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Right now, in 2022, we can read with perfect understanding mathematical articles and books
written a century ago. It is indeed remarkable how the way we do mathematics has stabilised.
The difference between the mathematics of 1922 and 2022 is small compared to that between the mathematics of 1922 and 1822.
Looking beyond classical ZFC-based mathematics, a tremendous amount of effort has been put
into formalising all areas of mathematics within the framework of program-language implementations (for instance Coq, Agda) of the univalent extension of dependent type theory (homotopy type theory).
But Coq and Agda are complex programs which depend on other programs (OCaml and Haskell) and frameworks (for instance operating systems and C libraries) to function. In the future if we have new CPU architectures then
Coq and Agda would have to be compiled again. OCaml and Haskell would have to be compiled again.
Both software and operating systems are rapidly changing and have always been so. What is here today is deprecated tomorrow.
My question is: what guarantee do we have that the huge libraries of the current formal mathematics projects in Agda, Coq or other languages will still be relevant or even "runnable" (for instance type-checkable) without having to resort to emulators and computer archaeology 10, 20, 50 or 100 years from now ?
10 years from now will Agda be backwards compatible enough to still recognise
current Agda files ?
Have there been any organised efforts to guarantee permanent backward compatibility for all future versions of Agda and Coq ? Or OCaml and Haskell ?
Perhaps the formal mathematics project should be carried out within a meta-programing language, a simpler more abstract framework (with a uniform syntax) comprehensible at once to logicians, mathematicians and programers and which can be converted automatically into the latest version of Agda or Coq ?
How can I find a list of open problems in Homotopy Type Theory and Univalent Foundations ?
People may have different names for ”the infinite related very small numerical things (infinitesimals)”, it doesn’t matter what they are called, they are there in our mathematics, but what are their positions as numbers or non-numbers or something else theoretically and practically, ontologically and formally?
The newly discovered modern Harmonic Series Paradox is one of family members of ancient Zeno’s Paradox, it discloses relentlessly a fact that we human still don’t know what infinitesimals are!
This problem has close relationship with whole fundamental part of infinite related area in our mathematics:
1, theoretical and practical infinite concept system
2, theoretical and practical infinite related number system
3, theoretical and practical infinite related number treating system
The problem disclosed by Zeno’s Paradox is still there and the exactly same idea is still working well. Let’s see one of the modern versions of Zeno’s Paradox
1+1/2 +1/3+1/4+...+1/n +... (1)
=1+1/2 +(1/3+1/4 )+(1/5+1/6+1/7+1/8)+... (2)
>1+ 1/2 +( 1/4+1/4 )+(1/8+1/8+1/8+1/8)+... (3)
=1+ 1/2 + 1/2 + 1/2 + 1/2 + ...------>infinity (4)
Such an antique proof (given by Oresme in about 1360), though very elementary, can still be found in many current higher mathematical books written in all kinds of languages.
Here, with limit theory and technique, we see a “strict mathematically proven” modern version of ancient Zeno’s Paradox:
1, in Harmonic Series, we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 100000000000000000000 or… from infinite infinitesimals in Harmonic Series by “brackets-placing rule" to change an infinitely decreasing Harmonic Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant (such as Un’ >10000000000000000000000000) ;
2, the “brackets-placing rule" to get 1/2 or 1 or 100 or 100000 or 100000000000000000000 or… from infinite items in Harmonic Series corresponds to different runners with different speed in Zeno’s Paradox while the items in Harmonic Series corresponds to those steps of the tortoise in Zeno’s Paradox. So, not matter what kind of runner (even a runner with the speed of modern jet plane) held the race with the tortoise he will never catch up with it.
Lacking the systematic cognition to “infinitesimal”, no one in the world now can answer following question scientifically and this is the very reason for many “suspended infinite related paradox families” in present classical infinite related mathematics:
Are “dx--->0 infinitesimal” in calculus and “Un--->0 infinitesimal” in Harmonic Series the same things? If ”yes”, why we have totally different operations on them? If ”no”, what are the differences and how to treat them differently and why?
--------Could anyone tells how many items of “Un--->0 infinitesimal” in Harmonic Series you use to produce the first Un’ >10000000000000000000000000, how many items of “Un--->0 infinitesimal” in Harmonic Series you use to produce the second Un’ >10000000000000000000000000, how many items of “Un--->0 infinitesimal” in Harmonic Series you use to produce the third Un’ >10000000000000000000000000?
There are two “reasonable” limit operations in convergent- divergent proof of Harmonious Series:
1+1/2 +1/3+1/4+...+1/n +... (1)
=1+1/2 +(1/3+1/4 )+(1/5+1/6+1/7+1/8)+... (2)
>1+ 1/2 +( 1/4+1/4 )+(1/8+1/8+1/8+1/8)+... (3)
=1+ 1/2 + 1/2 + 1/2 + 1/2 + ...------>infinity (4)
(1) During the whole process in dealing with infinite substances (infinitesimals) in limit calculations, no one dare to say “let them be zero or get the limit”. So, the infinitesimals in the calculating operations would never be too small to be out of the calculations and the calculations dealing with infinitesimals would be carried our forever. This situation has been existing in mathematics since antiquity------ those items of Un--->0 never be 0 all the time and Harmonious Series is divergent, so we can produce infinite numbers bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite Un--->0 items in Harmonious Series and change an infinitely decreasing Harmonious Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant or any infinitely increasing series with the property of Un--->infinity. Here, with limit theory and technique, we see a “strict mathematically proven” modern version of ancient Zeno’s Paradox.
