I with a question of Foundations

[matt grime]

Unrestricted comprehension is that thing, naive set theory, that you mentioned in Cantor's Paradox (a paradox that still remains if there are no infinities. Please state where in the Paradox, and Cantor's Theorem anyone uses the word infinite?).

Naively a set is any collection of objects with a rule for beloning to the set (eg, the set of all sets), that an object either satisfies (is in the set) or doesn't satisfy (is not in the set).

 

[master_coda]

I agree completely. I am only advocating the exploration of my Practicl Number approach. I don't propose at all discarding any prior work, including Cantor's. It's just that Cantor's approach has had a century of work by now, and to be fair and balanced, I think Kronecker's approach should be explored in as much depth.

That sounds much more reasonable. What you were saying before sounded a lot like "infinite sets are absurd; they should be abandoned". That sounds very bad, since a lot of cranks tend to advocate discarding all previous work in favour of their own.

Yes. I'm sure that's because I do have a very poor understanding of the history of set theory. I am here to learn.

I have heard the term 'naive set theory' many times but I never knew what it meant. Can you explain it to me at my level of understanding of the subject? I would be most grateful.

Naive set theory was the idea that set theory could just be based on unrestricted comprehension. So all you had to do to specify a set was to specify a predicate that all its elements had to satisfy. However this led to a number of paradoxes, so modern set theories were developed that used a more restrictive set of axioms.

This makes the modern theories weaker, since there are sets which can be specified with unrestricted comprehension that cannot be specified in the newer set theories, but this also means that the newer theories do not contain the sets that create paradoxes. Of course, it took a lot of work to create these newer theories, since mathematicians didn't just say "you can't do this, and you can't do that, etc.". Instead they tried to replace unrestricted comprehension with weaker operations that gave us as much power as was possible without introducing paradoxes.

Interesting! Thank you. (BTW, what are NF and NBG?) I know that ZFC defines infinite sets. Does ZF also? I am particularly reassured that the presence of antinomies is sufficient cause to discard a theory. I'll take your word that that is true.

NBG is con Neumann-Bernays-Gödel set theory and NF is New Foundations set theory. They're other set theories that were created which allow different operations than ZFC.

And yes, a theory with antinomies would be discarded; incompleteness is acceptable, inconsistency is not. Of course, there would still be some study of the inconsistent theory as people tried to construct a weaker (but consistent) replacement for it. Obviously when developing modern set theory it's important to understand the problems that naive set theory had.

I agree. I don't demand completeness. I only insist on consistency and rigor.

Well, I'm sure most mathematicians will be very happy to see someone demand rigor; when rigor is abandoned, it's usually by people with more "pracitical" concerns than all that tedious logic.

Please explain how, or give me an example of a calculation using a value of pi with an infinite number of digits.

Thank you again for your help, Master_Coda. You are giving me what I have been looking for.

Paul

Well, if I ask maple to tell me what the value of arcsin(1) is, it tells me pi/2. So it seems like a computer doesn't have any problem working with an exact value of pi.

Of course, if I ask it to produce a decimal representation of pi it won't be able to give me all the digits; but what's so special about decimal representation?

 

[Hurkyl]

I propose that a rigorous, consistent theory be developed assuming a largest integer.

The problem is, as stated, it just can't work, because every integer has a successor.

Alternatively, because whenever m and n are both integers, so is m+n. As a side note, + doesn't even have to be a function: even m+n is merely taken as a particular string of symbols, it sufficies for the addition operation, as long as the appropriate axioms are satisfied.

(and, since 1 is positive, m+1 > m)



You -might- be able to do what you propose by making some unusual modifications to the concept of theory, but what you really want to do, probably, is to develop some new theory about something that is merely analogous to the integers.

And, from what little I know, and from what I have learned here, I suspect that the principle of transfinite induction is the source of my distrust and concern.

The principle of transfinite induction is sound, its proof is similar to that of ordinary induction.

Definition: < is said to be a well-ordering of a set S iff < is an ordering of S that has the following property: whenever T is a nonempty subset of S, T has a smallest element (according to <).

Thm: Let < be a well-ordering of S. Let e be the smallest element of S. Let P be a logical proposition. Suppose also that:

Whenever P(x) is true for all x < y, then P(y) is true.

