According to the current dogma, Aleph-0 is less than Aleph-1, but is there evidence to the contra... more According to the current dogma, Aleph-0 is less than Aleph-1, but is there evidence to the contrary? Is it really true that something with an unlimited upside is less than another thing with an unlimited upside? This paper offers proof to the contrary.
Assuming different infinities are unequal leads to strange and counter-intuitive mathematical res... more Assuming different infinities are unequal leads to strange and counter-intuitive mathematical results such as Ramanujan's finite solutions to infinite divergent series. On the flip side, assuming different infinities are equal leads to infinite solutions to divergent infinite series. This last assumption, however, flies in the face of the current dogma: that different infinities are unequal. This paper offers proofs that show that different infinities are in fact equal.
This paper shows how an infinite series yields both an infinite solution and a finite solution an... more This paper shows how an infinite series yields both an infinite solution and a finite solution and how to determine which one is correct and which one is just plain crazy!
Here is a short proof that shows that the set of natural numbers has as many or more members than... more Here is a short proof that shows that the set of natural numbers has as many or more members than any other set.
This paper reveals a paradox within our current understanding of gravity that makes problems invo... more This paper reveals a paradox within our current understanding of gravity that makes problems involving more than two bodies seemingly unsolvable. By introducing the alpha factor, the paradox is resolved and the solutions to various n-body problems become clear.
Quantum entanglement can be explained using decks of cards. This paper shows how these decks of c... more Quantum entanglement can be explained using decks of cards. This paper shows how these decks of cards violate Bell's inequality, and that "hidden operators" determine a quantum measurement outcome. If these hidden operators could be known prior to measurement, one could predict the outcomes of quantum events. The herein thought experiment demonstrates that no faster-than-light communication takes place between entangled pairs. Further, when teleportation is scrutinized, there is no evidence that information passes through a wormhole.
According to R.P. Kerr, a black hole need not contain a singularity. Such an assertion prompted ... more According to R.P. Kerr, a black hole need not contain a singularity. Such an assertion prompted this author to explore alternate black-hole models where a singularity is unnecessary. The models presented here show that a sufficiently large black hole could contain a universe; that an average black hole could contain a neutron star. This paper also shows why micro-black holes are untenable and why a typical black hole has a mass of at least approximately three solar masses. As an additional bonus, the models presented conserve quantum information inside the black holes, and, are consistent with black-hole entropy and temperature.
Hawking radiation has not been directly observed. Maybe it exists, maybe it doesn't. Even it it... more Hawking radiation has not been directly observed. Maybe it exists, maybe it doesn't. Even it it exists, some very fundamental physical laws prevent black hole evaporation. At the quantum level, Hawking radiation can be turned on its head. It could just as easily add mass to a black hole.
According to Einstein's theories of relativity, nothing is faster than light; yet, observations m... more According to Einstein's theories of relativity, nothing is faster than light; yet, observations made by Newton, Laplace and Van Flandern led them to believe gravitational information is much much faster than light, virtually instantaneous. To thicken the plot further, LIGO observed gravitational waves propagating within the light-speed limit. Then there's the vacuum energy problem where the vacuum seems to have up to infinite energy! By making use of quantum physics and changing an initial assumption about the vacuum energy, it is possible to connect the dots between quantum physics and General Relativity. By exposing a fundamental flaw in the rubber-sheet model for curved spacetime, it is possible to create a superior model that reconciles the speed of gravitational waves with the illusion of faster-than-light gravitational information.
This paper shows mathematically and experimentally why it is highly unlikely that the speed of gr... more This paper shows mathematically and experimentally why it is highly unlikely that the speed of gravity was successfully measured. Consider the rubber-sheet analogy. If you place an iron ball on a rubber sheet, you will see the ball depress and curve the rubber sheet. If you roll the ball accross the sheet, you will see the sheet flatten out at the ball's previous position and the sheet will begin to curve at the ball's current position. Over a given distance, it takes time for the curve to flatten and reform. We can calculate the speed of this process by dividing the distance by the time. One might assume we can calculate the speed of gravity in an analogous manner. In fact, I had an email exchange with Sergei Kopeikin who claimed that during the Jovian Deflection Experiment he observed Jupiter's gravity fading and reforming as Jupiter moved through space. The speed of this process turned out to be the speed of light or close to it. He also informed me that the gravity was stronger at Jupiter's previous (retarded) position and weaker at Jupiter's current position due to the light-time delay.
This paper shows why entanglement is not limited to the quantum realm, and shows how entanglement... more This paper shows why entanglement is not limited to the quantum realm, and shows how entanglement and superluminal speed is not only possible, but consistent with special and general relativity.