(2) During the process in dealing with infinite substances (infinitesimals) in limit calculations, someone suddenly cries “let them be zero or get the limit”. So, all in a sudden the infinitesimals in the calculations become too small to stay inside the calculations, they should disappear from (be out of) any limit calculation formulas immediately. This situation has been existing in mathematics since antiquity-------those items of Un--->0 must be 0 from some time and Harmonious Series is not divergent, so we cannot produce infinite numbers bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or…from infinite Un--->0 items in Harmonious Series. But if it is convergent, another paradox appears.
But when and to which should or should not people treat infinitesimals appearing in infinite numeral cognitions that way?
Does limit theory need basic theory, what is it?
Many people agree that we are really able to produce infinite items each bigger than 1/2, or 100, or 1000000, or 100000000000000000000000000000,…by the brackets-placing rule with limit theory from Un--->0 items in Harmonic Series and change Harmonic Series into an infinite series with infinite items each bigger than any positive constants (such as 100000000000000000000000000000).
Can anyone prove how many Un--->0 items in Harmonic Series we use to produce the first three (just the first three!) positive constants of 100000000000000000000000000000----------how many Un--->0 items for the first 100000000000000000000000000000, how many Un--->0 items for the second 100000000000000000000000000000, how many Un--->0 items for the third 100000000000000000000000000000?
(1) How can we find the partial sum of n1000 instantly ?
(2) is there is a simple method to find partial sum of the sequence f(n) ?
(3) Any general method to compute partial sum of sequence ?
(4) What is the value of Method , if we have good approximation for all differentable sequence ?
Following six unavoidable suspended conundrums by newly discovered Harmonic Series Paradox have been troubling us human since antiquity:
1, what are those Un--->0 items in Harmonic Series, are they infinitesimals?
2, if they are not, what are they and what is infinitesimal?
3, if they are infinitesimals, are they potential infinitesimals or actual infinitesimals, why?
4, if they are potential infinitesimals, how to have numerical cognition to them; if they are actual infinitesimals, how to have numerical cognition to them?
5, what kind of mathematical tool “limit theory” is, does it treat “potential infinite related mathematical contents (number forms)” or “actual infinite related mathematical contents (number forms)”?
6,can we really produce infinite items each bigger than 1/2, or 100, or 1000000, or 1000000000000000000,…by that “Brackets-Placing Artifact” with limit theory from Un--->0 items in Harmonic Series and change Harmonic Series into an infinite series with infinite items each bigger than any positive constants (such as 100000000000000000000000000000)?
So many people say “infinite thing is not a number”, but we are able to take very freely all the infinite things (such as infinities and infinitesimals) as numbers in the field of analysis and set.
There are three different ways to deal with “infinite things” quantitatively in our actual limit operating behaviors:
(1) During the whole process in dealing with infinite substances (such as infinitesimals) in limit calculations, no one dare to say “let them be zero or get the limit”, because they are believed to have infinite property and they are existing forever for the whole of infinite. So, the infinitesimals in the calculating operations would never be too small to be out of the calculations and the calculations dealing with infinitesimals would be carried out forever. This situation have been existing in mathematics since antiquity, only in Zeno’s time he discovered something unscientific and created his Zeno’s Paradox to criticize this phenomenon bravely and sharply, the newly discovered Harmonic Paradox is its “modern strict mathematical proof version”.
(2) During the process in dealing with infinite substances (such as infinitesimals) in limit calculations, someone suddenly cries “let them be zero or get the limit”, because they are suddenly believed to lost infinite property and they only exist for the portion of infinite. So, all in a sudden the infinitesimals in the calculations become too small to stay inside the calculations, they should disappear from (be out of) any limit calculation formulas immediately. This situation have been existing in mathematics since antiquity, only in Berkeley’s time he discovered something unscientific and created his Berkeley’s Paradox to criticize this phenomenon bravely and sharply; our studies have proved that The Second Mathematical Crises can never be solved in present “potential infinite-actual infinite” related classical science system.
(3)Whenever infinite substances (such as infinitesimals) appear in limit calculations, someone cries “let them be zero or get the limit”. So, no calculations are needed and there would be no mathematical analysis at all. But we all know it is never allowed. Why?
The problem is: no one in the world now can tell self-justificationly why should or should not people treat “infinite things” appearing in limit calculations that way and how to understand and explain our behaviors relating to the “process and results of limit calculating operations”!
Are there really fatal defects disclosed by those suspended paradox families in limit theory related quantitative cognition ideas and behaviors to “infinite things”?
