Traditional paradoxes have been viewed as paradoxes due to our having been convinced of binary myopia as a valid and useful formal thought structure, which it sometimes is. How to counter-argue paradoxes is via mathematical logic and physics.
Paradox A situation allegedly containing a contradiction, and supposedly contradictions cannot exist. Paradoxes show deficiencies in theories and cause attention to new effects and further studies.
Sophism A deliberately presented wrong conclusion masking another error.
Zeno's Achilles and Tortoise Achilles (A) and the Tortoise (T) go east at speeds v and v/100 respectively starting with 100 meters between them. Zeno's 5th century BC idea was that A cannot catch T due to having to go through an allegedly infinite sequence of points in a finite time, which Zeno imagined impossible. The construction in Zeno's argument is Achilles' midpoint sequence: C1 = AT/2; C2 = C1T/2; CK = C(K-1)T/2 for an integer K; and the Tortoise's corresponding midpoint sequence given by moments t(Ci) for i = 1, ... K.
Achilles' sequence of midpoints would be infinite were Achilles a physically existent human. A human has the property of omitting the hypothetical option of shrinking while navigating an infinite convergent sequence, therefore the limit is forced by physics to obtain since the human gets so close to his or her target that separation is clearly of zero distance. For example, if Achilles' speed is 200 meters per minute, then his sequence of midpoints converges to obtaining the 100 meter distance in half a minute. Slightly trickier is when Achilles catches his tortoise, which does happen since both omit shrinking into infinitesimals in our biosphere: equate 200t with 2t + 100 to obtain their collision place as 101.01... meters from where Achilles started, which happens at time t = 1/1.98 minutes.
At merely the 5th century BC, Zeno had every reason to know to avoid forming a cyclic argument, amphiboly and demonstrate ignoratio elenchi - missing the point. Presuming what he purports to prove, Zeno sets up Achilles in an infinite sequence approaching T and then claims A cannot obtain T plus the distance T displaced in the real time of real A catching real T. Zeno's idea of continuity had a structural defect the correction of which is: the limit point obtained has to be included in the path, which is what physics does. Zeno's era was sufficiently advanced, he could have repeatedly experimented with a variety of runners and turtles to form a theory bearing some cognitive connection with physical observation. Had Zeno constructed his theory with intellectual honesty and human integrity, then he would have derived the predictable conclusion from the premises rather than forming an unrelated conclusion, that of the limit of a convergent sequence in a path being (according to Zeno) unobtainable.
We may conjecture Zeno had absence of thought for the hydrogen atom, which does seem to avoid limit convergence. A hypothetical hydrogen atom having assigned quantum number so large that the atom's radius is half a centimetre tends to agree with classical physics in terms of conjectured light emission, yet lost its real existence agreement with the biosphere.
In A. G. Dragalin's response to Zeno's first paradox in the section on Antinomy, A. G. Dragalin acknowledges the Newtonian mechanics response, omits insight from quantum mechanics and challenges the Archimedes Principle which states, for a pair of real numbers a, b, > 0, there exists a natural number n such that an > b. A. G. Dragalin challenging this analytical construction surprises me since taking the ceiling of b/a and adding one suffices for selecting n. A. G. Dragalin further surprises me by subsequently claiming Zeno's proposed paradox presents a real problem by demonstrating the possibility of denying Newtonian mechanics inside the valid domain of Newtonian mechanics (I think possibilities are by definition unreal, since were they constructible they would be labelled as such). A. G. Dragalin and I are agreed as to infinity being a concept worth revisiting, however the pragmatic impulse for reading non-Archimedian ordered fields is non-obvious to me.
Zeno's Sandpile is presented by A. G. Dragalin as:
Adjusting mathematics and physics for mathematical unification and for a clear explanation of quantum gravity causes many of us to ask questions we would not normally ask. For example, A. G. Dragalin's response to the Sandpile Paradox questions the inductive proof method in mathematics by raising consideration of undefined volumes which makes sense when considering the quantum number associated with a hydrogen atom, and remains difficult to interpret in classical mathematics. A. G. Dragalin introduces indefinite volumes handled by mathematical logic by exact methods which differs from indiscriminate interpretations of classical analytical methods; I agree with A. G. Dragalin as to the absence of mathematical induction inside unification on the basis of constructive selection.
