I with a question of Foundations

[Paul Martin]

I would like anyone to comment on a proposal I have written (it's short and intuitive) for a system of "Practical Numbers". The idea is to deny the Axiom of Choice, thus eliminating, or obviating, the notion of infinity and accepting the consequences of a largest integer and a smallest interval. I think all useful theorems of analysis will still hold in this system. The proposal can be found at http://paulandellen.com/essays/essay089.htm

My thanks to you in advance.

Paul Martin

 

[matt grime]

The axiom of choice has nothing to do with whether or 'infinity exists' (an ill defined phrase, which we will take to mean there is a set whose cardinality is not finite, the existence of which in any model of ZF is guaranteed by the axiom of infinity, which would seem an 'infinitely' more obvious thing to deny if one didn't want an infinite set. Of course there is a difference between the axioms and a model of the axioms...)

Exactly what does 'deny' mean? Assume it is false? That still doesn't mean that the set of natural numbers has a largest element, as is rather trivial to see, even if one were to 'deny' the axiom of infinity in the sense of not include it in the axioms of ZF, that doesn't stop there being a model in which there is an infinite set. (ever heard of Skolem's Paradox? That's a beauty to frazzle your brain.)

None of the usual results in analysis will hold is my initial prediction since you've given no method to construct a model of the real numbers, what ever that might be, and of course in your system 1/n doesn't even tend to zero, so how it would be a useful place to do any analysis is a mystery.

Are you attempting to enter the annals of crankhood?

If I were you I'd go away and learn what the axiom of choice is before you start making false claims about it.

 

[robert Ihnot]

Matt grime seems pretty right on that. The Axiom of Choice has nothing to do with the set of all integers. If n is an integer, so in n+1, I don't see how you are going find any kind of upper bound there.

There have been serious mathematical studies which have denied the axiom of choice, indeed Cohen in 1963 showed that the Axiom of Choice is independent of Z-F set theory. The intuitionists have argued that all math must be constructed from the basics which is the set of integers.

As I was told in College years ago by a logic professor, nobody denies the existence of the set of all integers, after all, that is intuitive!

But, then again, maybe there is something in such an idea! I am tempted to start my own thread: Relativity and the counting process! https://www.physicsforums.com/showthread.php?p=354350#post354350

 

[Hurkyl]

Furthermore, even if you took as an axiom that no infinite set exists, it does not follow that there are only finitely many integers (and thus a largest): it merely means that the integers don't form a set.

 

[matt grime]

There are reasons for assuming the axiom of choice, and for not doing so. WIthout it not every vector space has a basis, but with it we have the Banach Tarski Paradox. If we have the axiom of constructablity too (I think that's what I mean) then every set can be well ordered, and in particular there is a way to construct a theoretical game for two players where both have 'probability' one of winning. Of course, this is just theoretical and of no direct implication to the real world since you couldn't actually make such a game.

 

[Paul Martin]

More specifics, please.

The axiom of choice has nothing to do with whether or 'infinity exists' (an ill defined phrase, which we will take to mean there is a set whose cardinality is not finite, the existence of which in any model of ZF is guaranteed by the axiom of infinity, which would seem an 'infinitely' more obvious thing to deny if one didn't want an infinite set. Of course there is a difference between the axioms and a model of the axioms...)

Exactly what does 'deny' mean? Assume it is false? That still doesn't mean that the set of natural numbers has a largest element, as is rather trivial to see, even if one were to 'deny' the axiom of infinity in the sense of not include it in the axioms of ZF, that doesn't stop there being a model in which there is an infinite set. (ever heard of Skolem's Paradox? That's a beauty to frazzle your brain.)

None of the usual results in analysis will hold is my initial prediction since you've given no method to construct a model of the real numbers, what ever that might be, and of course in your system 1/n doesn't even tend to zero, so how it would be a useful place to do any analysis is a mystery.

Are you attempting to enter the annals of crankhood?

If I were you I'd go away and learn what the axiom of choice is before you start making false claims about it.

Hi Matt,

Thanks for your response. I will cave in for now and accept your statement that the Axiom of Choice has nothing to do with infinity. The issue that is important to me is that there are no infinities in mathematics. If denying (by that I mean excluding it from the ZF axioms) the Axiom of Infinity will do that, then that is what I would propose.

