Lukasz T. Stepien

By semantic problem I mean this. If the hypothesis is that p ⇒ q, and also p, then by modus ponens it is a logical consequence that q. Now, if it is otherwise established that ¬ q (i.e., as some matter of fact), then the hypotheses are semantically invalid. There is nothing wrong with the deduction as a logical matter, the problem is with it as a matter of fact (or of contradiction with respect to other hypotheses).

I notice that, in the chapter on "A problem with validity," in David Sanford's book, "If P, then Q" the examples are all of that nature when one distinguishes the deduction from the contradictory semantics of the hypotheses. That one can hide contradictions in the construction of "plausible" hypotheses is not news, including when one is being shifty in the use of natural language.

Geng Ouyang in my browser, I cannot read "(such as T={x|x📷x})". This might be a ResearchGate formatting problem. Also, I am not clear what the question is. It seems that you are proposing that these paradoxes all have the same underlying origin. Is that the case?

Lukasz T. Stepien 10 hours ago Lukasz T. Stepien added a reply

>My comment is such that the classical logic suffers from the paradoxes of material implication

These paradoxes arises from unrestricted modus ponens rule and nothin more

>>>>The unavoidable fatal defect of “’potential infinite--actual infinite’ confusion” in present classical infinite idea inevitably leads to the unceasingly production of “paradox events” (different in forms but same in nature).

One of the most pervasive occurances of the paradoxes related to infinity, is Cantor's famous Diagonal Method (DM). While I have not found DM to be defined as such in mathematical circles, apparently Theoretical Computer Sciences (TCS) sees DM as the literal definition of cardinality (i.e. set theoretic size). Given that DM is the definition of transfinite size, it is less than inconvenient that DM can't be used to measure size in computational applications such as P vs NP. The problem is also directly related to the issue of "Potential vs. Actual" infinity.

It has been well understood that the Diagonal method can not resolve P vs NP due to what is known as the "relativization barrier". Based on my inquiries, the conventional wisdom is that there is nothing wrong with DM, it just doesn't do everything CST would like!

Computability theory is necessarily in a finitist domain. However the DM, the very definition of cardinality, embodies the "actual infinity". DM is not only suspect, but can be proven to be the culprit. So it is also no coincidence that the Continuum Hypothesis(CM) which is inextricably linked to DM is also problematic for the same reasons.

So yes, these issues with infinity are pervasive in the standard results of the axiomatic set theory and is a fundamental barrier to resolving the infamous P vs NP. Whether the standard results of multiple infinities are logically correct under some stipulated modus ponens , the fact remains that the current theories of multiple infinities is a major impediment to the furtherance of CST and computational complexity.

Lukasz T. Stepien

"In [57] one can read that Lewis told that from his very first contact with the logic of “Principia Mathematica”, he had been bothered by the paradoxes of material implication. As Whitehead and Russell have it written, a true proposition is implied by arbitrary (true or false) proposition, while a false proposition implies arbitrary (true or false) proposition"

It is commonplace to hear this asserted about material implication. But it's origin is not in a correct reading of Principia Mathematica.

I am looking at Principia Mathematica (1910 edition), volume I, p.7 where Whitehead and Russell explain their choice of material implication. They do not fall for the reading that a false proposition can imply anything. They are clear about what it is to assert that p -> q is true and it says nothing about q if ~p also holds. The difficulty seems to be one of shifting around in natural language use of "implies" and taking Whitehead and Russell as getting it wrong.

Whitehead and Russell clearly build the notion of the implication operation without any prior meaning to "implies" and are making a technical simplification in saying "p implies q" is to be (in Principia Mathematica at least) taken exactly the same as saying ~p v q holds and nothing more. The rule for modus ponens is p, ~p v q |- q and that is all. All you can get from ~p, ~p v q is |- ~p as a valid deduction (well, or simply replay either one of the hypotheses). The rest strikes me as a language game, but the symbolic logic of it is perfectly clear (and probably a good thing that symbolic logic was arrived at).

Thank you dear Mr. Lukasz T. Stepien, Mr. Dennis Hamilton, Mr. Jaykov Foukzon and Mr. Jim Moore!

I just come back from a business trip, very sorry for my late reply.

1) Studies to the history of infinite related science proved that people have constructed since antiquity the concepts of “potential infinite” and “actual infinite” and affirm the important roles these two concepts have plaid in present infinite theory system and there will be no present classical infinite theory system without the concepts of “potential infinite” and “actual infinite” --------the whole present infinite related science theory system is under the restriction of the concepts of “potential infinite” and “actual infinite”. But for thousands of years, people have never been offering scientific definitions to these two important basic concepts, never been understanding the different essences between them and what on earth the roles they play during the cognizing process of “infinite things”.

