-light travel (This states a version of such an objection in section VI of my "Conflict between QM and SR" paper.), and (3) a comment about the triviality of quantum field theory.

(7) My puzzlement about the reason for the seemingly obsessive and irrational insistence by many physicists that there is, and must be, information conservation in quantum mechanics, which is, or was, considered by them to be a problem due to the apparent loss of information in black holes

Added after writing the above: The "Information Paradox" in QM is almost always stated by physicists to be that, according to QM, information is never lost, due to unitarity (of the time transformation of the QM wavefunction), but it seems to be lost when something enters a BH. This claim is incorrect, because according to QM such unitarity doesn't hold when a measurement is made, and measurement is an absolutely necessary part of QM. However, I have come to realize that what (at least some) physicists mean by "never lost" in their statement of the paradox is "never lost except when measurements are made". Physicists irritatingly often don't mean what they say. This modified supposed paradox is more nearly truly a paradox, but even it isn't really very paradoxical, since while unitarity of the time transformation of the wavefunction obeying the Schrodinger equation makes the past and future wavefunction calculatable in principle from, or at least determined by, the present wavefunction (in the absence of intervening measurements), the present wavefunction of a heretofore unknown system, which most systems in the universe are, in principle can't be determined. For systems whose wavefunction is known, the seeming loss of information when they enter a BH may be in conflict with what QM without general relativity says, but GR perhaps should be considered to modify QM in such a way that according to it unitarity sometimes does not hold even in the absence of observations, just because of the possible failure of unitarity for systems that enter a BH (as some physicists in fact say).

(8) A question: Is there really an elementary error at the beginning of volume III of Whitehead and Russel's Principia Mathematica because what is said there to be an equivalent of the definition of "well-ordered set" trivially isn't actually equivalent to that definition, or is the appearance of that to me due to my misunderstanding of PM's terminology? (PM has "series" for "totally ordered set", but that isn't what I'm referring to.)

Added later- Neither. The standard definition of "well-ordered set" A given by PM in Vol. III is that every non-empty subset of A contains a smallest (first) element. The definition offered by PM as equivalent to this is that every subset of A which has a successor (an element of A which is larger than every element of the subset) has a sequent (a smallest successor). I thought the integers Z would be a counterexample to this, since every non-empty subset of Z which has a successor has a smallest successor, but Z isn't well-ordered, since it doesn't have a smallest element. (I was assuming the restriction "non-empty" applied also to such subsets with successors.) Two members of Mathematics Stack Exchange pointed out that PM's equivalent definition included no such "non-empty" restriction, so Z isn't a counterexample, since the empty set is a subset of Z and has a successor (every element of Z is larger than every element of the empty set), but doesn't have a sequent, a smallest successor. PM proves the equivalence later. A simple proof goes like this: Let W be a set such that every subset S of W has a sequent in W. For each subset S of W, let L be the set of all elements of W which are less than every element of S. Then, by the assumption on W, L must have a sequent p in W. (For W = S = Z, and under the "non-empty" requirement on subsets S, since L would be empty, it wouldn't be required to have a sequent p.) Then p is the smallest element of S, so we have shown that W is well-ordered.

Perhaps the absence of "non-empty" in the PM equivalent condition should have been pointed out in PM just after the statement of equivalence, because assuming it was meant to be there is an easy mistake to make, since it is in the standard definition.

(9) The question of what the Axiom of Reducibility in Principia Mathematica really says. There are unclear but seemingly conflicting versions of it in PM. Some people have claimed that the A. of R. allows so much freedom in constructing sentences that it shorts out the Theory of Types and allows Russell's Paradox into PM, while others say that it is so restrictive that it would prevent much of present mathematics from being done. Is any form of it really necessary for logic and mathematics?