I suspect that your prediction that "none of the usual results of analysis will hold" was made without having read my proposal. If you had read it, you would have seen that Real Numbers are replaced by Practical Numbers. For computational purposes, Practical Numbers are every bit as useful as Real Numbers.

In my system, 1/n would tend to zero, but in the sense of a new and different definition of the limit which I sketched out in the essay. With new, but roughly equivalent, definitions for the limit, continuity, and convergence, I think all useful theorems (i.e. those applicable to real phenomena) of analysis could be proved in this system.

If you still don't think so after reading my proposal, please be specific as to why.

Paul Martin

 

[Paul Martin]

Matt grime seems pretty right on that. The Axiom of Choice has nothing to do with the set of all integers. If n is an integer, so in n+1, I don't see how you are going find any kind of upper bound there.

There have been serious mathematical studies which have denied the axiom of choice, indeed Cohen in 1963 showed that the Axiom of Choice is independent of Z-F set theory. The intuitionists have argued that all math must be constructed from the basics which is the set of integers.

As I was told in College years ago by a logic professor, nobody denies the existence of the set of all integers, after all, that is intuitive!

But, then again, maybe there is something in such an idea! I am tempted to start my own thread: Relativity and the counting process! https://www.physicsforums.com/showthread.php?p=354350#post354350

Hi Robert,

You said, "If n is an integer, so in n+1, I don't see how you are going find any kind of upper bound there."

Here's how I see it: To even state the premise, "If n is an integer", you must first have the integers. Of course, if you do have the customary infinite set of integers, then I agree that n+1 is also an integer. But I want to back up and ask how do we get the integers in the first place?

You mentioned the set of integers as the "basics" of all mathematics. With my apologies to your professor, I am one person who denies the existence of the set of all integers. I say it is not intuitive.

In my view, mathematics deals only with concepts which are either taken as primitive, or they are defined, or they are derived. (And, of course, the definitions and the derivations must obey strict rules.)

It would be fair to take the infinite set of integers as primitive, and then you would have the familiar mathematics of real numbers. But, starting with Peano and his "successors", The integers are typically defined instead of taken as primitive I claim that you cannot legitimately define an infinite set of integers (or of anything else for that matter).

In your statement that I quoted, you implied that you can define the next integer after n simply by forming the sum n+1. But how is the sum itself defined? Addition is defined as an operation which is a function taking each pair of integers to another integer. In other words, you have to have the integers first before you can define the operation of addition.

But, no matter how you do the definition of the integers, in order to get an infinite set you have to assume that there is some automatic process that works blindingly fast and which has been working away for an exceedingly long time in order to produce it. Even if there conceptually were such a process, how could we be sure it has completed by now and actually produced the entire infinite set of integers?

There is nothing intuitive about this picture at all. It is preposterous. Yet, mathematicians accept it without much hesitation.

What seems more reasonable to me is to accept only definitions that have been explicitly made and written, either by human hands or by human made machines. So I'll readily accept the current approximate definition of Pi out to a trillion decimal places because it has actually been produced by a man-made machine. But I don't accept the existence of an approximation of Pi out to 10^trillion decimal digits. When it's finally computed out that far, I will.

By far the most common definitions of numbers today are found in the computer chips which embody algorithms to define and manipulate them. In each case, there is a practical limit on the size of the largest useful number. The usefulness of the larger of these numbers follows a pattern like the umbra and penumbra of a shadow as I described in my proposal (Thank you for reading and citing it, Robert.). Nothing prevents us from defining numbers as big as we need. But in all cases, the number of integers remains finite and the set of integers has an upper bound.

(There are actually two important upper bounds. In my proposal, the largest integer is equivalent to the largest integer for a given word size, which is the first of these bounds. The second bound, the largest Practical number, is the largest number which will not give an overflow when added to or multiplied by any smaller or equal number.)

When I expressed my doubts and concerns on this issue to my professors, I was told that the Axiom of Choice was what justified the assumption that an iterative inductive process like the definition of the succession of integers would ultimately produce an infinite set. It seemed like baloney to me then and it still seems like baloney to me now. If I'm wrong, I'd sure like someone to show me where and why.

It's fun talking to you, Robert.

Paul Martin

 

[Paul Martin]

Furthermore, even if you took as an axiom that no infinite set exists, it does not follow that there are only finitely many integers (and thus a largest): it merely means that the integers don't form a set.

Hi Hurkyl,

Thanks for your response. Please see my reply to Robert for a discussion of how we get the integers.