According to Relativity Theory, everything propagates through spacetime at light speed. However, ... more According to Relativity Theory, everything propagates through spacetime at light speed. However, a mass at rest propagates solely through time and experiences zero velocity. A massless photon propagates solely through space and experiences no time. Other objects propagate through time and space, and, experience both time and subluminal velocities. This paper demonstrates that both gravity (the fundamental interaction) and gravitational waves propagate at light speed through spacetime, but with varying degrees through time and space, i.e., they each have different velocities through space.
According to Relativity Theory, everything propagates through spacetime at light speed. However,... more According to Relativity Theory, everything propagates through spacetime at light speed. However, a mass at rest propagates solely through time and experiences zero velocity. A massless photon propagates solely through space and experiences no time. Other objects propagate through time and space, and, experience both time and subluminal velocities. This paper demonstrates that both gravity (the fundamental interaction) and gravitational waves propagate at light speed through spacetime, but with varying degrees through time and space, i.e., they each have different velocities through space.
This paper reveals the flaws in the thought experiment used to demonstrate that the speed of grav... more This paper reveals the flaws in the thought experiment used to demonstrate that the speed of gravity is no faster than light speed.
Many P vs. NP discussions leave out what is perhaps the most important question any problem solve... more Many P vs. NP discussions leave out what is perhaps the most important question any problem solver could ask: "Are there any clues?" This paper shows how clues can lead to solving the most complex problem, and, how the lack of clues prevents one from solving a seemingly simple problem. Sudodu is deemed NP-complete, but the player is provided clues, and these clues cut the complexity of Sudoku down to size, a size that can be managed by a typical personal computer or a player that employs three deduction strategies. According to Arto Inkala, the Sudoku puzzle below is the hardest ever. It took him three months to design it, but I was able to solve it … ... in less than 10 seconds ... with the help of a python app I created. Here
This paper offers a solution to the sample NP problem that can be found at the Clay Mathematics I... more This paper offers a solution to the sample NP problem that can be found at the Clay Mathematics Institute's website. A python program is included along with commentary on whether or not P = NP.
Re: the Continuum Hypothesis: Is there any set which has more members than the set of natural num... more Re: the Continuum Hypothesis: Is there any set which has more members than the set of natural numbers (N), but fewer members than the set of real numbers (R)? The short answer is no. The long answer, that includes mathematical proof, shall be set forth in this paper.
This paper reveals why Cantor's diagonalization argument fails to prove what it purportedly prove... more This paper reveals why Cantor's diagonalization argument fails to prove what it purportedly proves and the logical absurdity of "uncountable sets" that are deemed larger than the set of natural numbers. Cantor's diagonalization proof (CDP) is used to prove various things like Gödel's Incompleteness theorem (GIT) and the uncountability of real numbers (URN). In the case of GIT, it is assumed that you can have a complete list of provable statements. You use CDP to prove a statement that is not on the list, then conclude the statement is not a provable statement; otherwise, it would be part of the list of all provable statements. However, if your complete list is not really complete, then it makes sense that you can prove a new provable statement not on the list. In the case of proving the URN, you use proof by contradiction: assume the real numbers are countable and assume a complete list of them between, say, .000 ... and .111 ... The list is believed to be complete because an infinite number of counting numbers (1 to n) each allegedly have a oneto-one correspondence to each real number on the list. CDP is then used to show there is a real number that is not on the list (a contradiction). Therefore, according to the proof, there is no bijection between the real numbers and the counting numbers. Below, we will work through an example proof using CDP and demonstrate why it comes up short. Let's begin with an nXn matrix of real numbers in base 2. We go along the diagonal and flip each bit from 0 to 1 or 1 to 0 to create a new number not on the list. If we think the number is on the list, say, at row r, we can check it and see that its bit at column r doesn't match. We can repeat this exercise for all rows and see that we truly have a new real number not on the list.
This paper shows how Gödel failed to prove his first incompleteness theorem and shows a standard ... more This paper shows how Gödel failed to prove his first incompleteness theorem and shows a standard of proof that completes a system containing genuine, not contrived, undecidable statements. Gödel's First Incompleteness Theorem is often stated as follows: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are disproved in F."
This paper shows what happens when Gödel's theorem is turned on itself, and the logical inconsist... more This paper shows what happens when Gödel's theorem is turned on itself, and the logical inconsistency that occurs when selfreferencing statements are converted to Gödel numbers. Gödel's First Incompleteness Theorem (GFIT) is often stated as follows: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F." Here's another way GFIT is stated: "In any reasonable mathematical system there's always true statements that can't be proved."