The exactly same situation happens in “non-standard analysis”.
Our science history tells that we human have been arguing and debating on the nature and definition of “potential infinite and actual infinite” for at least 2500 years since Zeno’ time, and there are still no results nowadays. So, when facing “infinite things” in present classical infinite related science theory system, no one is sure whether they are “potential infinite things” or “actual infinite things” and are not sure how to treat them scientifically------- many suspended old and new infinite related paradox families are produced (such as 2500-year old Zeno’s Achilles--Turtle Race Paradox and so many modern versions of Zeno’s Achilles--Turtle Race Paradox, Russell's Paradox and so many modern versions of Russell's Paradox, …)by the confusing of “potential infinite and actual infinite” and these paradox family members are surely unsolvable in present classical infinite related mathematics where they were produced and nourished.
One may say “look at the history of mathematics and don’t mind the arguing and debating on the nature and definition of potential infinite and actual infinite, our mathematics goes well without the definition of infinite, just close one eye and open another eye in the field of infinite and we are used to this since Zeno’ time”.
But the facts are: more and more suspended infinite related paradox families are produced by the “confusing of potential infinite and actual infinite” such as infinitesimal relating paradoxes in analysis, infinity relating paradoxes in set theory, both infinitesimal and infinity relating paradoxes in the ideas and skills of “Cantor’s diagonal-contradictory proofs and the conclusion on “Real Number Set has more elements than Natural Number Set (infinite elements in Real Number Set is more infinite than that in Natural Number Set-------Infinite R is more infinite than Infinite N)”.
Can we really just close one eye and open another eye in the field of infinite and force us be used to the fundamental defects in our mathematics?
Trying our best to have scientific foundation for our mathematics is a must we should shoulder sooner or later beyond our will.
Let me recall that, for a positive integer n, Cut(n) is the statement that, for each sequence of n-element sets, the union of all sets of this sequence is at most countable. Cut(fin) is the statement that countable unions of finite sets are at most countable. I am unable to deduce whether it is true in ZF that if Cut(n) holds for each positive integer n, then Cut(fin) also holds. Perhaps, there exists a mathematician who knows a model for ZF in which Cut(fin) fails and, simultaneously, Cut(n) holds for each positive integer n. I would be grateful for any helpful hint to give a satisfactory answer to my question. Regards, Eliza Wajch
In our present infinite related mathematics theory system, no “infinite things” can run away from “actual infinite” and “potential infinite”. So, are there “actual infinite many (big)” and “potential infinite many (big)” in our mathematics?
The exactly same question goes to infinitesimals: are there “actual infinite small (few)” and “potential infinite small (few)” in our mathematics? --------- Take Un--->0 in Harmonic Series and dx --->0 in calculus for example: (1) are they same infinitesimals under the same infinite concept with the same infinite definition? (2) are they “actual infinitesimals” or “potential infinitesimals”?
The thousand—year old suspended infinite related paradoxes tell us: something must be wrong in present classical “actual infinite” and “potential infinite” related philosophy and mathematics!
At least from Zeno’s time, most people (except Zeno with his creative “actual infinite--potential infinite” related paradox families) have been ignoring a fact that our whole present infinite related mathematics theory system has been based on “actual infinite--potential infinite”. The indefinable and confused “actual infinite” and “potential infinite” have been troubling us human for at least 2500 years with all kinds of debates, magics, paradoxes,… ------ the “2500-year-old huge black cloud of infinite related paradoxes over mathematics sky”.
Take Un--->0 in Harmonic Series and dx --->0 in calculus for example: (1) are they same infinitesimals under the same infinite concept with the same infinite definition? (2) are they “actual infinite small (few)” and “potential infinite small (few)”?
The choice to be made is similar to the choice between set theory and category theory.
There have been two opposite phenomenon about the definition for infinite in present classical infinite related science system: (1) plenty of different mathematical definitions for infinite and (2) no mathematical definition for infinite.
On the one hand, we really have had many different mathematical definitions for just half of the infinite: infinities (infinite bigs) in set theory such as “super (higher, more, less…) infinite, super (higher, more, less…) unbounded, super (higher, more, less…) unlimited, super (higher, more, less…) endless,…”, the plenty of non-self-justification “half definition” for infinite which is nothing to do with infinitesimals strongly prove that we human in fact are unable to know how to have mathematical definition for “infinite” at all------it is impossible to have mathematical definition for “infinite” in present classical infinite related science system.
On the other hand, we really have had many different mathematical definitions for another half of the infinite: non-number infinitesimal variables, actual infinitesimals,… in standard and non-standard analysis, but those suspended infinitesimals related paradoxes (the 2500—year suspended black cloud over mathematical sky), especially the newly discovered Harmonic Series Paradox in present classical infinite related mathematics make us have to admit that till now we human in fact are unable to know how to have mathematical definition for “infinite” at all------it is impossible to have mathematical definition for “infinite” in present classical infinite related science system.
It is the time for us human to work at the integration of mathematical definition for “infinite” now?!