Paul

 

[Hurkyl]

First, there are some subtle points that I think need addressed.


The first is the issue of denying an axiom -- this means much more than simply excluding the axiom from your theory. If I take ZFC, but omit the axiom of infinity, I'm left with a theory in which the set of natural numbers may or may not exist.

To deny the axiom, you have to take a new axiom that asserts the old axiom is false.

For your purposes, however, I think this may not even be sufficient -- you need an axiom that says, somehow, that all sets are finite.



The second is that of a class¹ vs a set. It can be hard to distinguish the two concepts because the idea of a set is very closely modeled after the idea of a class, and ZFC was designed to be general enough that we'd never have to resort to using classes when doing everyday mathematics.

This is relevant to the current discussion -- while you can deny the existence of an infinite set, that does not mean there do not exist infinite classes. In particular, you can simply take the class of all sets!



1: I'm using the term "class" to refer to the objects that satisfy a logical proposition. For example, there is a class of "everything", because everything satisfies x = x.

 

[Hurkyl]

Next, I would like to suggest some topics you might find interesting. (and these topics are certainly not disjoint!)


You'll probably find studying at least some formal set theory worth your while. It would certainly help you develop your version of set theory, and it would probably be illuminating to see precisely how mathematics deals with these issues at the most fundamental level.


Next, I'd like to suggest formal logic, particularly first-order logic. IMHO, it just feels more constructive. It also has the particularly nice feature that everything in a theory that must be true has a proof!


I've found the basic theory of real closed fields (in particular, the fact the theory is complete) to be somewhat illuminating.


Nonstandard analysis is also quite interesting. Basically, it deals with theories A and B, where theory A is basically contained in theory B. However, if you limit your perspective properly, theories A and B will become identical, which allows you to do all sorts of interesting things. While I don't think the subject material will be relevant, per se, I think you might be able to extract some ideas and understanding from it.

 

[master_coda]

What seems more reasonable to me is to accept only definitions that have been explicitly made and written, either by human hands or by human made machines. So I'll readily accept the current approximate definition of Pi out to a trillion decimal places because it has actually been produced by a man-made machine. But I don't accept the existence of an approximation of Pi out to 10^trillion decimal digits. When it's finally computed out that far, I will.

By far the most common definitions of numbers today are found in the computer chips which embody algorithms to define and manipulate them. In each case, there is a practical limit on the size of the largest useful number. The usefulness of the larger of these numbers follows a pattern like the umbra and penumbra of a shadow as I described in my proposal (Thank you for reading and citing it, Robert.). Nothing prevents us from defining numbers as big as we need. But in all cases, the number of integers remains finite and the set of integers has an upper bound.

But what's the value in restricting math in such a way? It's like arguing that computer scientists should only study computer programs that we can prove will halt or not halt, and should pretend that other programs do not exist until it has been proven that they halt (or do not).

Even if you feel that current concept of the integers is "absurd", that doesn't change the fact that the math and logic behind them is perfectly solid; this is all that mathematicians care about anyway.

If you want to study a specific number system, then do it; don't waste you time telling everyone that your system is better because it satisfies your personal prejudices about what is intuitive and what is not.

 

[Hurkyl]

Now, some practical stuff!


First off, have you seen Cantor's model of the natural numbers in set theory? Intuitively speaking, every Cantor natural number is defined to be the set of all smaller Cantor natural numbers. In particular:

0 := {} (the empty set)
1 := {0} = { {} } (the set containing the empty set)
2 := {0, 1} = { {}, {{}} }
3 := {0, 1, 2} = { {}, {{}}, { {{}, {{}} } }
...

In ZF, there's a class Ord of ordinal numbers which is defined by a logical proposition that says whether or not a set is of this form. However, since we only have finite sets in your set theory, the logical condition that defines this class will simply define the natural numbers.

So, even in this "finite set theory", we can define the class of natural numbers.


As an aside, the practical difference between a class and a set is that sets can be treated as objects, but classes cannot.


When dealing with set theory on this basic of a level, this model of the natural numbers is very convenient, because some of the basic operations are given by elementary set manipulation. Some examples:

The successor of N is given by N U {N} (U is union)
M <= N if and only if M is a subset of N
M < N if and only if M is an element of N
If Y is a set of (some) natural numbers, then the interesction of all the elements of Y is precisely the smallest element of Y.