Russell's Paradox Construct the set T whose elements are sets each of which is not an element of itself. Is T an element of itself? In traditional binary logic without considering the meaning of the word 'not', each of the two available answers contradicts itself. However, 'not' could validly mean approximately, expected to be, asymptotically approaching, an inversion of, or a successor of.
My refutation of Russell"s Paradox is A = {A} which sidesteps interpretations of the word 'not' and is supported by my two birth certificates issued by the government of Canada.
JES is not an element of JEO and JEO is not an element of JES, yet both are me thus I am.
Responding to A G Dragalin's response to Russell's Paradox, I disagree with the traditional response to Russell's Paradox of prohibiting sets which are members of themselves. Recursion happens via containing a defined entity within its definition. A G Dragalin guides us to consider the question whether an exactly-described set of properties therefore causes a set of objects to exist in possession of the described properties.
Inside binary myopia, contradictions fallaciously yield nothing. "Left + right = nothing. Say sing = nothing. Possible impossibility = nothing. Two birth certificates in semblance of disagreement = nothing. Us + them = nothing. You + me = nothing. Vancouver Canada + Logic = nothing. One red shift star + one blue shift star = nothing since shift cancels to 0." Inside honest ternary logic, the one which works as this biosphere works, "Left one step + right one step = two steps. Us + them = us all together. You + me = us. Vancouver Canada + Logic = a new ten billion dollar industry. One red shift star + one blue shift star = two stars. Two birth certificates = one legal name change without red tape."
Russell's Paradox is constructed with semblance of reasonableness due to double negatives cancelling in traditional binary logic, also due to the absence of self-inclusion in arithmetic sets normally studied in undergraduate mathematics. However, two 180 degree rotations in R^3 differs from absence of rotation. Consider the complementary definition, a set Y has property A iff Y is self-inclusive; let Z be the set of all such Y then ask whether Z is in Y - Z being self-inclusive suffices.
Russell's Paradox raises the valid question of what a mathematician is. This mathematical logician excludes anyone relying on vacuous claims, lies, or harm. Formal fallacies I have observed among some mathematicians practising what could be interpreted as embezzlement include False Cause, Enthymeme, Composition, Accident, Existential Fallacies, Exhortation, Illicit arguments, Bandwagon argument, Appeal to Ignorance, Appeal to Authority, Ignoratio Elenchi, Red Herring, Petitio Principia, Appeal to Force, Suppressed Evidence, Ad Hominem Abusive, Ad Hominem Circumstantial, Amphiboly; Informal Fallacies I observed include blaming victims for what was done to victims of real crimes, insisting interpretations be dishonestly low, agreeing to semblances of fairness too little too late, stealing intellectual property in lieu of teaching, insistence upon blind faith within mathematics, or living inside a 24/7 hour party.
Who belongs in this world? Some say the definition for existence is to be able to make a product which sells profitably. Some say our definition for existence is in being able to define matter or energy. I prefer inclusive capitalism in our dual economy provide our existence conditions, giving each of us supported liberty to relate within our range of skilled interests. I prefer standards we establish in our range of logical disciplines exclude anyone that relies on fallacies to obtain results, plus includes whom contributes positively to our discipline, our community, humanity, the people of Earth plus our biosphere. I prefer our fields stop being hideaways for anyone shy in this world. We would lack strategy were we to follow Russell's lead by defining mathematicians by a particular attribute since the logical complement of the definition qualifies, too; however, denying obvious presence of skill in the hope of secrecy could be assessed as a weak strategy.
A G Dragalin seeks a domain protected from paradox. Consider a form G/H where H is being mapped to the empty set rather than to zero. In binary, (G, H) could have traditional truth values (T, T), (T, F), (F, T) or (F, F). Add a dimension for manoeuvrability of information, while mapping G/H to G (since H becomes empty) arrange truths strictly via honest communications to T. G at truth value T is a final solution, as A G Dragalin sought. Reference, my corrections of Aristotle in 1996.
