Room for Dejan Govc and Wolfgang Mueckenheim - 2013-01-21

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11:01 AM

Good morning Dejan. I just have seen your thoughts and will not hesitate to answer because this is a very interesting discussion. (Unfortunately the community is not so liberal and has meanwhile deleted all my corresponding questions.)

The Binary Tree contains all possible FISs (which are the generally used tool to approximate real numbers). If we agree that the infinite path of a real number does not contain anything than its FISs, then we see that also Cantor's argument only concerns FISs and proves that their countable set is uncountable. Something like the liar-paradox or Hessenbers's …

11:27 AM

@Dejan: Here I try to answer your question: "So the diagonal differs from every element in the list, but is still in the list?"

Consider the sequence 0.9, 0.99, 0.999, ...
Every term differs by some finite amount from the limit 0.999... = 1. Nevertheless there is no finite amount by which all terms differ from 1. And if there is no finite difference, how could we distinguish? In particular every set of FISs can be replaced by one single FIS. By induction we can show that never more than one FIS is necessary to contain all FIS of a given set of FISs (of the same sequence). Of course inducti…

5 hours later…

4:33 PM

Hi, Wolfgang. I will probably be quite busy this week, but I will try to answer you as soon as I manage to think the issues through. I agree that the discussion is quite interesting.

5:24 PM

I am very glas to hear that. Because, although my students (who are not mathematicians) in general understand very easily, mathematicians only seldom reach that point of understanding. They are so convinced by Cantor's simple (and in fact so much convincing proof) that they refuse to understand. Please do not hurry! Take the time to think it over. Have a nice week! Regards, WM

1 hour later…

6:27 PM

@BelsaZarkin: Just a question (so I can understand your thought more easily): if I understand you correctly, you more or less completely reject set theory. How do you define real numbers then? (I'm interested in the complete definition, from scratch.)

6:47 PM

@Dejan, I do not "reject" set theory, but I don't see a way to save the transfinite parts. I use the finite parts, as you can see here

amazon.de/Mathematik-Physik-10-2012-ersten-Semester/dp/…

The real numbers are on p.35ff.

7:30 PM

@BelsaZarkin: Unfortunately, I do not seem to have access to page 35. Do you accept the construction of real numbers by Dedekind cuts? (It does not use anything transfinite, I believe.) Or do you use some other construction/definition?

Sorry, I did not check that. In fact page 34 is the last one. Here we have a field defined by the usual axioms. The reals are defined by Dedekind-cuts. Note that every cut needs a finite Definition. (But I do not eleaborate on that topic in the book. In the preface I qoute Robinson, the inventor of non-standard analysis:

(i) Infinite totalities do not exist in any proper sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally /meaningless/.
(ii) Nevertheless, we should continue the business of Mathematics "as usual," i.e., we should act as /if/ infinite totalities really existed.

)

Nice. What axioms of set theory do you accept as valid?

(More to the point: which axioms of ZFC do you reject?)

7:46 PM

In principle all nine axioms (or eight, depending on the scheme). But the axiom of infinity that Zermelo has been taking from an idea of Dedekind (who credits Bolzano to have invented it first) can only be interpreted as establishing potential infinity, namely as Dedekind and Bolzano explained it. many "first elements", but we can only denote countable many.

Then however, it would not be required at all because induction is sufficient to establish potential infinity. The power set axiom then has also potential character only. And AC is natural and correct, because no uncountable sets exist. For uncountable sets we would have to choose uncountably

many "first elements", but we can only denote countable many.

But can we denote countably many at once?

*choose

So, how exactly would you state the axiom of infinity, then?

If we have only potential infinity, then "countable" is a meaningless notion. There cannot be a bijection with the complete set N because it does not exist. (I use these words in discussion with set theorists, when I assume transfinity correct.)

The axiom of infinity would be stated as in current set theory. But the interpretation of "there is a set" does not concern a complete set, but does only establish: I you have n, then you can get n+1 or 2n or n^n and so on. If you have actual infinity, then you arrive at the paradox of Tristram Shandy that every n can be taken off N and nothing remains. That is in contradiction with analysis.