Then, we can conclude that P(s) is true for all s in S.

Proof: Let T be the subset of S of all elements such that P(s) is false. Formally: T := { s in S | P(s) is false }. Suppose T is a nonempty set. Because < is a well ordering, T has a smallest element; call it t. Because of the way T and t are defined, P(x) is true for all x < t. By the hypothesis, this means P(t) is true, which is a contradiction. T must be an empty set, so P(s) is true for all s in S.


So, your problem is probably with the well-ordering principle which asserts that every set can be well-ordered. (which is logically equivalent to the axiom of choice)


An important thing to know is that the axiom of choice has been proven independent of the ZF axioms. This means that if ZF-C (the ZF axioms, plus the denial of the axiom of choice) is a consistent theory, then so is ZFC (the ZF axioms, plus the axiom of choice), and vice versa.



Oh, it may help to see examples of well-orderderings.

The natural numbers are well-ordered by their usual ordering.
The integers are not well-ordered by their usual ordering. {..., -3, -2, -1, -0} doesn't have a smallest element.

The integers are well ordered by an alternate ordering, <<, that puts the integers in this order:
0, -1, 1, -2, 2, -3, 3, ...
More precisely:
a << b iff |a| < |b| or (|a| = |b| and a < 0 < b)

Also, I'd like to note that, while I stated the well-ordering principle in terms of sets, it is more general (but requires a lot more care to use properly).

 

[Paul Martin]

You Guys are Great!

To Hurkyl, Master_Coda, Matt Grime, StatusX, CrankFan, and Robert Ihnot,

Thank you all for your responses to my original post. You have been most helpful and you have given me what I was looking for. I was looking for someone who understands mathematics who was willing to take a look at my proposal and tell me what they think. Some of you obviously understand the mathematics involved and you have helped me understand things a lot better than I did before. That has given me a delightful bonus. Thank you.

I apologize for not having the time it would take to respond to each of you individually and close off all the threads I have left dangling, but, you probably won't mind being spared another load of words from me anyway.

Before I leave, though, I will try to close off a few of those dangling threads.

You have introduced me to the concept of "unrestricted comprehension". Thank you for that. Thank you, Hurkyl, for your explanation that "unrestricted comprehension ... is, saying "Let S be the set of all blah" where "blah" could be anything." After thinking about this, I think the difficulty I am having comes down to the meaning of the word 'all'. I think my basic objection is that we have not rigorously defined the term 'all', so when we use it in a construct like "the set of all blah", I think we introduce a fatal fuzziness into the dialog. As Matt Grime mentioned, it comes down to the question of the meaning of existence itself. It seems to me that "unrestricted comprehension" is too general and leads to paradoxes.

Master_Coda, it looks like you agree with me here. You seemed to imply that unrestricted comprehension implies paradoxes when you said, "Naive set theory was the idea that set theory could just be based on unrestricted comprehension. So all you had to do to specify a set was to specify a predicate that all its elements had to satisfy. However this led to a number of paradoxes".
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Hurkyl, I asked you for a rigorous definition of the ellipsis, and you answered me with English vernacular. You said, " The ellipsis was, I thought, obvious, and I even spelled it out elsewhere in my posts: to get the next number, you simply apply the successor operation S(x) := x U {x}."

That's OK because I'd rather discuss this in English vernacular anyway. As I see it, the concept symbolized by the ellipsis is anything but obvious. I want to look carefully at what you said.

First of all, by "get the next [Cantor natural] number" I take it that you mean this is how the next Cantor natural number is defined. Or, in other words, how the next Cantor natural number is produced. The idea is that until we have done this, we don't "have" that next number. After we have done this, we have an additional number which is then included in a growing set of Cantor natural numbers.

Next, the definition of the successor operation is clear enough. It says that you form a new set which is the union of the original set and the set containing the original set as its only member.

So to produce the next Cantor natural number, in your words, "you simply apply the successor operation", and in my words, "you form a new set". The end result, which you left unsaid, is that somehow you get the infinite set of Cantor natural numbers. I want to slow down and figure out exactly how we get that result by performing that operation.