2) To Mr. Dennis Hamilton, it is true that I am proposing these paradoxes all have the same underlying origin.

3) To Mr. Lukasz T. Stepien, Mr. Dennis Hamilton and Mr. Jaykov Foukzon, I agree with you “that the classical logic suffers from the paradoxes of material implication and these paradoxes arises from unrestricted modus ponens rule”---------due to the unavoidable fatal defect of “’potential infinite--actual infinite’ confusion”.

4) To Mr. Jim Moore, my idea on Cantor's famous Diagonal Method (DM):

The promiscuous situation of “potential infinite--actual infinite” confusion in present classical set theory and its “one to one correspondence” tool not only triggers many insoluble problems and paradoxes in our science, but also produces some meditative and absurd phenomena: “Russell’s proof and its result by self-contradiction of T={x|x📷x}” produced Russell’s Paradox triggering the alarming Third Mathematical Crisis while the identical “Cantor’s proof and its result by self-contradiction of T={x|x📷x}” produced important fundamental laws for set theory and became the particularized and refulgent “Cantor’s Law of Uncountable Real Number Set” and “Cantor’s Power Set Theorem of 📷” (in potential infinite thinking, an infinite set should have a “self-contradiction subset” while in actual infinite thinking, an infinite set should not have a “self-contradiction subset”). So, in many situations of present classical set theory, people have been being always in a typical state of “not know what's what” when cognizing some infinite related mathematical things: you can take them either as “unknown ‘potential infinite’ things” or “unknown ‘actual infinite’ things”, even more, you can take them as mathematical things with both “potential infinite and actual infinite” characters and decide when they are “potential infinite” character and when they are “actual infinite” as you like. Let us see two typical members in Russell’s Paradox Family” (1) famous Hilbert's Paradox of the Grand Hotel-------- telling people with potential infinite thinking that the Grand Hotel is always full and there never have any vacancies then telling people with actual infinite thinking that the Grand Hotel always have vacancies and never being full; (2) famous Cantor’s proof of Uncountable Real Number Set-------- telling people with potential infinite thinking that all the elements in Real Number Set and Natural Number Set are all infinite “abstract mathematical things with same characteristic, same existence condition, same pattern of manifestation, same relationship, same quantitative meaning and prove they should be bijective during “one to one correspondence” then telling people with actual infinite thinking that all the elements in Real Number Set and Natural Number Set are with very clear characteristic differences, very clear existence condition differences, very clear pattern of manifestation differences, very clear relationship differences as well as very clear quantitative meaning differences and prove they are impossible bijective during “one to one correspondence” because infinite elements with characteristic of real number, existence condition of real number, pattern manifestation of real number, relationship of real number, quantitative meaning of real number are left. This artful and arbitrary “potential infinite--actual infinite” confusing working way produced a very woefully result: creating a huge and suspending “Infinite Paradox Family” (a 2500-year old huge black cloud of “Infinite Paradox Family” enveloping over the sky of human science). Because of lacking scientific foundation theoretically in present classical set theory and its quantitative cognizing tool of “one to one correspondence”, people have been unable to realize a harsh fact that all the efforts we have been striving to dislodge Russell’s Paradox and the Third Mathematical Crisis are in fact wholly refusing “Law of Uncountable Real Number Set” and “Power Set Theorem of 📷” in Cantor’s Cardinal number theory and continuum theory, because they are different version of Russell’s Paradox in the same paradox family.

Mr. Geng Ouyang

I am a relative newcomer to set theory. This is not a normal part of engineering training. However to the extent that I understand , I think we are in agreement.

In paragraph "2.5 Cantor’s Diagonal Method and the Continuum Hypothesis" of my recent Preprint ComputableUniversePneNP Preprint2019Mar5

I present a simple example that demonstrates this issue of potential vs actual infinity and how the DM generates a contradiction in a very concrete example of two transfinite sets. In fact I describe the paragraph as demonstrating the following points.

• A counter example to the Cantor’s Continuum Hypothesis.
• Bijections between sets alone cannot determine cardinal order.
• The Diagonal Method can only determine that cardinalities are equal or unequal.
• The existence of a Cardinality Paradox for standard Well-Ordered sets.

Regards

Jim

Thank you dear Mr. Jim Moore!