The Village Barber The traditional form is, a village barber opts to shave only those villagers who do not shave themselves; does the village barber shave himself? Avoiding the traditional trap, form the positively constructive interpretations of "not" which do exist:
Richard's Paradox
In August 2003, I announced I hold life's clock, by which I referred to (a) a binary pair of opposites of a binary pair of opposites trapping semblances of nothing within, (b) relativity without time, (c) ternary logic in agreement with living entities in this biosphere coordinated with physics. I have since been surprised by some interpretations of my announcement, including the expectation of explaining where our universe comes from. The correct definition of if (the positive parts) forms part of part (c) of my 2003 announcement.
Texts which communicate in retrospect have some meaning. Landmark Education provides an interesting alternative response to Richard's Paradox in which words can be viewed as meaningless and results as possible to create from nothing. Landmark Education's technology works without belief yet also without disbelief. One could invite a representative of Landmark Education to explain the phenomenon of distance education in accredited universities inside a theory of meaningless text since distance education in accredited universities also works.
Eubulides' Paradox Euclid's student who was an adversary of Aristotle (4th century BC) announced, "What I say now is a lie." The paradox arises from assuming whatever is supposed by saying ABC is therefore true. We have a format for supposition in mathematics which lets us work with sensible hypothesis via logic to hold our work inside a range of sensible conclusions. While working, we implement logic without supposing a declared supposition be true or false - what would motivate us to implement decisions prematurely? Observe Eubulides reportedly avoided formats equivalent to, "I am lying now." Thus the alleged paradox in the supposition, "what I say now is a lie," has absence of existence as a paradox. Eubulides could equivalently have said, "Suppose I am lying now."
References
The Soviet Encyclopaedia of Mathematics, 2nd edition, chief editor I. M. Vinogradov. ISBN 1 55608 010 7.
Physics by Halliday and Resnick. LCCC 66 11527.
Paradox A situation allegedly containing a contradiction, and supposedly contradictions cannot exist. Paradoxes show deficiencies in theories and cause attention to new effects and further studies.
Sophism A deliberately presented wrong conclusion masking another error.
Zeno's Achilles and Tortoise Achilles (A) and the Tortoise (T) go east at speeds v and v/100 respectively starting with 100 meters between them. Zeno's 5th century BC idea was that A cannot catch T due to having to go through an allegedly infinite sequence of points in a finite time, which Zeno imagined impossible. The construction in Zeno's argument is Achilles' midpoint sequence: C1 = AT/2; C2 = C1T/2; CK = C(K-1)T/2 for an integer K; and the Tortoise's corresponding midpoint sequence given by moments t(Ci) for i = 1, ... K.
Achilles' sequence of midpoints would be infinite were Achilles a physically existent human. A human has the property of omitting the hypothetical option of shrinking while navigating an infinite convergent sequence, therefore the limit is forced by physics to obtain since the human gets so close to his or her target that separation is clearly of zero distance. For example, if Achilles' speed is 200 meters per minute, then his sequence of midpoints converges to obtaining the 100 meter distance in half a minute. Slightly trickier is when Achilles catches his tortoise, which does happen since both omit shrinking into infinitesimals in our biosphere: equate 200t with 2t + 100 to obtain their collision place as 101.01... meters from where Achilles started, which happens at time t = 1/1.98 minutes.
At merely the 5th century BC, Zeno had every reason to know to avoid forming a cyclic argument, amphiboly and demonstrate ignoratio elenchi - missing the point. Presuming what he purports to prove, Zeno sets up Achilles in an infinite sequence approaching T and then claims A cannot obtain T plus the distance T displaced in the real time of real A catching real T. Zeno's idea of continuity had a structural defect the correction of which is: the limit point obtained has to be included in the path, which is what physics does. Zeno's era was sufficiently advanced, he could have repeatedly experimented with a variety of runners and turtles to form a theory bearing some cognitive connection with physical observation. Had Zeno constructed his theory with intellectual honesty and human integrity, then he would have derived the predictable conclusion from the premises rather than forming an unrelated conclusion, that of the limit of a convergent sequence in a path being (according to Zeno) unobtainable.