According to the current dogma, Aleph-0 is less than Aleph-1, but is there evidence to the contra... more According to the current dogma, Aleph-0 is less than Aleph-1, but is there evidence to the contrary? Is it really true that something with an unlimited upside is less than another thing with an unlimited upside? This paper offers proof to the contrary.
Assuming different infinities are unequal leads to strange and counter-intuitive mathematical res... more Assuming different infinities are unequal leads to strange and counter-intuitive mathematical results such as Ramanujan's finite solutions to infinite divergent series. On the flip side, assuming different infinities are equal leads to infinite solutions to divergent infinite series. This last assumption, however, flies in the face of the current dogma: that different infinities are unequal. This paper offers proofs that show that different infinities are in fact equal.
This paper shows how an infinite series yields both an infinite solution and a finite solution an... more This paper shows how an infinite series yields both an infinite solution and a finite solution and how to determine which one is correct and which one is just plain crazy!
Here is a short proof that shows that the set of natural numbers has as many or more members than... more Here is a short proof that shows that the set of natural numbers has as many or more members than any other set.
This paper reveals a paradox within our current understanding of gravity that makes problems invo... more This paper reveals a paradox within our current understanding of gravity that makes problems involving more than two bodies seemingly unsolvable. By introducing the alpha factor, the paradox is resolved and the solutions to various n-body problems become clear.
Quantum entanglement can be explained using decks of cards. This paper shows how these decks of c... more Quantum entanglement can be explained using decks of cards. This paper shows how these decks of cards violate Bell's inequality, and that "hidden operators" determine a quantum measurement outcome. If these hidden operators could be known prior to measurement, one could predict the outcomes of quantum events. The herein thought experiment demonstrates that no faster-than-light communication takes place between entangled pairs. Further, when teleportation is scrutinized, there is no evidence that information passes through a wormhole.
According to R.P. Kerr, a black hole need not contain a singularity. Such an assertion prompted ... more According to R.P. Kerr, a black hole need not contain a singularity. Such an assertion prompted this author to explore alternate black-hole models where a singularity is unnecessary. The models presented here show that a sufficiently large black hole could contain a universe; that an average black hole could contain a neutron star. This paper also shows why micro-black holes are untenable and why a typical black hole has a mass of at least approximately three solar masses. As an additional bonus, the models presented conserve quantum information inside the black holes, and, are consistent with black-hole entropy and temperature.
Hawking radiation has not been directly observed. Maybe it exists, maybe it doesn't. Even it it... more Hawking radiation has not been directly observed. Maybe it exists, maybe it doesn't. Even it it exists, some very fundamental physical laws prevent black hole evaporation. At the quantum level, Hawking radiation can be turned on its head. It could just as easily add mass to a black hole.
According to Einstein's theories of relativity, nothing is faster than light; yet, observations m... more According to Einstein's theories of relativity, nothing is faster than light; yet, observations made by Newton, Laplace and Van Flandern led them to believe gravitational information is much much faster than light, virtually instantaneous. To thicken the plot further, LIGO observed gravitational waves propagating within the light-speed limit. Then there's the vacuum energy problem where the vacuum seems to have up to infinite energy! By making use of quantum physics and changing an initial assumption about the vacuum energy, it is possible to connect the dots between quantum physics and General Relativity. By exposing a fundamental flaw in the rubber-sheet model for curved spacetime, it is possible to create a superior model that reconciles the speed of gravitational waves with the illusion of faster-than-light gravitational information.
This paper shows mathematically and experimentally why it is highly unlikely that the speed of gr... more This paper shows mathematically and experimentally why it is highly unlikely that the speed of gravity was successfully measured. Consider the rubber-sheet analogy. If you place an iron ball on a rubber sheet, you will see the ball depress and curve the rubber sheet. If you roll the ball accross the sheet, you will see the sheet flatten out at the ball's previous position and the sheet will begin to curve at the ball's current position. Over a given distance, it takes time for the curve to flatten and reform. We can calculate the speed of this process by dividing the distance by the time. One might assume we can calculate the speed of gravity in an analogous manner. In fact, I had an email exchange with Sergei Kopeikin who claimed that during the Jovian Deflection Experiment he observed Jupiter's gravity fading and reforming as Jupiter moved through space. The speed of this process turned out to be the speed of light or close to it. He also informed me that the gravity was stronger at Jupiter's previous (retarded) position and weaker at Jupiter's current position due to the light-time delay.
This paper shows why entanglement is not limited to the quantum realm, and shows how entanglement... more This paper shows why entanglement is not limited to the quantum realm, and shows how entanglement and superluminal speed is not only possible, but consistent with special and general relativity.
