I do not think that your example with dates (2015-12-10) is about infinity. I would say that it is lexicographic; see https://en.wikipedia.org/wiki/Lexicographical_order . But maybe I misunderstood your point.

If we want to talk about infinity in mathematics, I just know that there are two types of infinities: countable set (with the power of normal infinity of unbounded sequence) and uncountable set (all real numbers).

I do not think that your example with dates (2015-12-10) is about infinity. I would say that it is lexicographic; see https://en.wikipedia.org/wiki/Lexicographical_order . But maybe I misunderstood your point.

If we want to talk about infinity in mathematics, I just know that there are two types of infinities: countable set (with the power of normal infinity of unbounded sequence) and uncountable set (all real numbers).

Dear Mr. Yuri Yegorov, please see Cantor’s “first generating law”, “second generating law”, “third generating law”,…for "super numbers" super super numbers, super super super numbers, … as well as the “power set theory”-------endless (infinite elements), more endless (more infinite elements), more more endless (more more infinite elements), more more more endless (more more more infinite elements), more more more more more endless (more more more more infinite elements),….

Thank you.

 Best regards

Dear Geng, the problem is extremely nice, and i remember an observation made by my prof. of analytical geometry, about the train from X station.

"When the train arrive in the station X an infinite passengers come down from the train, another infinite passengers come up in the train and inside the train, remain an other infinite passengers."

I'm sorry, but your question make me remember his lighty memory.

Dear Mr. Ioan Sorin V Leoveanu， thank you.

Your professor’s story is exactly another version of that famous story of “Hilbert Hotel”: the Hotel which has infinite rooms is full with infinite tenants one night but it still has infinite empty rooms for the other infinite tenants-------what is the meaning of full and not-full (whole and portion).

The chaos and confusing of “infinite”, “potential infinite”, “actual infinite” in present classical infinite related science theory frame created a lot of contradictions and paradoxes in analysis and set theory challenging us human as a suspended “comprehensive syndrome”. 

And, Cantor applied the very same chaos and confusing idea of whole and portion, potential infinite and actual infinite to prove real set has more elements than natural set------the number of elements in real set is more infinite than that in natural set.

Yours sincerely, 

Geng

Dear Geng, this time the question is not clear at all... And: such structures as you speak about, that allow cuts, have not only cardinality but also order type. The order type is the equivalence class under order isomorphisms. Order isomorphisms are bimonotone (increasingly) bijections. So the order information is inherent here, the question is not only about cardinalities. So please reformulate the question in the appropiate language.

Dear Mr. Mihai Prunescu, so happy to meet you again and thanks for your mentioning.

You are right the story of 2015-12-1 "H" and another day 2015-12-10 "b" … is not mathematical enough. I just use "b, H, …" in stead of the marks of those “cuts” in mathematics. In fact, we can choose any “marking system with different features” in such “infinite thing cutting process”.

Yours sincerely, 

Geng

My dear Mr. downvoter, would you please be so kind as to give your frank view points on all of my posts you have downvoted in this thread? I am all ears to your voice and I will answer all of you doubts sincerely.

RG is a discussing platform for scientific researchers, but someone is unable to present his ideas and view points, hide somewhere, takes the advantage of anonymous and enjoy 'down-voting' as mischief games for fun, watch his victim’s RG score getting lower and lower by his anonymous 'down-voting' and build his happiness on others’ sadness?

What are you my dear Mr. downvoter?  If you are a real scientific researcher, I sincerely hope you can ask some questions rather than just play anonymous 'down-voting' game in this thread.

Geng, I am NOT the downvoter.

But the question is still very unclear. 

However I think that your question is well answered by the so called \eta_\alpha ordering types. See:

Erdös, Paul; Hajnal, András; Milner, E.C.

Title: Partition relations for \eta\alpha and for \aleph\alpha-saturated models. (In English)

Source: Theory Sets Topology, Collection Papers Honour Felix Hausdorff, 95-108 (1972)

"""In fact, results more general than these are established. These are applied to show, in particular, that results corresponding to I and II hold if the set R\alpha of order type \eta\alpha is replaced by certain \aleph\alpha-saturated models. Other partition relations for \eta\alpha-sets have appeared in an earlier paper of P. Erdös, E. C. Milner and R. Rado [J. London math. Soc., II. Ser. 3, 193-204 (1971)]"""

Dear Mr. Mihai Prunescu, I know you are a very serious, sincere scientific researcher always presents your frank ideas in many threads. I am very lucky to have such researcher friend like you in my heart.