We may conjecture Zeno had absence of thought for the hydrogen atom, which does seem to avoid limit convergence. A hypothetical hydrogen atom having assigned quantum number so large that the atom's radius is half a centimetre tends to agree with classical physics in terms of conjectured light emission, yet lost its real existence agreement with the biosphere.
In A. G. Dragalin's response to Zeno's first paradox in the section on Antinomy, A. G. Dragalin acknowledges the Newtonian mechanics response, omits insight from quantum mechanics and challenges the Archimedes Principle which states, for a pair of real numbers a, b, > 0, there exists a natural number n such that an > b. A. G. Dragalin challenging this analytical construction surprises me since taking the ceiling of b/a and adding one suffices for selecting n. A. G. Dragalin further surprises me by subsequently claiming Zeno's proposed paradox presents a real problem by demonstrating the possibility of denying Newtonian mechanics inside the valid domain of Newtonian mechanics (I think possibilities are by definition unreal, since were they constructible they would be labelled as such). A. G. Dragalin and I are agreed as to infinity being a concept worth revisiting, however the pragmatic impulse for reading non-Archimedian ordered fields is non-obvious to me.
Zeno's Sandpile is presented by A. G. Dragalin as:
One grain of sand does not form a sandpile. If n grains of sand still do not form a sandpile, it follows that they will not form a pile after another grain of sand has been added. Accordingly, no number of grains of sand can form a sandpile.Obviously whatever we agree a small pile of sand is, it has a finite number of grains of sand, as do beaches. Disagreement about a definition's detail differs from disagreement regarding the meaning of words. For example, in my own life, rather than me being required to argue as though in an endless sequence of PhD defences and argue against an endless sequence of PhDs all the while having the experience of Jonathan Borwein unresponsively indicating each argument insufficient for him, instead, end that useless sequence and replace that with requiring Jonathan Borwein to defend his written assertion of May 1996 that I have an absence of mathematical skill, together with an acceptable grade in his analysis class in spring 1995, together with an honours mathematics degree from Dalhousie University. We could agree a pile of sand has two million grains, ten thousand grains or one hundred grains. Let some natural number m be the number upon which we agree (observe the absence of the axiom of choice in this construction since prerequisite to two of us agreeing, communication happens and communication resulting in agreement is constructive) is the defining minimum number of grains of sand in order to form a sandpile, disambiguated from a bunch of sand insufficient to pile. m > 1. Proof by induction is irrelevant to Zeno's paradox as presented and the entire paradox depends on having not yet agreed upon a definition among relevant participants in the conversation. The erroneous conclusion exemplifies forming a conclusion based on an absence of definition; due to that absence, equivocation and amphiboly are evaded however, leaping to conclusions based on full absences of relevant material exemplifies appeal to ignorance.
Adjusting mathematics and physics for mathematical unification and for a clear explanation of quantum gravity causes many of us to ask questions we would not normally ask. For example, A. G. Dragalin's response to the Sandpile Paradox questions the inductive proof method in mathematics by raising consideration of undefined volumes which makes sense when considering the quantum number associated with a hydrogen atom, and remains difficult to interpret in classical mathematics. A. G. Dragalin introduces indefinite volumes handled by mathematical logic by exact methods which differs from indiscriminate interpretations of classical analytical methods; I agree with A. G. Dragalin as to the absence of mathematical induction inside unification on the basis of constructive selection.
Russell's Paradox Construct the set T whose elements are sets each of which is not an element of itself. Is T an element of itself? In traditional binary logic without considering the meaning of the word 'not', each of the two available answers contradicts itself. However, 'not' could validly mean approximately, expected to be, asymptotically approaching, an inversion of, or a successor of.
My refutation of Russell"s Paradox is A = {A} which sidesteps interpretations of the word 'not' and is supported by my two birth certificates issued by the government of Canada.
JES is not an element of JEO and JEO is not an element of JES, yet both are me thus I am.