In fact, there are many troubles in “Bijections” due to potential--actual infinite confusion. Just see another example:

It is believed that there are the bijection relationships between Natural Number Set and Rational Number Set, but following simple story tells us that Rational Number Set has far more elements than that of Natural Number Set:

The elements of a tiny portion of rational numbers from Infinite Rational Number Set (the sub set : 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 (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 and they are not bijective at all.

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 two infinite sets and it can be nothing to do with the term of “proper subset”.

Can we have many different bijection proofs with different one-to-one corresponding results between two infinite sets? If we can, what conclusion should people choose in front of two opposite results, why?”

Regards

Geng

If we put those infinite related paradoxes in present mathematics together and study them, we discover that these paradoxes all have the same underlying origin because of the same fundamental defect of “potential--actual infinite confusion” at least since Zeno’s time 2500 years ago.

In fact, infinitesimal has to be a “number of non number” or “zero of non zero”. The present classical infinite related number system basing on “potential infinite --actual infinite” have produced infinite related paradox families since Zeno’s time at least from 2500 years agro.

So many infinite related paradox family members in our mathematics and philosophy have been strongly reminding us that something must be done.

Geng Ouyang I think citing Zeno's paradox is a bit foolish. Simple physical mechanics does not suffer from it, nor does any of our observed experiences. Given that d = v * t is observed to work. Traveling at constant rate of velocity, v, for duration t, will traverse distance d. The empirical matter is rather solid.

What Zeno did not account for is that the rate of movement does not change. We imagine it, the way Zeno expressed it, as some kind of impulse travel, and that is simply not the case where this theory of motion applies. There is no intermediate stopping, and Zeno had no account for inertia. One can, of course, diminish t and travel a shorter d in that interval, yet the rate of motion is always there. Extending to complete Newtonian Mechanics does not compel a metaphysical commitment to infinitesimals, just because the mathematical expression of the theory relies on continuity and somewhat informal arguments about limits.

I think actual versus potential infinity is not at issue here. Consideration that there are uncountable points along the dimension from 0 to distance d fails to defeat the velocity, with traversal in that time. The distance is measurable and so it goes.

Of course Newtonian mechanics is more elaborate than that, and the calculus is needed to deal mathematically with acceleration and other non-uniform motions. In that respect, the Zeno thought experiment simply fails to account for observable reality. It's not a paradox. Simply mistaken.

Thank you dear Mr. Dennis Hamilton!

According to my studies, Zeno's great creation of “Achilles--Tortoise paradox” is not only a simple mistake but is a huge paradox family and its typical modern family member is the newly discovered Harmonic Series Paradox.

Let’s see following divergent proof of Harmonic Series which can still 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）

Because of not knowing what infinitesimals are (number of non-number), the unavoidable practical problem has been troubling us ever since is how many items in infinite decreasing Harmonic Series can be added up by “brackets-placing rule" to produce infinite numbers each bigger than 1/2?

This kind of “infinite relating paradox” tells us:

1, in Harmonic Series, we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite items 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 the constant of 100000000000000000000000);

2, the different “brackets-placing rule" to get 1/2 or 1 or 100 or 100000 or 1000000000000000 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.

By the way, Robinson's non-standard analysis can do nothing to solve any of those suspended “infinite-infinitesimals paradoxes” either.

Sincerely yours,

Geng

Geng Ouyang Umm, I think that a careful mathematician would simply suggest that the harmonic series is divergent. It is monotonically increasing even though the rate of increase declines from addition of term to term, 1/n, 1/(n+1), etc.. The symbolic use of an "infinity" symbol is more an indication of boundlessness there, since there is no such rational number. It has been difficult to stamp out the "infinity" informality even though it can be misleading.

Of course, all divergent series of this nature are equivalently boundless upward and all are denumerable (or denumerably infinite, in a manner of speaking), even though 1+i/2 grows far more (i.e., exponentially) rapidly with increasing I, using simple arguments about denumerability.

I think the leap to comparison of the speed of runners is awkward. That does not seem any more apt than comparing {..., i, i+1, ...} with {..., 2i, 2(I+1), ... }. Eventually, the first always achieves a value in the second. We are sort of comparing with { ..., 2^i, 2^(I+1),...} in the reasoning about the Harmonic series, in number of terms clumped together, but the same seems to apply. We are counting terms; physical analogies with runners seems very contrived even though we speak of rates of growth of series terms.

One can, of course, be unhappy about mathematical denumerability. I don't think mathematics will give it up, not even for the portion that is essential to computer science.