Limit theory is a very useful cognizing tool in human science. It treats “infinite number forms”. But its foundation is empty:
1, what kind of “infinite number forms” it treats-------potential infinite number or actual infinite number or missing ups or …?
2, in present mathematical number system, there are no “infinite number forms” at all.
We should do something to make up the defects in the foundation of limit theory.
It is believed that there are the bijection relationships between Infinite Natural Number Set and Infinite Rational Number Set, but following simple story tells us that Infinite Rational Number Set has far more elements than that of Infinite Natural Number Set:
The elements of a tiny portion of rational numbers from Infinite Rational Number Set (the sub set : 0, 1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …) map and use up (bijective) all the numbers in Infinite Natural Number Set (0,1, 2, 3, 4, 5, 6, …, n …); so,infinite rational numbers (at least 2,3,4,5,6,…n,…) from Infinite Rational Number Set are left in the “one—to—one element mapping between Infinite Rational Number Set and Infinite Natural Number Set (not the integer set )------- Infinite Rational Number Set has infinite more elements than Infinite Natural Number Set.
This is the truth of a one-to-one corresponding operation and its result between two infinite sets: Infinite Rational Number Set and Infinite Natural Number Set. This is the business just between the elements’ quantity of two infinite sets and it can be nothing to do with the term of “proper subset, CARDINAL NUMBER, DENUMERABLE or INDENUMERABLE”.
Can we have many different bijection operations (proofs) with different one-to-one corresponding results between two infinite sets? If we can, what operation and conclusion should people choose in front of two opposite results, why?”
Such a question needs to be thought deeply: there are indeed all kinds of different infinite sets in mathematics, but what on earth make infinite sets different?
There is only one answer: unique elements contained in different infinite sets -------the characteristics of their special properties, special conditions of existence, special forms, special relationships as well as very special quantitative meaning! However, studies have shown that, due to the lack of the whole “carriers’ theory” in the foundation of present classical infinite theory, it is impossible for mathematicians to study and cognize those unique characteristics of elements operationally and theoretically in present classical set theory. So, it is impossible to carry out effectively the quantitative cognitions to the elements in various different infinite set scientifically -------a newly constructed Quantum Mathematics.
The article《On the Quantitative Cognitions to “Infinite Things” (IX) ------- "The Infinite Carrier Gene”, "The Infinite Carrier Measure" And "Quantum Mathematics”》 has been up loaded onto RG introducing the working ideas. https://www.researchgate.net/publication/344722827_On_the_Quantitative_Cognitions_to_Infinite_Things_IX_---------_The_Infinite_Carrier_Gene_The_Infinite_Carrier_Measure_And_Quantum_Mathematics
There have been two suspended questions challenging us human in present classical infinite related science system at least since Zeon’s time:
1, is it unavoidable that we mix (jump between) “potential infinite” and “actual infinite” whenever we cognize any infinite relating things?
2, do we treat potential infinite things or actual infinite things or jump between in mathematics quantitatively?
Can we have “potential infinite sets”? If yes, can anyone give an example of “potential infinite set”?
How can we develop a list of micro foundations for organizational capabilities? is such a list different for different capabilities?
This paper provides a simplified exposition (no real analysis) of the economic theory presented in the second part of my 1999 book, Axiomatic Theory of Economics. It makes no mention of the first part of my book about the foundations of economics. In this question we will discuss my three-term system of formal logic, specifically with comparison to the attached paper by Steve Faulkner, which was posted in reply to another question that I recently asked.
In Section 4.3 of my 1999 book I write:
“The great crack in the foundation of mainstream logic where first-sense and third-sense truth are confused has been resolved. Whenever mainstream logic speaks of affirmation they refer to phenomena having been observed that conform to a definition (truth in the first sense) and whenever they speak of negation they refer to the impossibility of phenomena conforming to a definition (falsity in the third sense). The three senses of truth must be strictly separated…”
The three sense of truth are defined in an earlier section of my book, but suffice it to say that I was not bothered so much by the paradoxes that Gödel addressed but by the fact that, if p is impossible, the statement “some p are q” is false while the statement “all p are q” is true. This is absurd. If I told you that all red-headed Eskimos can foresee the future, a logician would have to admit that, within his science, this statement is true. But everybody else would denounce me as a lunatic: Eskimos do not have red hair and nobody – regardless of the color of their hair – can foresee the future. The logical truth value of my statement will not inspire anybody to travel to Alaska to find Sibyl the Eskimo with her flaming red hair.
“A new system of formal logic will now be introduced. The three terms of this system of logic are P for possible, I for impossible and M for maybe (similar to Zen Buddhism’s mu.) Following are eleven logical relations concerning the definitions p and q. These statements are followed by a truth table which shows, in each of the four situations with which one could be presented when observing phenomena’s conformance to p and q, whether the statement affirms its possibility, its impossibility, or says nothing about that situation.”
While I do not have space here to print the entire list of eleven logical relations, I will print the truth table for “p is possible unless q is possible” to give a taste of what I am doing:
Do phenomena conform to definition p? T T F F
Do phenomena conform to definition q? T F T F
p is possible unless q is possible I P M M
I then use an example from Willard Quine’s Methods of Logic (p. 196) to illustrate how my method works:
Premises:
1) The guard searched all who entered the building except those who were accompanied by members of the firm.