As for functions, you are familiar with the set-theoretic definition. In particular, a function is supposed to be a set of ordered pairs. Well, you can also speak of a logic-theoretic definition of functions -- a (logic) function can be represented as a class of ordered pairs.


So, for example, I can speak of the function S which maps each natural numbers to its successor. It's defined by:

S(N, M) := M = N U {N}

But we would more conventionally write this as:

S(N) := N U {N}

even though the notation is the same, here I am denoting a logic function, not a set function.



Now, even though we cannot speak of the set of all integers, that's usually irrelevant. The nifty trick is to reduce our scope of inquiry to the relevant integers, but do it in an arbitrary way, so it's still valid for all integers!

That was a convoluted and impenetrable statement, wasn't it? Let me try via example.


Suppose we're still trying to define the relation <. We can give these two axioms:

N < S(N)
N < M and M < L implies N < L

(we really need more axioms to do this properly)

Now, I want to prove that 0 < M if M is any integer other than 0.


Normally, this would be a straightforward proof by induction:

0 < 1.
Suppose 0 < M. Then, 0 < M and M < S(M), therefore 0 < S(M).
Thus, 0 < M for all M >= 1.


But induction works by using the set of all integers, right? The usual proof of induction goes as follows:

Suppose there is a counterexample to the theorem. Then, there must be a least counterexample, X. However, there is a Y such that X = S(Y). Since the theorem is true for Y, it must also be true for X, which is a contradiction!


Actually, we don't need to work with the set of all integers to prove induction! We can use this slightly modified proof:


Theorem: Suppose P(0) is true and P(n) implies P( S(n) ). Then, P(n) for all natural numbers n.

Suppose the theorem is false. Then, there exists a natural number, M, such that P(M) is false.

Now, take the set {0, 1, 2, ..., M}. (This is a set, remember?) Now, among the elements of this set, there is a least counterexample! Call it X. X is not zero (because P(0) is true) Thus, there must be a Y such that X = S(Y). However, P(Y) is true (because Y < X, and X was the least counterexample), so P( S(Y) ) = P(X) must be true, which is a contradiction.


I've been sort of rambling on this, so I'll stop here and give things a chance to sink in.

 

[Hurkyl]

Well, I was slightly misleading... my example involving induction was entirely unnecessary -- given any class of natural numbers, you can always find the least element!

Recall the axiom of the subset: if I have some class of sets, C, then I can define the intersection of the entire class as follows:

Let a be some element of C.
Define I := {x in a | for all b: C(b) --> x in b }

By the axiom of subsets, I is, in fact, a set, and it is precisely the intersection of all of the sets in the class C. So, I didn't need to apply my little trick to prove that induction works, I can just use this.


It's still a useful trick, I just didn't select a good example.

 

[matt grime]

You don't actually give any proper definitions to make it work. Your notion of penumbra is hazy (what is it for heaven's sake?) and requires the arbitrary removal of closure from the additive property of the natural numbers, and cartesian products (why should I be allowed to take the prodcut of two sets if I can't sum two numbers?) Why does Q even possesses a square root?

nb Q is the 'largest integer' not the rationals.


Point 15 wrong. If every set is closed every set is open too, or would you need to redefine what a topology is?

Obviously all axioms are arbitrary ultimatelym but why must I forgo the fact that Z is a ring in order to satisfy your need to have "no infinities", especially since as we've seen nothing you've said forces there not to be some infinite object

You claim that you don't accept the set of integers if finite. Do you accept that there is at least 1 integer, say 1? Do you accept that 1+1 is also an integer, and that it is 2 and proceeding can we say 2+1 is an integer?

If the set is finite there is a maximal element, agreed? Say M, which is gotten by adding 1 M times, so why can't I logically add one more to M and get a bigger integer contradicting the assumption of finiteness.

Which of those steps is wrong in your opinion?

Note we do not assume that the whole set of integers exists and is infinite. We have 1 'existing' in some sense and we can add to it as we want. Purely a mathematical idea this nothing to do with what can be constructed (construct the number 1?) in any physical sense.


Why is it important to you that in your mathematics you have no infinities? They cause no problems.

And you can simply look at the statement of the axiom of choice and see that at no point does it "make an infinity exist" whatever the hell that might mean, and simply removing it from ZFC does not create a system that possesses a model that has only finitely many elements in it. The system without C is ZF and is well known indeed preferred by many for the philosophical issues this alleviates, but it still contains a copy of the natural numbers, or rather an inductive set.