Thank you very much for your view points.

Yours sincerely, 

Geng

I'm not sure what you're after. As you say about Cantor, he is able to show higher infinities can be presented consistently as part of mathematics. But when you're interested in applying the math, not all of it seems to apply. Generally in physics, we admit there is no infinitely divisible and no uncountable but we "tolerate" both because of their ease in analysis. If we are doing integrals, we even use things like delta functions (as long as they're behind an integral) and even this has a consistent justification in generalized function theory.

 Dear Mr. James Langworthy ，thank you for your ideas from practical physics world.

1, I call those infinite related mathematics used in physics “typical applied infinite”.

2, In fact, the title of this thread can be changed into: Can we really have infinite, higher infinite, higher higher infinite, higher higher higher infinite, … in our mathematics?

Thank you again.

I am sorry; a very “Mr. downvoter” is waiting there in the dark hungrily and eagerly for the anonymous 'down-voting' to my post.

Yours sincerely, 

Geng

The confusion of universal “time—space” (objective, absolute “time—space”, belonging to theoretical philosophy) and knower’s “time—space” (human’s “time—space”, relative “time—space”, belonging to applied philosophy) results in the confusions of “potential infinite and actual infinite”, “whole and portion”, “infinite law and the carriers of infinite law”, “ontology of infinite and form of infinite”, … thus produces “higher infinite”, “super infinite”, “super infinite numbers”, “the infinite of more infinite”, (more infinite, more more infinite, more more more infinite,…) through following two ways:

1, the “cuts” marked with “different marking system of different features” in single infinite carrier cutting process to prove the existence of “more infinite, more more infinite, more more more infinite,…” (such as producing those super infinite numbers by the “generating laws”).

2, the whole and portion confusion on different infinite carriers by “mathematical magic” to prove the existence of “more infinite, more more infinite, more more more infinite,…” (such as the “power set theory”, real set is bigger than natural set------the number of elements in real set is more infinite than that in natural set).

In our daily life, we have “a story of Hilbert Hotel”: Hilbert Hotel has infinite rooms, but as time goes, the Hotel will have more infinite rooms by each “cut”------ infinite rooms, more infinite rooms, more more infinite rooms, more more more infinite rooms, more more more more infinite rooms, …

Of cause these are not the faults of any people, they are produced un-avoidable and naturally in present classical infinite related science theory system frame-------a “2500 years old huge black cloud of infinite related paradoxes over mathematics sky”.

If you have infinite set then set of all subset of given set will be infinity of higher order. You can move this way many times.

Dear Mr. V. A. Babaitsev ，thank you.

So, does it produces “higher infinite”, “super infinite”, “super infinite numbers”, “the infinite of more infinite”, (more infinite, more more infinite, more more more infinite,…)?

Yours sincerely, 

Geng

“…So, does it produces “higher infinite”, “super infinite”, “super infinite numbers,…”

- yea, and if this process is repeated “absolutely infinite number” of times, the absolutely infinite Set appears.

And this Set – the “Information” Set indeed exists, there is nothing besides this Set.

When the problem “actual/ potential infinity” should be forgotten…

More – see https://www.researchgate.net/publication/260930711_the_Information_as_Absolute  (or http://viXra.org/abs/1402.0173  ) .

Cheers

Article the Information as Absolute

The old and new suspended infinite related paradoxes strongly prove that it is the chaos “actual/ potential infinite” that produces all the defects in present classical infinite related science theory system and the very “2500 years old huge black cloud over mathematics sky”.

To the very Mr. downvoter who dare not to ask any questions in our scientific discussion but can only hide somewhere in the dark playing downvoting as mischief games for fun:

Men behave, GOD watches.

I think these paradoxes are poor for me.. So, I  have no  any comments. Sorry!

I'm interested in Inequalities.

Dear Mr. Miguel Ángel Montes , happy to hear from you again and thank you for your frank view points.