Geng Ouyang OH OH. I made a mistake. You cannot say ">" in (3). That is an incorrect inference about the two series. That is the case about the compared groups of sums after the first two terms, but the complete sums are divergent in both cases and comparing them is meaningless.

Try defining the summation that is obtained by subtracting (3) from (2). Is it convergent?

Thank you dear Mr. Dennis Hamilton!

(1) How many numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from items in Harmonic Series by “brackets-placing rule" can we gain?

(2) You can find ">" in (3) from above proof in many current higher mathematical books written in all kinds of languages but not I say ">" in (3).

Sincerely yours,

Geng

Geng Ouyang I have been thinking about the ">" in (3) some more. It is more precise to observe that the corresponding groupings, in a progression of lengths 2^i, are such that the 2^i-length group sum in (2) is never less than the corresponding 2^i-length grouped sum of 1/2 in (3). So one can infer that if (3) is divergent, (2) must be also. And (3) is obviously divergent, as is apparent from the simplified form (4).

The "shorthand" use of ">" between (2) and (3) is an unfortunate way of expressing that observation, no matter how often it might be used in books. I see that it is used at https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Divergence in Wikipedia, also. However, the "More precisely, ..." statement there avoids the leaping between the ">" that applies to finite cases of the sums and an argument about fancied totalities of divergent series.

Cauchy gives us a valuable generalization that one can apply directly: https://en.wikipedia.org/wiki/Cauchy_condensation_test to the Harmonic Series and more.

I think we can forgive the medieval mathematician, if the leap from ">" over the definite ordering of rationals to that of an unending summation was actually made at that time and not introduced later.

Thank you dear Mr. Dennis Hamilton !

Is Harmonic Series Paradox a typical modern family member of Zeno's great creation of “Achilles--Tortoise paradox”, why yes and why not?

Sincerely yours,

Geng

Geng Ouyang asks "Is Harmonic Series Paradox a typical modern family member of Zeno's great creation of 'Achilles--Tortoise paradox', why yes and why not?"

No and Yes

Zeno's thought experiment proposes a condition on motion that is inaccurate in terms of how travel happens.

With regard to series that are extended without limit, it would appear that the terms reflecting distance covered in a particular amount have Achilles always farther along than the Tortoise.

However, when they travel without ever stopping, one cannot suppose anything. If we fix the position of Achilles at any definite moment, that is at a place that the Tortoise will always reach eventually. Because neither of them ever stops, we must conclude that they each cover all of the same territory, whatever that is. The notion of one catching the other becomes, in some sense, meaningless when both run for eternity.

It is the second notion that arises in mathematics concerning divergent and convergent series, apart from applications to physics. Consider these divergent series:

{0, 1, 2, 3, 4, 5, 6, 7, 8, ..., n, ... }

{0, 2, 4, 6, 8, 10, 12, 14, 16, ..., 2n, ... }

{0, 1, 1, 2, 3, 5, 8, 13, 21, ... Fib(n), ... }

{1, 2, 4, 8, 16, 32, 64, 128, 256, ... 2n, ...}

{1, 1, 2, 6, 24, 120, 720, 5040, 40320, ... n!, ...}

These are denumerable, as demonstrated by correspondence to the enumeration of the integers in the first series. At some point in the series (n = 8 here), the

corresponding terms achieve persistent rankings that do not change thereafter. The order of listing the sequences reflects that ranking.

Ultimately, the first sequence will transit all of the values that are found in any of the other sequences. By virtue of the 1-to-1 correspondences (the columns), all of these sequences are denumerable (the first, trivially so), and there is no sense in which there are more or fewer items in any of the series.

All of those examples are divergent, and none of them has an entry that cannot be exceeded by every other of them at some point in their unlimited enumeration.

Thank you dear Mr. Dennis Hamilton !

How many numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from items in Harmonic Series by “brackets-placing rule" can we gain ?

Sincerely yours,

Geng

Geng Ouyang asks "How many numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from items in Harmonic Series by “brackets-placing rule" can we gain ?"

The bracketing in the technique (3) sums groups of 2^n terms of value 1/2^(n+1) each for a group sum of 1/2, so 1 + n/2 is the diverging sum of all the groups up to the nth. The sequence of values of (3) for increasing n is denumerable and divergent.

The corresponding bracketing in (2) sums the 2^n terms of value 1/(2^n-i) where i runs from 1 to 2^n. Each such group sum is never less than 1/2. So the sum of these grouped-sums, +1, is also divergent. The sequence of values of (2) for increasing n, taken this way, is denumerable and divergent.