2) Some of Fiorecchio’s men entered the building unaccompanied by anyone else [unaccompanied by non-Fioreccio men].
3) The guard searched none of Fioreccio’s men.
Using my system, by filling in a truth table with P (possible), I (impossible) and M (maybe), we can quickly determine if the statement, “Some of Fioreccio’s men are members of the firm” is proven. There is no room to print this here, but it is a sixteen-column truth table with four rows of P, I or M for each of the three premises and the relation, “people who work for Fioreccio.” Below this is another row labeled “result.”
“Now, filling in an I wherever we see one, a P wherever we see one that is not dominated by an I, and an M only where no statement is made either way, we get the result.”
This is in contrast to Dr. Quine’s method (p. 199), which only proves or disproves one statement at a time. I write:
“From this result [the three-term truth table] one can test the truth of any conclusion one is interested in… If we were interested in knowing whether the statement ‘All of Fiorecchio’s men entered the building unaccompanied by non-Fiorecchio men’ is implied by the premises, we would need [elaborate what is needed that we do not know] so the conclusion is not proven; it is a maybe. This is a more insightful ‘maybe’ than we had before analysis, however, as we now know where our investigation must lead.”
REFERENCES
Quine, W.V. 1982. Methods of Logic. Cambridge, MA: Harvard University Press
Whether exist an paraconsistent version of PA which can prove own paraconsistency?
“Potential infinite” and “actual infinite” are really there in our mathematics and science, but it seems very difficult to understand and express these two concepts clearly and logically ever since.
In present infinite related science and mathematics, people have been admitting the being of “potential infinite, actual infinite” concepts, unable to deny their qualitative differences and the important roles they play in the foundation of present classical infinite theory system and, unable to deny that the present infinite related classical mathematics is basing on present classical infinite theory system. The fact is: on the one hand, present classical mathematics can not avoid the constraining of “potential infinite--actual infinite” concepts and their relating “potential infinite mathematical things --actual infinite mathematical things”; on the other hand, no clear definitions for these two concepts of “potential infinite--actual infinite” and their relating “potential infinite mathematical things --actual infinite mathematical things” have been constructed since antiquity , thus naturally lead to following two unavoidable fatal defects in present classical mathematics:
(1)it is impossible (unable) to understand theoretically what the important basic concepts of “potential infinite, actual infinite” and their relating “potential infinite mathematical things --actual infinite mathematical things” are. So, in many “qualitative cognizing activities on infinite relating mathematical things (infinite relating number forms)” in present classical mathematicis, many people actually don’t know or even deny the being of “potential infinite, actual infinite” concepts and their relating “potential infinite mathematical things --actual infinite mathematical things”--------the “qualitative cognizing defects on infinite relating mathematical things (infinite relating number forms)”.
(2)it is impossible (unable) to understand operationally what kind of relationship among the important basic concepts of “potential infinite, actual infinite”, their relating “potential infinite mathematical things --actual infinite mathematical things” and all the “infinite mathematical things as well as their quantitative cognizing operations” are. So, in many “quantitative cognizing activities on infinite things (infinite mathematical things)” in present classical mathematics, many people have been unable to know whether the infinite relating mathematical things being treated are “potential infinite mathematical things” or “actual infinite number forms”, no one has been able to avoid the confusing of “potential infinite mathematical things” and “actual infinite mathematical things”, no one has been able to know whether or not treating “potential infinite number forms” or “actual infinite number forms” with the same way or different ways. What is more, many people actually don’t know or even deny the being of “potential infinite mathematical things” and “actual infinite mathematical things”--------the “quantitative cognizing defects on infinite relating mathematical things (infinite relating number forms)”.
Since “infinite” concept came into our science, the “infinite” related concepts and theories such as “potential infinite”, “actual infinite”, “countable infinite”, “uncountable infinite”, “infinity”, “infinitesimal”, “infinite set”, “variables” were introduced; still, some other mathematicians (such as G. Kantor and A. Robinson) have tried to develop some different “infinite” theories specially (only) for set theory or analysis …. The question of “What is potential infinite and actual infinite?” has been analyzed, discussed and debated and this situation is sure to be “endless” in present classical science theory frame--------our science history strongly proved!
Our studies prove that when facing and treating the “infinite related beings” in present cluttered, unsystematic classical “infinite” theory system, we are unavoidable to meet following two unexplainable arguments: (1) what on earth are “infinite”, “potential infinite”, “actual infinite”, “higher infinite”, “lower infinite”, “the ‘infinite’ of more infinite”, …? Can we really have many different definitions for “infinite”? Are different definitions for “infinite” the same mathematical things in our science? Why? (2) What kind of “infinite related number forms” should we have to demonstrate and cognize so many different “infinites”? Can we use just one kind of “infinite number form” forthem (several “infinite related number forms” in Harmonic Series Paradox is a typical example)? Why? Cardinality, continuum hypothesis and non-standard analysis theories help nothing here.