Responding to A G Dragalin's response to Russell's Paradox, I disagree with the traditional response to Russell's Paradox of prohibiting sets which are members of themselves. Recursion happens via containing a defined entity within its definition. A G Dragalin guides us to consider the question whether an exactly-described set of properties therefore causes a set of objects to exist in possession of the described properties.
Inside binary myopia, contradictions fallaciously yield nothing. "Left + right = nothing. Say sing = nothing. Possible impossibility = nothing. Two birth certificates in semblance of disagreement = nothing. Us + them = nothing. You + me = nothing. Vancouver Canada + Logic = nothing. One red shift star + one blue shift star = nothing since shift cancels to 0." Inside honest ternary logic, the one which works as this biosphere works, "Left one step + right one step = two steps. Us + them = us all together. You + me = us. Vancouver Canada + Logic = a new ten billion dollar industry. One red shift star + one blue shift star = two stars. Two birth certificates = one legal name change without red tape."
Russell's Paradox is constructed with semblance of reasonableness due to double negatives cancelling in traditional binary logic, also due to the absence of self-inclusion in arithmetic sets normally studied in undergraduate mathematics. However, two 180 degree rotations in R^3 differs from absence of rotation. Consider the complementary definition, a set Y has property A iff Y is self-inclusive; let Z be the set of all such Y then ask whether Z is in Y - Z being self-inclusive suffices.
Russell's Paradox raises the valid question of what a mathematician is. This mathematical logician excludes anyone relying on vacuous claims, lies, or harm. Formal fallacies I have observed among some mathematicians practising what could be interpreted as embezzlement include False Cause, Enthymeme, Composition, Accident, Existential Fallacies, Exhortation, Illicit arguments, Bandwagon argument, Appeal to Ignorance, Appeal to Authority, Ignoratio Elenchi, Red Herring, Petitio Principia, Appeal to Force, Suppressed Evidence, Ad Hominem Abusive, Ad Hominem Circumstantial, Amphiboly; Informal Fallacies I observed include blaming victims for what was done to victims of real crimes, insisting interpretations be dishonestly low, agreeing to semblances of fairness too little too late, stealing intellectual property in lieu of teaching, insistence upon blind faith within mathematics, or living inside a 24/7 hour party.
Who belongs in this world? Some say the definition for existence is to be able to make a product which sells profitably. Some say our definition for existence is in being able to define matter or energy. I prefer inclusive capitalism in our dual economy provide our existence conditions, giving each of us supported liberty to relate within our range of skilled interests. I prefer standards we establish in our range of logical disciplines exclude anyone that relies on fallacies to obtain results, plus includes whom contributes positively to our discipline, our community, humanity, the people of Earth plus our biosphere. I prefer our fields stop being hideaways for anyone shy in this world. We would lack strategy were we to follow Russell's lead by defining mathematicians by a particular attribute since the logical complement of the definition qualifies, too; however, denying obvious presence of skill in the hope of secrecy could be assessed as a weak strategy.
A G Dragalin seeks a domain protected from paradox. Consider a form G/H where H is being mapped to the empty set rather than to zero. In binary, (G, H) could have traditional truth values (T, T), (T, F), (F, T) or (F, F). Add a dimension for manoeuvrability of information, while mapping G/H to G (since H becomes empty) arrange truths strictly via honest communications to T. G at truth value T is a final solution, as A G Dragalin sought. Reference, my corrections of Aristotle in 1996.
The Village Barber The traditional form is, a village barber opts to shave only those villagers who do not shave themselves; does the village barber shave himself? Avoiding the traditional trap, form the positively constructive interpretations of "not" which do exist:
- Some people who do not shave their heads are bald.
- Some people who do not shave their heads avoid all haircuts with preference for long hair.
- Some people prefer to keep some hair, thus go to hairdressers or neighbours for hair cuts.
- Some people shave their own heads.
Contrary to one of A G Dragalin's assertions, such barbers may exist. Contrary to a second of A G Dragalin's assertions, most real-life situations could be exactly formulated or reliably defined. We agree as to the importance of internal consistency in reliable systems.