(10) My uncertainty about the significance and correctness of saying that Zermelo-Fraenkel set theory is a first-order theory, and also about a similar problem for those theories which use ZF

(11) The much-debated question of whether there is an energy-time uncertainty relation in quantum mechanics

(12) The problem of why there is a mental direction ("arrow") of time (we can remember some past things, but no future things), and why it is oriented the way it is with respect to the perceived temporal direction of causation, and why both are oriented the way they (almost always) are with respect to the temporal direction of increase of entropy of closed systems (It turned out that I couldn't, and can't, solve this problem, which I decided I needed to solve in order to properly write my Ph.D. dissertation in Philosophy of Science on my chosen subject of causation; this was the principle reason that I never got a doctorate.)

(13) The falsity of the common belief that there are still major unsolved paradoxes, indeed actual contradictions, in logic, set theory, and mathematics

(14) A slight criticism of the common claim, by even technically qualified people, that laser radiation is monochromatic and coherent

(15) Impossibility theorems, and their proofs, in mathematics and logic: Gödel's First Incompleteness Theorem, abbreviated "G1IT" in my Gödel paper discussed above, as far as I know currently has no known (correct) proof, but two other impossibility theorems, Alan Turing's Halting Problem theorem, and Andrey Kolmogorov's complexity theorem, both have known valid proofs. Turing's theorem says that there does not exist a Turing machine which, for each Turing machine X and input to it Inp, can solve the halting problem for X with input Inp, that is, compute whether X with input Inp eventually stops, after a finite number of steps, instead of continuing forever. Kolmogorov's theorem says that there does not exist an algorithm (recursive function, Turing machine) which, for each programming system P and positive integer N, can compute the Kolmogorov complexity in P of N (maybe it should be "numeral" N instead of "positive integer" N). Turing's theorem, in fact, has two quite different proofs that I know of. One of these was suggested but not actually given in Turing's 1937 paper, the other, a very simple and non-technical one, is given in the YouTube video "Proof that computers can't do everything", https://www.youtube.com/watch?v=92WHN-pAFCs, Neither is the [maybe] proof actually given by Turing in that 1937 paper, which I don't completely understand. The Turing theorem has been used to prove other impossibility theorems, including the Tiling the Plain theorem and a theorem similar to, but different from, G1IT, which similar theorem I state and give an outline of the proof of in my Gödel paper. (The theorem and proof outline are adapted from a Wikipedia article.)

(16) My road to the Planck mass, length, and time

(17) General, necessarily brief, musings on the probably unsolvable problems of (a) Why anything physical or mental exists, (b) The problem of personal mental identity, which problem is difficult even to state, (c) What some philosophers call the "hard problem"-- generally, problems related to consciousness, especially explaining (at least, predicting- and this, at least probabilistically) the details of consciousness on the basis of physical theory (Actually, I think there is a slight possibility that this is solvable.), (d) Finding a justification for scientific induction, other than that so far induction has worked fairly well, which justification itself is justification by (perhaps a higher level, -meta) induction, and (e) Finding a satisfactory understanding of the meaning of "the probability of (a type of) event" (which intuitively seems to refer to an extremely important concept) in terms of the frequency of (that type of) event's actually occurring

(18) Free will versus determinism- there is no conflict

(19) Skolem's paradox- complete confusion by Skolem?

(20) The quantum mechanical measurement problem- I discuss some seldom-mentioned aspects, but don't propose any solution

(21) A change in the reference point for potential energy produces a change of the potential energy term in the non-relativistic Schrödinger equation, which changes, in a non-trivial way, what wavefunctions are solutions of it, without changing the physical situation the SE applies to. I just realized this; I should have done so long ago. Such sensitivity to the gauge of the potential may not be a problem in current gauge-invariant quantum field theories, but presently I don't know whether this is the case.

(22) A discussion of the extremely precise (within less than 4 pS) worldwide time synchronization required by the Event Horizon Telescope for it to be able to image the supermassive black hole in M87, why this requirement couldn't be met by existing systems at the time the M87 BH image data was recorded by the EHT project, and how EHT nevertheless created the BH image.