Our science history since Zeno’s time tells us clearly: there are serious fundamental defects in present infinite related classical science theory system-------both in philosophy and mathematics. Our science history since Zeno’s time also proved that not matter how we have tried, all the paradoxes and troubles produced by present infinite related classical science theory system are impossible to be solved (unsolvable) inside this very system itself.
For some small defects, the diminutive mendings are very much ok; but for the serious fundamental defects, those diminutive mendings do not only of no help but produce more troubles------errors plus other errors. So, the challenge is: to be or not to be staying in the foundation of present infinite related classical science theory frame.
The following proof (given by Oresme in about 1360), very elementary but important, can be found in many current higher mathematical books written in all kinds of languages.
1+1/2 +1/3+1/4+...+1/n +... (1)
=1+1/2 +(1/3+1/4 )+(1/5+1/6+1/7+1/8)+... (2)
>1+ 1/2 +( 1/4+1/4 )+(1/8+1/8+1/8+1/8)+... (3)
=1+ 1/2 + 1/2 + 1/2 + 1/2 + ...------>infinity (4)
Each operation in this proof is really unassailable within present science theory system. But, it is right with present modern limit theory and technology applied in this proof that we meet a “strict mathematical proven” modern version of Ancient Zeno’s Achilles--Turtle Race Paradox[1-3]: the “brackets-placing rule" decided by limit theory in this proof corresponds to Achilles in Zeno’s Paradox and the infinite items in Harmonic Series corresponds to those steps of the tortoise in the Paradox. So, not matter how fast Achilles can run and how long the distance Achilles has run in “Achilles--Turtle Race”, there would be infinite Turtle steps awaiting for Achilles to chase and endless distance for him to cover, so it is of cause impossible for Achilles to catch up with the Turtle; while in this acknowledged modern divergent proof of Harmonic Series, not matter how big the number will be gained by the “brackets-placing rule" (such as Un’ >100000000000000000) and how many items in Harmonic Series are consumed in the number getting process by the “brackets-placing rule", there will still be infinite Un--->0 items in Harmonic Series awaiting for the “brackets-placing rule" to produce infinite items each bigger than any positive constants, so people can really produce infinite items each bigger than 1/2, or 100, or 1000000, or 100000000000000000,… from Harmonic Series and change Harmonic Series into an infinite series with items each bigger than any positive constants (such as 100000000000000000), “strictly proving” that Harmonic Series is divergent. In so doing, the conclusion of “infinite numbers each bigger than any positive constants” can be produced from the Un--->0 items in Harmonic Series by brackets-placing rule and Harmonic Series is divergent" has been confirmed as a truth and a unimpeachable basic theory in our science (mathematics) while “the statement of Achilles will never catch up with the Turtle in the race” in Ancient Zeno’s Achilles--Turtle Race Paradox has been confirmed as a “strict mathematical proven” truth and a unimpeachable theorem------it would be Great Zeno’s Theorem but not Suspended Zeno’s Paradox! ?
If the decision between two choices is to be made, but neither choice is preferred over the other, because there is perfect symmetry between the two, then no information separates the choices and the only decision that can be made is a random one.
What kind of imperative can force such a decision, and does such a decision resist the imperative?
I. On Fri Oct 5 23:35:03 EDT 2007 Finnur Larusson wrote
I confirm that Larusson "proof" under corrections can be formalized in ZFC.
II. On Sun Oct 7 14:12:37 EDT 2007 Timothy Y. Chow wrote:
"In order to deduce "ZFC is inconsistent" from "ZFC |- ~con(ZFC)" one needs
something more than the consistency of ZFC, e.g., that ZFC has an
omega-model (i.e., a model in which the integers are the standard
integers).
To put it another way, why should we "believe" a statement just because
there's a ZFC-proof of it? It's clear that if ZFC is inconsistent, then
we *won't* "believe" ZFC-proofs. What's slightly more subtle is that the
mere consistency of ZFC isn't quite enough to get us to believe
arithmetical theorems of ZFC; we must also believe that these arithmetical
theorems are asserting something about the standard naturals. It is
"conceivable" that ZFC might be consistent but that the only models it has
are those in which the integers are nonstandard, in which case we might
not "believe" an arithmetical statement such as "ZFC is inconsistent" even
if there is a ZFC-proof of it.
So you need to replace your initial statement that "we assume throughout
that ZFC is consistent" to "we assume throughout that ZFC has an
omega-model"; then you should see that your "paradox" dissipates.".
J.Foukzon.Remark1. Let Mst be an omega-model of ZFC and let ZFC[Mst] be a ZFC with a quantifiers bounded on model Mst. Then easy to see that Larusson "paradox" valid inside ZFC[Mst]
III. On Wed Oct 10 14:12:46 EDT 2007 Richard Heck wrote:
Or, more directly, what you need is reflection for ZFC: Bew_{ZFC}(A) -->
A. And that of course is not available in ZFC, by L"ob's theorem.
J.Foukzon.Remark2 However such reflection is .available in ZFC[Mst] by standard interpretation of Bew_{ZFC}(A) in omega-model Mst
When we want to find out the gap between upper bound and lower bound, we can use Gap to see how much upper bound deviates from lower bound.