Cantor's Paradox The traditional form has been to ask whether the power set of the set of all sets could be contained inside the set of all sets, without examining whether the set of all sets may be constructed by what specific definition. (The power set of a set S is the set of subsets of S, generally denoted P(S).) Clearly in English, of course the power set of the set of all sets, being merely one set, must be in the set of all sets in order for the sentence to make sense; however there is more to mathematics than English.
What constructability constraints have we? During autumn 2011, we specified formal language L (publicly distributed) has the existential quantifier without universal quantification. Adapting from binary logic to ternary, we place strong emphasis on the word some, an emphasis which may wane as we habituate to ternary. In the formal language L, Cantor's Paradox remains unconstructable meaningfully.
For reference, the formal language L to which we refer is {x, y, z, inclusion, description, implication, or, existence}, with the empty set having a positively constructive definition as the description of whatever variables ostensibly exist which accept anything included in the variable.
Richard's Paradox
In the formulation of G. D. W. Berry (1906), consider the set of natural numbers each one of which can be uniquely defined by a meaningful text containing not more than 1000 syllables. Obviously, the number of such texts is finite, since the collection of all texts with 1000 syllables is finite. Consider the smallest natural number that is not a member of the set defined above.
The above paragraph is a meaningful text, containing not more than 1000 syllables, which uniquely determines some natural number which, by definition, cannot be characterized by a set of this type. Obviously, this paradox can be avoided if this text is declared to be not meaningful (or to be a text which does not define a natural number), but in such a case, as before, there arise difficult problems concerning the criteria for a text to be meaningful.
(Page 207, Volume 1, Encyclopaedia of Mathematics 2nd edition)A finite subset of natural numbers which may be uniquely defined, each set member in one-to-one correspondence with one specific meaningful text containing up to 1000 syllables (as per Berry's formulation) may have distinct numbers corresponding to different meanings conveyed by the same n syllables for some natural number n defined in a traditional way. The smallest meaningful paragraph has one monosyllabic verb in the imperative form thus intuitively could correspond to the natural number one however each imperative monosyllabic verb makes a distinct meaningful paragraph of normal size one, therefore one such paragraph could define whichever natural number we select, including the option of selecting zero. The first quoted paragraph defines an unspecified natural number, say m, without contradicting the provided definition contrary to the claim in the second paragraph. Words have meaning. Pi exists. The text of Richard's Paradox conveys meaningful information. What criteria establish a text as meaningful?
In August 2003, I announced I hold life's clock, by which I referred to (a) a binary pair of opposites of a binary pair of opposites trapping semblances of nothing within, (b) relativity without time, (c) ternary logic in agreement with living entities in this biosphere coordinated with physics. I have since been surprised by some interpretations of my announcement, including the expectation of explaining where our universe comes from. The correct definition of if (the positive parts) forms part of part (c) of my 2003 announcement.
Texts which communicate in retrospect have some meaning. Landmark Education provides an interesting alternative response to Richard's Paradox in which words can be viewed as meaningless and results as possible to create from nothing. Landmark Education's technology works without belief yet also without disbelief. One could invite a representative of Landmark Education to explain the phenomenon of distance education in accredited universities inside a theory of meaningless text since distance education in accredited universities also works.
Eubulides' Paradox Euclid's student who was an adversary of Aristotle (4th century BC) announced, "What I say now is a lie." The paradox arises from assuming whatever is supposed by saying ABC is therefore true. We have a format for supposition in mathematics which lets us work with sensible hypothesis via logic to hold our work inside a range of sensible conclusions. While working, we implement logic without supposing a declared supposition be true or false - what would motivate us to implement decisions prematurely? Observe Eubulides reportedly avoided formats equivalent to, "I am lying now." Thus the alleged paradox in the supposition, "what I say now is a lie," has absence of existence as a paradox. Eubulides could equivalently have said, "Suppose I am lying now."
References
The Soviet Encyclopaedia of Mathematics, 2nd edition, chief editor I. M. Vinogradov. ISBN 1 55608 010 7.
Physics by Halliday and Resnick. LCCC 66 11527.