So, your theory of sets would be syntactically completely the same as ZFC? (And differ only in the semantics?) Doesn't it then prove the same things?

It is only the difference in interpretation. Can we exhaust N or not? Cantor in his writings has very often and very clearly discussed this point, namely the difference between actual and potential infinity. He claimed that actual infinity is required for his theory. But set theorists of today are rather reluctant to accept that there is a difference at all. I have met many who are proud not to know the difference!

Sorry, I have to leave for an hour. But if you continue, I will respond in the evening. Best regards, WM

8:10 PM

Ok. Actually, what bothers me, is this. We prove statements by providing proofs of them. A proof is a finite sequence of formulas where every formula is either an axiom, or follows from the preceding formulas by a rule of inference. This means a proof is a completely syntactical object and thus *independent* of any interpretation. Therefore your theory will prove precisely the same things.

More precisely, any proof that is valid in classical set theory, will have a corresponding statement in your set theory. This means that there is *no* mathematical difference between the theories. If tha…

1 hour later…

9:31 PM

Hi, Dejan. First of all, I don't claim to have any theory of mine, neither mathematical nor set theoretical. I do mathematics as Euler and Weierstraß did - although not as brillant, of course.
Second: Potential infinity never exhaust an infinite set. Hence, if set theory proves what Fraenkel told and what I described in
planetmath.org/?op=getobj&from=objects&id=12607
then there is an influence of the interpretatation. Probably you don't recognize it because you naturally have been taught to assume that actual infinity is the only possible way to deal with infinity.

9:41 PM

In the formalism of ZFC a bijection of N and Q can be proved. This fact shows that actual infinity is the basis. In classical analysis such a bijection cannot be proved because there only convergent sequences are meaningful and have limits. The sequence of cardinal numbers n in N and indices i of q_i in Q will never be exhausted. Both are simply infinite.

Lt me give an example: In classical analysis we will never sum all terms of geometric series (1/2^n). We are always aeare of the fact that we have only the limit in that sense that addition of arbitrarily many further terms will never exceed 1 (or 2, depending, where we start), but that every real number below that limit can be surpassed.

By enumerating the rationals, however, it is not assumed that there always remain infinitely many not enumerated, how far we ever go, but it is assumed that we can enumerate all rationals and that beyond this "all", there appear new horizonts. This is so fundamental a difference that it must be possible to formalize both positions in different ways.

You say "Potential infinity never exhaust an infinite set." Do you believe that there exists an actual infinite set?

(Just that our minds cannot grasp it?)

And, classically, the sum of the series is defined to be the limit. In this way, we sum it completely. Nothing remains.

10:00 PM

I do not believe that there exists an infinite set in that sense that all elements are "there" (where ever that might be), because then it would not be infinite. I believe that there exists the set , or better say sequence or collection, of natural numbers, such that every desired natural number can be taken and that beyond it a number of natural numbers can be taken, that surpasses every threshold.

By the way, the paradox of Tristram Shandy is a very interesting one. A few years ago when I first heard of it, I thought a lot about it. I was hoping - quite naively - it might yield a contradiction in ZFC. (Which would be a very interesting thing to find, in case it exists.) I could find no such contradiction.

The sum of a series is defined to be its limit. But this is a sloppy terminology - although the easiest way for beginners. I use it in my book too. The sum is complete only in the sense of actual infinity. Potential infinity of naturals (and therefore indices of the terms of the series) is always finite though not limited by a finite threshold.

Do you believe 0.999... differs from 1?

Tristram Shandy might not yield a contradiction in ZFC, but it contradicts the idea that ZFC is the basis of analysis, because the analytical limit is different. By the way, it is a matter of definition what is a contradiction. Well-ordering requires identification and naming of the ordered elements. Since we can only name countably many, the well-ordering of an uncountable set is already a contradiction in my opinion.