What is exactly the meaning of deviate in here?
In pure mathematics there is no absolute truth [Stabler]; we invent rules then see what they prove or see what is consistent with them. So in physics, what kind of truths are we looking for? Are we looking for absolute truths in physics?
(Note that the premise of my question immediately contradicts itself -- saying there is not absolute truth is an absolute statement. Maybe it is not a mathematical statement, I'm not sure. Apologies for this mess.)
Ref: Edward Russell Stabler, An introduction to mathematical thought, Addison-Wesley Publishing Company Inc., Reading Massachusetts USA, 1948.
I trying to extend some results that hod true for reflexive spaces, and I'm wondering whether this assumption could be too strong in a general LCS.
As we may know, in the standard approach to the $q$-calculus there are two types of $q$-exponential functions $e_{q}=\sum_{n=0}^{\infty}\frac{z^{n}}{\left[ n\right]_{q}!}$ and $E_{q}(z)=\sum_{n=0}^{\infty}\frac{q^{\frac{1}{2}n\left(n-1\right) }z^{n}}{\left[ n\right]_{q}!}$. Based on these $q$-exponential functions, also, some new functions are defined whose most of their properties are similar to the exponential function in calculus. Despite having interesting properties similar to the exponential function in calculus, for none of them the above property holds. So, can we define a new $q$-exponential function with similar characteristics to the exponential function in calculus which satisfies the aforesaid property?Any help is appreciated.
People believe that 0 (zero) has played different rolls as a kind of special form of number (substantiality) in our science, such as:
(1) A kind of reference ------- generator, middle, neutral, beginner, origin, placeholder, marker,…,
(2) Absolutely non-existent ------- without numerical value meaning, the negation of being, objectively nothingness,
(3) Relatively non-existent ------- with numerical value meaning, subjectively nothingness, the approximate nothingness, the result of infinitesimal limit , ….
Lacking the systematic cognition to 0 (zero), no one in the world now can answer following question scientifically and this is the very reason for many “infinite related paradox families” in present classical infinite related mathematics:
Are “dx--->0 infinitesimal” in Calculus and “Un--->0 infinitesimal” in Harmonic Series the same things? If ”yes”, why we have totally different operations on them? If ”no”, what are the differences and how to treat them differently and why?
We focus on the “deep structural relationship” between “nonstandard one” and “standard one”. Let’s exam following facts:
1, as “monad of infinitesimals” has much to do with analysis; nonstandard analysis is much more a way of thinking about analysis, as a different analysis------simpler than standard one.
2, CONSERVATIVE is the nature and a must for Nonstandard Analysis or Nonstandard Mathematics, it is called a conservative extension of the standard one.
3, because of the “deep structural CONSERVATIVE”, the “provable” equivalence are guaranteed.
If there are “no defects” in the “standard one”, the “CONSERVATIVE guaranteed nonstandard” work would be really meaningful.
Now the problem is “nonstandard one” inherits all the fundamental defects disclosed by “infinite related paradoxes” from “standard one” since Zeno’s time 2500years ago------guaranteed by the “deep structural CONSERVATIVE” .
Theoretically and operationally, “nonstandard one” is exactly the same as those of “standard one” with suspended infinite related defects in nature.Simpler or not weights nothing here.
A transitive action of the automorphism group means that for any two nonzero elements there is an automorphism taking one element to another.
We're working on finite modules over finite rings.
The following statement is well known in distributive lattices (D, 0, 1, ., +):
(1) Let F be a filter disjoint from an ideal I. Then there exists a prime filter F’ extending F and disjoint from I.
Usually the proof of (1) follows by an application of Zorn Lemma. But then the proof yields a stronger version:
(2) Let F be a filter disjoint from an ideal I. Then there exists a prime filter F’ extending F and disjoint from I and for every x not in F’ there exists y in F’ such that x.y is in I.
I am interested if (1) and (2) are equivalent (or not ) in ZF. Any references?
“Infinite”, “potential infinite”, “actual infinite”, “potential infinitesimal”, “actual infinitesimal”, “potential infinite-big”, “actual infinite-big”, “Infinite related numbers”,…; What are they?
Some people insist we can we have only one definition of “infinite” in science but others argue that we can have many definitions of “infinite” with different natures in science (at least two: “potential infinite” and “actual infinite”), what can we do?
In literature, for real vectors, the relationship between higher order cumulants and moments can be found.
How does this relationship look for complex vectors?
For example, how does E[y*1 y*2 y*3 y4 y5 y6] relate to all the higher order cumulants? Is there a general formula for this?