Followings are my frank ideas, very different from your points:

a) To me, Potential--Actual lnfinite must be a very practical scientific theory and skill in the field of mathematics (such the practical operation of divergent proof to Harmonic Series) and it should not belong to (nothing to do with) the field of "Mathematics Education".

b) it is logical to say A is bigger (more) than B if A has more identified contents than B------an infinite power set A has more elements than its original set B so it is logical to say infinite A is bigger (more infinite) than infinite B. ln fact, this is one of the very defects of Potential--Actual lnfinite in present classical infinite related science system disclosed by the infinite related paradox family.

c) I believe there are many infinite related paradoxes in our present classical infinite related science but you do not.

During my 40 years studies on the defects in present classical infinite related science system disclosed by the infinite related paradox family, I really read many papers and books to detect what have been done in our science history but not as you believe “paying no attention to what other people, other very important people in many fields have done”.

I am sorry; so much differences of above basic ideas make us always argue.

Yes, you are right “Do not pay too much attention to the downvoter”. I have endured so many kinds of sufferings because of my “Anti-Potential-Actual-lnfinite” ideas and work in pass 40 years, the anonymous mischief down-voting in RG is really nothing; I just think such anonymous “mischief games for fun” should not happened in RG. I never did and will never do such ‘anonymous RG down-voting' to anyone during scientific discussions because personally, I think it only create threaten, panic, disorder (it is a scientific discussion, why not give your frank view points to improve our work?) and I regard such anonymous behaves by adults in the field of scientific discussions as immorality, incivilization, maleducato and wicked. 

Men behave, GOD watches.

Thank you again.

Yours sincerely, 

Geng

Merry Christmas and Happy New Year!

Geng:

I will pull out of my toolkit two items - the concept of number and the representation of the concept of number.

There are numbers and there are representations of numbers.  In mathematics, it can be difficult to separate these.  In most other disciplines the representation rules and there is generally no attempt to separate the representation from the concept.

Cantor had difficulty separating these (and did try in a limited way) and ended up with categories of infinite sets.  As Miguel notes, there is a type of order to infinite sets, however this is not the same as the ordering of numbers.

Note that Cantor puts the Integers and Rationals in the same infinite category.  He does this by showing a method for putting the representation of Integers 1-1 with representations of Rationals.  He then goes on to present a proof-by-contradiction that the representation of Real numbers cannot be put 1-1 with the representation of Integers or Rationals.   Using concepts like 'aggregates', Cantor attempts to separate representation from concept.  Likewise Dedekind attempted to define the infinite without reference to numeric representations (an infinite set can be put 1-1 with a subset of itself and the concept of a Dedekind Cut).

The main problem is that matching things 1-1, when we are discussing numbers, generally ends up using our (current) representation of number to match 1-1.  If our representation of number changes, then this matching (or our ability to match) might also change.  If our representation of number is that of ratios, then we are really only able to directly represent Rational numbers.  Historically, 'transcendental' or 'irrational' did not refer to numbers that could be represented.  It is only with the introduction of the decimal numeric system that we can represent them and then perform a 1-1 matching using them.

We do not have a more powerful numeric representational system than the decimals today - even though we have different representations of Reals that involve powers of calculation beyond that of decimals (eg. logarithms).  By analogy, while the greeks could be said to be limited to ratios and thereby to Rational numbers, we are limited to positional numeric systems and thereby to Reals.  Just as the greeks could identify numbers they could not represent (eg. the famous Pythagorean proof that sqr(2) is not a rational number), we have been clever enough to represent numbers we cannot represent as a number and even calculate (in a limited fashion) with these representations (complex numbers and quaternions are examples - which always involve an unknown entity such as 'i', 'j', or 'k' and hence are not complete representations of numbers).

Note that Cantor's 1-1 mapping of Integers with Rationals uses the fractional representation of Rationals for its proof.  And his diagonal proof uses properties of the decimal representation of Real numbers for his proof-by contradiction.  So his proofs implicitly assume the properties of the representational system apply to the properties of the numbers being represented.  Representations are always limited in how completely they represent the underlying concepts, so Cantor does not set his theories on a solid foundation - what if more powerful numeric systems can be found?

To continue the greek analogy as limited by ratios and Rationals, it is quite possible that there exist representations of numbers that can match more numbers 1-1 than we currently can.  We might find that, using a non-positional representational system that could fully represent complex numbers, we could put Reals 1-1 with Integers.  Then we would have to re-think our ordering of infinite sets.  Maybe Integers are one order, Rationals another, Reals another order, and quaternions another order.  Given the appropriate representation for a number, all could be put 1-1 with the Integers.  Then we would only have one concept for infinity - just with (potentially infinitely) many representations of numbers.