The sequences enumerated by succession of n achieve innumerable values beyond any specified finite integer that we choose. There is no end to it.

Also, striking out a fixed number of initial groupings from both (2) and (3), by setting the terms in the groups to 0, leaves (2') and (3') that remain denumerable and divergent over n, so any fixed value still have innumerable values beyond it in either series.

Thank you dear Mr. Dennis Hamilton !

So, are you trying to prove following two facts?

1, in Harmonic Series, we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite items 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 the constant of 100000000000000000000000);

2, the “brackets-placing rule" to get 1/2 or 1 or 100 or 100000 or 10000000000 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.

Sincerely yours,

Geng

Geng Ouyang asks "So you are trying to prove the following two facts."

No. I am not treating the bracket-placing rule as any source of facts about nature or an analogy about runners that is unrelated to motion in Newtonian mechanics.

Geng Ouyang remarks "2, the 'brackets-placing rule' to get 1/2 or 1 or 100 or 100000 or 10000000000 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."

The problem is about "never" when the runners continue for eternity and both paces are divergent and denumerable in the mathematical formulation of the two series. This is a case of "never say never."

The proof of divergence of the summed harmonic series also depends on a change of coordinates. The bracket grouping technique that arrives at multiples of 1/2 involves step-down functions and exponentially-longer groups of smaller values. It is important to notice that there is a kind of change-of-coordinates involved in the series that has steps that sum to 1+m/2.

Geng Ouyang remarks "1, in Harmonic Series, we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite items in Harmonic Series by 'brackets-placing rule' to change an infinitely decreasing Harmonic Series ... "

That is not the change that is made. It is the sum of the terms of the Harmonic Series that is compared with the sum of a bracket-placing case. The bracket-placing procedure demonstrates that the progressive sums of decreasing harmonic values is a divergent series.

MORE INTERESTING QUESTION

I think a better question, in reference to the demonstration of divergence, is this.

At the point where the bracket groupings of stepped-down values sum to 1+m/2, m non-negative, what is the farthest term in the stepped-down sequence included in that sum?

a. The value at the term stopped on is 1/2^m in both summations.

b. In the summation of the Harmonic series, this is the only term with that value and it is at position 2^m in that series.

c. We know, at this point, that the sum of decreasing Harmonic series values out to that position is not less than 1+m/2

d. We can determine, now, the number of terms that are needed in the sum of decreasing harmonic-series terms to exceed any specified numerical value.

For any desired Harmonic sum S, derive m = 2(S-1). The accumulation of 2^m terms of the decreasing Harmonic series will not be less than S.

Thank you dear Mr. Dennis Hamilton !

1, how do you know that the Harmonic Series is a divergent series without the 'brackets-placing rule to get 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite items in Harmonic Series '?

2, if there is no "never" as you say, do you really can prove that the runner (even a runner with the speed of modern jet plane) in the race will catch up with the tortoise some time, when?

Sincerely yours,

Geng

1. I am not objecting to the bracketing technique. It works fine as a demonstration that the sum of the Harmonic Series terms is divergent.

I even demonstrated how to determine, based on the bracketing technique, when the sum of the Harmonic Series will not be less than a given number, S.

2. With regard to "never say never" I was being a smart-ass. I mainly don't treat these comparisons of divergent series as being meaningful at all about natural events, as when taken as analogs of a theory of motion of the sort Zeno offered in his thought problem. I've said enough about denumerable sets already.

I feel we are in an endless cycle here and I have nothing more to add.

Thank you for the challenge, though. I taught myself a lot in responding.

Thank you very much dear Mr. Dennis Hamilton !

I also benefit a lot in responding.

Sincerely yours,

Geng

The defects in the foundations of present science system inevitably lead to the existence of “self-refutation mechanism”. Our 2500 years unsuccessful “infinite relating paradoxes” fighting history (the most typical example is Zeno's “Achilles--Tortoise Paradox”) has proved clearly that in present science theory system, it is impossible to solve those self-refutation mechanism relating paradox family members produced by the fundamental defect of theory system itself (unsolvable). The underlying origin of self-refutation mechanism should be studied and the theory of “abstract concept and the carriers of abstract concept” should be studied and developed. This is the only way to eradicate thoroughly those paradox families produced by “self-refutation mechanism” in different fields of human science (including philosophy and mathematics).

Thank you very much for your ideas dear Mr. Sydney Ernest Grimm !

I am now working hard investigating the causes behind infinite relating paradoxes (infinite paradox families syndrome) in analysis and set theory.

Sincerely yours,

Geng