If, by "you" you mean either you (Hurkyl), or me (Paul) or any other human, or sentient entity, or even any machine, then you and I mean the same thing when we use the word 'you'. If you meant something else by the word 'you', then I don't understand what you said.

So we have "you" applying the successor operation (your words), or "you" forming a new set (my words) in order to get the next successor. This application, or forming, of course produces exactly one successor. After, and only after, that application or forming has been done, do we have a new successor. And only after we have the new successor can we produce its successor. What we have is an iterative process consisting of a well-ordered sequence of steps.

The entire infinite set of Cantor natural numbers does not emerge from this process at once. What emerges at each successive step is a set of Cantor natural numbers that is one larger than what we had before. At each such step along the way, the set is finite.

So, in order to produce an infinite set, "you" would have to perform this application or formation an infinite number of times. With the definition of 'you' that I thought we agreed on, this is not possible. It is not possible practically, or physically, or (I claim) even in principle. If you think otherwise, I would like to have you explain it to me, preferably again in English vernacular.
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(Matt Grime):"Do you understand the difference between the axioms of a set theory and a model of the axioms?" - I think I do, but I doubt that I have a grasp of the implications of that difference.

(Matt Grime):"How many times, Paul, must we tell you that it is the Axiom of infinity in ZF that means that in any MODEL of ZF there is an infinite set; it is nothing to do with the C in ZFC." - No more. I think I've heard it enough. Thanks for your patience and perseverance.
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(Master_Coda): "Well, if I ask maple to tell me what the value of arcsin(1) is, it tells me pi/2. So it seems like a computer doesn't have any problem working with an exact value of pi." - Who or what is "maple"? And what is the form of the answer it gives you, the symbol "pi/2"? It seems quite a stretch to call a symbol an "exact value". Let me define the symbol 'D' to mean the diameter of the earth. To what degree of precision can I say that the "exact value" of D represents the diameter of earth? I can use my symbol, 'D', in describing relationships among the Earth's diameter and other quantities, but it does nothing for me in terms of providing an "exact value" of the Earth's actual diameter. The same is true of the symbol, 'pi'. You can use the symbol in all sorts of manipulations of statements relating to the number which 'pi' represents, but it does nothing for you in terms of producing an "exact value".

(Master_Coda): "If we can produce a finite representation of pi on a piece of paper, then we can certanly do it on a computer." - If the "finite representation" is merely a symbol, then see above. If the finite representation is an "exact value" then I doubt that you can produce it on a piece of paper or in a computer or in any other way.
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(Hurkyl): infinite sets have proven themselves useful for solving "real problems". - Such as?
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(Matt Grime):"These [Cantor's Theorem and Cantor's Paradox] remain "true" even if all sets are finite. Where do you see the word infinite appear in them, or any reference made to the axiom of choice?" - I don't see the word or the reference explicitly, but the concept seems to creep in along with such words as "all". I tried to convince myself that both the theorem and the paradox remain "true" for a simple example of finite sets, and I failed. I couldn't even imagine an example that satisfies the conditions, "Let S be the set of all sets, and T the set of the subsets of S". If you define the set S first, then T turns out to be the power set of S which is bigger than S. If you define T first, e.g. T = {a}, then by definition, a is the one and only subset of S. But S is the set of all sets so it must include both the set a and the set {a}. So we have {a} as a subset of S and it is not in T. I think the word 'all' is the bugaboo.
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(CrankFan): "With your finite set of naturals, how do we construct Q and then more importantly R!? We can do real analysis with a complete ordered field in your "foundation", right?" - I think it would be a mistake to use the terms 'naturals', 'integers', 'rationals', 'reals', and 'complex' to describe the sets of numbers in my proposed system. What I propose would be finite subsets of the naturals, the integers, and the rationals. There would be no reals in my system and the complex numbers would all have rational components. Thus, there would also be no irrationals or transcendentals. Constructing the subset of the rationals would simply be by forming the quotients of integers. Real Analysis would not be complete by any means. There would be no real numbers. On the other hand, I think that all theorems of analysis that are involved in calculation algorithms would be provable in my system and that calculation results would be identical to any results producible by algorithms of real analysis. Obviously people disagree with me on this point. But I don't think the jury should deliberate until someone has done some work developing the theory. As far as I know, that has not been done. Discrete systems have been investigated, but I think that they have all been closed systems with respect to addition and multiplication. My proposed system is not.