According to Relativity Theory, everything propagates through spacetime at light speed. However, ... more According to Relativity Theory, everything propagates through spacetime at light speed. However, a mass at rest propagates solely through time and experiences zero velocity. A massless photon propagates solely through space and experiences no time. Other objects propagate through time and space, and, experience both time and subluminal velocities. This paper demonstrates that both gravity (the fundamental interaction) and gravitational waves propagate at light speed through spacetime, but with varying degrees through time and space, i.e., they each have different velocities through space.
According to Relativity Theory, everything propagates through spacetime at light speed. However,... more According to Relativity Theory, everything propagates through spacetime at light speed. However, a mass at rest propagates solely through time and experiences zero velocity. A massless photon propagates solely through space and experiences no time. Other objects propagate through time and space, and, experience both time and subluminal velocities. This paper demonstrates that both gravity (the fundamental interaction) and gravitational waves propagate at light speed through spacetime, but with varying degrees through time and space, i.e., they each have different velocities through space.
This paper reveals the flaws in the thought experiment used to demonstrate that the speed of grav... more This paper reveals the flaws in the thought experiment used to demonstrate that the speed of gravity is no faster than light speed.
Many P vs. NP discussions leave out what is perhaps the most important question any problem solve... more Many P vs. NP discussions leave out what is perhaps the most important question any problem solver could ask: "Are there any clues?" This paper shows how clues can lead to solving the most complex problem, and, how the lack of clues prevents one from solving a seemingly simple problem. Sudodu is deemed NP-complete, but the player is provided clues, and these clues cut the complexity of Sudoku down to size, a size that can be managed by a typical personal computer or a player that employs three deduction strategies. According to Arto Inkala, the Sudoku puzzle below is the hardest ever. It took him three months to design it, but I was able to solve it … ... in less than 10 seconds ... with the help of a python app I created. Here
This paper offers a solution to the sample NP problem that can be found at the Clay Mathematics I... more This paper offers a solution to the sample NP problem that can be found at the Clay Mathematics Institute's website. A python program is included along with commentary on whether or not P = NP.
Re: the Continuum Hypothesis: Is there any set which has more members than the set of natural num... more Re: the Continuum Hypothesis: Is there any set which has more members than the set of natural numbers (N), but fewer members than the set of real numbers (R)? The short answer is no. The long answer, that includes mathematical proof, shall be set forth in this paper.
This paper reveals why Cantor's diagonalization argument fails to prove what it purportedly prove... more This paper reveals why Cantor's diagonalization argument fails to prove what it purportedly proves and the logical absurdity of "uncountable sets" that are deemed larger than the set of natural numbers. Cantor's diagonalization proof (CDP) is used to prove various things like Gödel's Incompleteness theorem (GIT) and the uncountability of real numbers (URN). In the case of GIT, it is assumed that you can have a complete list of provable statements. You use CDP to prove a statement that is not on the list, then conclude the statement is not a provable statement; otherwise, it would be part of the list of all provable statements. However, if your complete list is not really complete, then it makes sense that you can prove a new provable statement not on the list. In the case of proving the URN, you use proof by contradiction: assume the real numbers are countable and assume a complete list of them between, say, .000 ... and .111 ... The list is believed to be complete because an infinite number of counting numbers (1 to n) each allegedly have a oneto-one correspondence to each real number on the list. CDP is then used to show there is a real number that is not on the list (a contradiction). Therefore, according to the proof, there is no bijection between the real numbers and the counting numbers. Below, we will work through an example proof using CDP and demonstrate why it comes up short. Let's begin with an nXn matrix of real numbers in base 2. We go along the diagonal and flip each bit from 0 to 1 or 1 to 0 to create a new number not on the list. If we think the number is on the list, say, at row r, we can check it and see that its bit at column r doesn't match. We can repeat this exercise for all rows and see that we truly have a new real number not on the list.
This paper shows how Gödel failed to prove his first incompleteness theorem and shows a standard ... more This paper shows how Gödel failed to prove his first incompleteness theorem and shows a standard of proof that completes a system containing genuine, not contrived, undecidable statements. Gödel's First Incompleteness Theorem is often stated as follows: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are disproved in F."
This paper shows what happens when Gödel's theorem is turned on itself, and the logical inconsist... more This paper shows what happens when Gödel's theorem is turned on itself, and the logical inconsistency that occurs when selfreferencing statements are converted to Gödel numbers. Gödel's First Incompleteness Theorem (GFIT) is often stated as follows: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F." Here's another way GFIT is stated: "In any reasonable mathematical system there's always true statements that can't be proved."
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