I think the situation might be worse, really.

We can only name finitely many things.

10:09 PM

No 0.999... is by definition equal to 1. 1 is here the limit of the series. By addition of 9*10^-n you can never surpass 1 but every real number that is smaller than 1. This is a typical limit of a series.

And I also believe that the integers might not even be potentially infinite. The usual argument is "we have the number n, now we increase it by 1." This might not be always possible: at some point you will be out of paper, so it will become impossible to write down such a number.

We can, if we assume actual infinity, name countaby many things, for instance all words 0, 1, 00, 01, 10, 11, 000, ... That is a big difference to uncountably many names. Zermelo and Fraenkel were of the opinion that R really can be well-ordered. They write only that "up to now" nobody has been able to accomplish it.

Zermelo's first paper has the headline "Proof that every set can be well-ordered". This is wrong as Feferman has shown. ZFC avoids contradictions only by re-interpreting as much as is necessary.

Yes, I agree. But on what basis do you assume potential infinity? In what sense is it real? Isn't the universe a finite place?

If you consider reality, then you get much more restricted. I have avoided to mention that because it has nothing to do with the difference between countably many names and uncountably many. If you are interested you may like to read my articles

"Physical Constraints of Numbers", arXiv, math.GM/0505649, 050530, 061211. http://arxiv.org/pdf/math.GM/0505649
http://arxiv.org/ftp/math/papers/0505/0505649.pdf
P4PP W. Mückenheim: "The Infinite in Sciences and Arts", arXiv:0709.4102v1 [math.GM], 070926 http://arxiv.org/abs/0709.4102

Thanks.

I think the main problem is assuming that there even is an infinity to start with.

If we assume that, the step from countable to uncountable is really a very small step.

10:18 PM

But that would make mathematics very difficult. There is no number with a Kolmogorov-complexity of 10^80 because we no more atoms in the universe. This is the same problem as with a pocket calculator that refuses numbers with more than 10 digits. But that would carry us too far from the original topic.

I think this is precisely the reason why most mathematicians stick with set theory.

It make things easy.

You are right. The step from countable to uncountable is small. But it requires the identification of "every" and "all". And it requires the solution of the $1000 qustion of the tree.

Right.

For the same reason Robinson has proposed to work as if infinite set would exist. And I join him.

So, was I right when I said that your position is mostly philosophical in nature?

10:22 PM

Yes, my position is mostly philosophical. But as long as the $1000 problem exists, I believe it is the only tenable position. Are you not disturbed by that problem? Or do you have a way to circumvent it?

I am not disturbed by the problem, because I am used to sets and their peculiarities. The problem is quite orthogonal to my current way of thinking, so it in a sense can't really disturb me at this moment. I must also admit that I still do not feel that I really understand the problem. I have a couple of further questions, though.

If I understand you correctly, by "all x such that P(x)" you mean the set {x| P(x)}.

So when you say something holds for all x such that P(x), you mean that it hold for the set {x| P(x)}.

Is this indeed what you mean?

In fact this is the main reason why your problem does not disturb me. I never felt the need to use the word "all" (as distinct from "every") in mathematics and I still don't understand what you mean by it. As this word does not appear in my mathematical thoughts, problems related to it cannot disturb me. I wish I could understand it, though.

10:38 PM

If your question concerns infinite sets, I do not believe in their existence and hence not in the possibility andsensibility of "all". But if I take the position of current set theory, by {x| P(x)} I assume the same as you, namely the set of all elements for which P(x) holds.

Now, Dejan, I have to leave. I you remain, please write your further thoughts. Tomorrow I will have plenty of time to answer. I wish you a good night.

Good night to you, too. I'll keep the questions in my subconsciousness and if any meaningful thoughts arise, I'll be sure to ask. Thanks for the discussion.

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Jan '1321

$Mathematics$ Room for Dejan Govc and Wolfgang Muec…

Continuation of a discussion about cardinality from math.SE

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