BTW, I'm puzzled by your choice of name. For a fan you seem pretty hostile.
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(StatusX): Well, would you deny fantasy writers the oppurtunity to write stories about unicorns because they haven't been observed? - Not at all. But I would object to anthropologists including unicorns (as distinct from the concept of unicorns) in their theories.

(StatusX): the fact is infinity CAN be studied. so why shouldn't it be? i must be missing something. - What you are missing is the fact that I never said, or proposed, or implied that infinity should not be studied. It should be studied. And it has been studied for over a hundred years now. I am only proposing that in addition to the vast amount of study of infinite systems, we should also study systems in which the magnitude of numbers has a finite limit.

This has been fun and enlightening, and once again I thank you all for your help. My time is limited so I can't devote as much time to this conversation as I'd like. I'll still check back from time to time to see if you have any last words for me. If you want to know where I spend my time, check out my web site. Thanks again.

Paul

 

[StatusX]

(StatusX): the fact is infinity CAN be studied. so why shouldn't it be? i must be missing something. - What you are missing is the fact that I never said, or proposed, or implied that infinity should not be studied. It should be studied. And it has been studied for over a hundred years now. I am only proposing that in addition to the vast amount of study of infinite systems, we should also study systems in which the magnitude of numbers has a finite limit.

I know you want to wrap this up, but I just have a couple quick remarks. first of all, "studied" was a bad word choice. I am basically saying that, infinity is there, even in your system, which is to say there are numbers bigger than Q, so how can you ignore them?

now i haven't read this whole topic, and i know very little about set theory, but I was wondering how you intend to define real numbers in your system. every number is rational, namely, a multiple of 1/Q, right? there is no number that will give exactly 2 when squared, but there is a number whose square is closer to 2 than any other, so will you being calling that the square root of 2? also you mentioned the dirac delta function in your essay, and it seems that in this system, integrals are calculated using the trapezoidal approximation. this would leave unwieldy expressions for integrals of even simple functions, like x^2. would you use the "approximation" of infinitessimal widths in these integrals, or stay true to your hypothesis? if you do use this approximation, youll be admitting that infinity exists. if you don't, even simple physical problems like simple harmonic oscillators will be reserved for numerical calculation, and i doubt physicists would tolerate that.

that being said, i think your onto something, because the universe does seem to be both quantized and finite in extent. however, i think your going about it the wrong way, applying the restriction when you define the number system where as you should be applying it in the physical theory. and your implication that pi does not really have an infinite number of digits, even outside your system (if I am reading that right), is simply incorrect. it is easy to prove pi is irrational, and therefore, in theory (theres nothing wrong with this), you will never reach the end of its decimal expansion (and the burden of proof would be on you to show that there is an end).

 

[matt grime]

You really haven't grasped the concepts in Cantor's proof that a power set of a set is strictly larger than the set (assuming the ordinary definitions).
That remains true even if the set is finite. If card(X)=n, then card(P(X))=2^n>n

The second bit, the paradox: it is you who said that simply removing tha axiom of choice and thus making all sets (in a model, my words) finite (which isn't true) would remove all contradictions and paradoxes. Well, that isn't true if we adopt the idea that there is a set of all sets (and all are finite), then the same "paradox" still holds.

In ZF there is no set of all sets because the class of all sets is too large to be a set. That doesn't stop in some formalist sense there being a class of all sets, it just doesn't follow the rules, and can't be called a set.


The axioms are a set of rules. A model of those rules is, roughly speaking a collection of objects that satisfy the rules. The 'elements' in the collection are what we call sets. In some models the Real numbers form a set, in another model they don't (I don't really understand the exact subtlety of this, it is called skolem's paradox).

So, the axioms do not stop anything "existing" nor for that matter do they create anything necessarily.

For instance I can create the axioms for a group:

(G,*) is a group if G is a set and * is a binary operation on G satisfying

identity, closure, associativity and inverse.

An example is the integers under addition (it is a model for the axioms)

If I now insist that the set is finite it stops Z being a model for this 'restricted' group axioms system, but doesn't mean that Z suddenly winks out of "existence", does it?