Back to my toolkit: Mathematics is a dynamic discipline that tends to have major shifts measured in much larger time frames than other scientific disciplines (it was about 2500 years before 3-D 'Euclidean Space' was found to be a sub-set of larger spaces).  We might be due for such a major shift and it might just come as a new way to represent numbers - and fully represent numbers we cannot today.  This, in turn, could change many mathematical notions near and dear to us today - including our concept of infinity and infinite sets.

(See how many down-votes I can get for this post...)

Dear Mr. Donald G Palmer, thank you very much for your kind, frank and constructive answer, I really enjoy such scientific and fruitful discussion.

1, your question of ”what if more powerful numeric systems can be found?” is right to the point.

I had the same question 40 years ago when I faced following situation: “can we really produce infinite items each bigger than 1/2, or 100, or 1000000, or 10000000000,… from Harmonic Series and change Harmonic Series into an infinite series with items each bigger than any positive constants (such as 10000000000)”?

In fact, 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?

As you have analyzed in your post, if the classical infinite related numeric system is unable to solve those suspended problems of numbers and representations of numbers, why not develop one? Although “it could change many mathematical notions near and dear to us today - including our concept of infinity and infinite sets” as you put it in your post.

2, during my 40 years work with the idea of “anti-present classical infinite related science system”, I have experienced much more “down-votes” in my life; those “anonymous down-votes without frank ideas in scientific discussion” are really nothing to me------some people are unhappy some where, something really need to be done but it is pity that no one knows what to do. I still present my thanks to those down-voters in RG and regard them as my friends because these not only train my willpower but also pass me the information of “it is really difficult for them to offer frank ideas”.

My heartfelt thanks to you again dear Mr. Donald G Palmer!

Happy New Year!

Sincerely, 

Geng

 Dear colleagues, anyone knows how to solve following problems:

1, According to the definition, the original Even Number Set and Odd Number Set is each half of the Natural Number Set (otherwise it can not be called Even Number Set and Odd Number Set). People have been using two deformed Natural Number Set of (1x2, 2x2, 3x2,...,nx2,...) and (1x2+1, 2x2+1, 3x2+1,...,nx2+1,...)  and just used the components of even numbers and odd numbers (natural numbers, but not the whole thing of“2,4,6,8…2n” and “1,3,5,7,9…nx2+1”) to map natural numbers, of cause they map very well because it is N ----> N itself but not 2n ----> N and nx2+1----> N.

What are mapping to natural numbers: the even numbers and odd numbers or the components of even numbers and odd numbers?!

2,The elements of tiny portion of rational numbers from Rational Number Set (1, 1/2, 1/3, 1/4, 1/5, 1/6, …, 1/n …) map and use up (bijective) all the numbers in Natural Number Set (1, 2, 3, 4, 5, 6, …, n …); so infinite rational numbers (at least 2,3,4,5,6,…n,…) from Rational Number Set are left in the “one—to—one element mapping between Rational Number Set and Natural Number Set”------- Rational Number Set and Natural Number Set are not bijective at all.

What decides the same cardinality of two sets?

In present classical infinite science system, people have been paying too much attention to the forms but too less to the ontology at least since Zeon’s time 2500 years ago.

Regards,

Geng

Five of my published papers have been up loaded onto RG to answer such questions:

1，On the Quantitative Cognitions to “Infinite Things” (I)

https://www.researchgate.net/publication/295912318_On_the_Quantitative_Cognitions_to_Infinite_Things_I

2，On the Quantitative Cognitions to “Infinite Things” (II)

https://www.researchgate.net/publication/305537578_On_the_Quantitative_Cognitions_to_Infinite_Things_II

3，On the Quantitative Cognitions to “Infinite Things” (III)

https://www.researchgate.net/publication/313121403_On_the_Quantitative_Cognitions_to_Infinite_Things_III

4 On the Quantitative Cognitions to “Infinite Things” (IV)

https://www.researchgate.net/publication/319135528_On_the_Quantitative_Cognitions_to_Infinite_Things_IV

5 On the Quantitative Cognitions to “Infinite Things” (V)

https://www.researchgate.net/publication/323994921_On_the_Quantitative_Cognitions_to_Infinite_Things_V