Nowadays, the consistency of theories is not demanded and in alternative we search for relative consistency. In the future things may change. In particular, in the answer 70 and more easily in answer 76, it was proved that set theory is consistent as a result of a relative consistency. There were published several datasets proving the consistency of set theory. In the last times it was publshed a paper in a journal, without success, since there is some inertia concerning the acceptance of the consistency of NFU set theory. It can be said that NFU set theory is consistent as the result of a relative consistency: since Peano arithmetic is consistent than NFU is consistent too. By a similar argument it can be prooved that set theory is consistent too: since NFU set theory is consistent then set theory is consistent. Thus, set theory is consistent, and since the related proof can be turned finite then we also prooved the Hilbert's Program, that was refered in many books on proof theory. There is an extension of set theory, the MK set theory, which is a joint foundation of set theory and category theory, two well known foundations of mathematics. Once again a paper by myself with title "Conssitency of Set Theory" was rejected without a valid reason. This agrees with an answer given by me 26 days ago. With set theory consistent we can replace the use of models to prove the independence of axioms (as did by Goedel and Cohen) by deduction in set theory.
(A) Let X\neq\emptyset and \tau\in P(x), where P(X) is the power set of X. Then if we have that:
i) X,\emptyset\in\tau
ii) i\in I, A_{i}\in\tau \Rightarrow \cup_{i\in I}{A_{i}} \in \Tau
iii) j\in J, J finite, A_{j}\in \tau \Rightarrow \cap_{j\in J}{A_{j}}\in \tau
We say that the collection \tau is a Topology in X and that (X,\tau) is a topological space.
(B) The minimum requirement for every algebraic structure is closure under the defined binary operation.
What is the essential difference between Algebra & Topology?
Both of them are guided by the concept of closure.
So, why have we defined two branches that are almost of the same philosophy?
In the attached paper it is a rather speculative position. I would like to have specialist opinion of the situations of today's position.
How can we introduce the idea of infinity to students? Its properties, relationship with zero etc.?
If we have a first order language for Real Numbers (this is exactly the language of rings), the using Dedekind cuts we can form a formula defining a real number. This is a syntactical determination of a real number. Some mathematicians stop here! They do not like models and interpretations. But if we try to find models, and interpret the formula, then the are infinitely many non-isomorphic models!
First let us take Boolean-valued models V^(B). Takeuti in “Two Applications of Logic to Mathematics” have prove that in a Boolean-valued model V^(B), if B is a complete Boolean Algebra, of projections in a Hilbert space, then real numbers are just self adjoint operators and if B is a probability algebra then the reals are “random variables”. In addition we have nonstandard models of reals that contain infinitesimals, and finally the are also tops theoretic models of real numbers.
Thus the real question is: How we can comprehend this infinitude of models of real numbers? We should note that completeness axiom is a second order axiom and thus if we want to stay in first order logic, we have to accept that there are infinitely many non-isomorphic models.
Are Automated theorem provers really useful to help students to know and understand how to proof? Is there any research about it?
Dear Colleagues, I am working on a new theory named Prelogic or Crealogic that uses the idea of sets and their creation for explaining the mechanisms that assemble intelligence and logic.
More interestingly, using Prelogic as the possible foundation of Set Theory, the Russell Paradox is no longer a contradiction or inconsistency of math and logic, but rather a clear illustration of the set forming dependency of an intellect.
Additionally, this so called Prelogic, can explain the reason why neurons connect and communicate the way they do to create intelligence.
Noteworthy, we have assembled a computational prototype of the above, that also uses sets for representing Language with mathematical-like proficiency, and soon, we hope to begin assembling the representations for Images.
For example, in language, Prelogic claims why is it that when we say "the red car and the yellow submarine," the car is red and not yellow, while the submarine becomes conceptually yellow but not red. What happens is that the meaningless entity representing "car" forms a SET with the meaningless entity representing "red." A similar thing happens to yellow with submarine, and therefore the reason why the submarine is not red nor the car is yellow. In this fashion Grammar becomes nothing more than binding rules (reason why grammatical elements are not freely exchanged) which only purpose is to acknowledge complex concepts (which concepts/words from SETS with other concepts/words)
So in the case of biological intelligence, neurons (meaningless cells representing concepts) form SETS which are identifiable only when communications travel along their connections (dendrites). Noteworthy, neurons can not draw areas around them like in Set Theory to create sets. The only way they can do this is by connecting with each other and identifying this connections by communicating.
Thanks
Dedekind cuts establish the set of real numbers. But like the well ordering theorem, the comparison of two arbitrary real numbers and linear ordering or law of trichotomy defies intuition. Of course rational numbers can be compared. Mostly epsilons can be replaced by 1/2^n etc.It is interesting to note that Dieuodenne's treatise on Analysis always uses rational numbers in the above form rather than using arbitrary Epsllons. Actually real numbers only give a universal set.
We need a few irrational numbers like Pi, e, Gamma and square roots or n'th roots of natural numbers. So my question has a lot practical relevance . The foundations of analysis depends upon set of real numbers but not the actual transcendental real numbers save those specified in earlier sentence. Also one can use Continued fractions to express Pi , sq root of 2 etc. Will some researcher throw light on this? It is interesting to note that that Dieuodenne actually uses only rational numbers in his proofs? What was his philosophy?