It is perfectly possible to 'discretize' and choose your number system to be Z[e], by adjoining an free indeterminate to the integers. It's still an infinite set but all sums in it are required to be finite. Algebra abounds with examples of infinite systems like this where only finitely many terms are allowed (see the direct sum indexed by an infinite set for instance, as opposed to direct product).



Many mathematicians do not accept the axiom of choice (exactly what did cantor do with it that bothers you?).

If we accept its negation then there are many things that fail to be true since there are known to be many equivalent statements.

Infinite dimensional vector spaces wouldn't possesses a basis springs to mind as the msot obvious one.

there are some good pieces of work by Conway et al that do set theory of infinite sets without the axiom of choice.

Here's an older example (Bernstein-Schroeder)

If A and B are sets and there are injections A to B and B to A then there is a bijection between them.

Proof is axiom of choice free.

The dual statement involving surjections has a proof that requires the axiom of choice.

 

[Paul Martin]

Hi Hurkyl,

Thanks for writing.

To deny the axiom, you have to take a new axiom that asserts the old axiom is false.

No, I don't think so. By "denying" the axiom I simply mean not including it with the axioms of the system.

For your purposes, however, I think this may not even be sufficient -- you need an axiom that says, somehow, that all sets are finite.

No, I don't need that either. In the system I propose, we wouldn't ever introduce the terms "infinite" or "finite".We would accept only explicitly defined terms, including the numerals which are terms denoting specific numbers. Thus, the complete set of defined terms, in particular the complete set of defined numbers, would always be, to use a currently vernacular term, "finite". That would serve my purpose.

The second is that of a class1 vs a set. ... While I don't think the subject material will be relevant, per se, I think you might be able to extract some ideas and understanding from it.

I agree that it is not relevant. I also agree that it might be interesting to pursue if I had the time and the inclination. Thanks for the suggestion all the same.

Paul

 

[Paul Martin]

have you seen Cantor's model of the natural numbers in set theory? Intuitively speaking, every Cantor natural number is defined to be the set of all smaller Cantor natural numbers. In particular:

0 := {} (the empty set)
1 := {0} = { {} } (the set containing the empty set)
2 := {0, 1} = { {}, {{}} }
3 := {0, 1, 2} = { {}, {{}}, { {{}, {{}} } }
...

Yes, I have seen that and I am familiar with it. The part of it which is of interest to me is the ellipsis at the end. I would like you to give me a rigorous definition of what you mean by the ellipsis.

In ZF, there's a class Ord of ordinal numbers which is defined by a logical proposition that says whether or not a set is of this form. However, since we only have finite sets in your set theory, the logical condition that defines this class will simply define the natural numbers.

First of all, I'm not sure what you mean by "ZF". I'm guessing you mean the system of arithmetic as axiomatized by Zermelo and Fraenkel, but that's only a guess. Secondly, when you say "the natural numbers" I'm assuming you mean the customary infinite set of natural numbers. If so, please explain to me what "logical condition" will define the infinite set of natural numbers. That's what I think cannot be done rigorously. If we restrict ourselves to definitions which must be explicitly made, we must always have a system which we commonly refer to as "finite".

Thanks for the energy you put into this, Hurkyl.

Paul

 

[Paul Martin]

Hi Master_Coda,

Thanks for your post.

But what's the value in restricting math in such a way?

Since our world is grainy, it seems likely to me that a grainy math would be better suited to
describe the world than a smooth, continuous math would.

Even if you feel that current concept of the integers is "absurd", that
doesn't change the fact that the math and logic behind them is perfectly solid;

But the "facts" are otherwise. To be "perfectly solid", math must be consistent. Currently
accepted math and logic failed this test as soon as Cantor encountered his first paradox. I stand
with Kronecker and Brouwer on this issue. It was evidently exciting enough for mathematicians
to see the consequences of Cantor's work that they were willing to sweep the inconsistencies
under the rug. They euphemistically called the paradoxes and inconsistencies "antinomies".
Russell tried systematically to eliminate specific sets and propositions, with his Theory of Types,
in an attempt to isolate the inconsistencies. But, the way I see it, Kurt Goedel served up the fatal
blow. Goedel's theorem says that any system of math that includes the infinite set of integers
cannot be consistent and complete. In my opinion, that news should have caused mathematicians
to abandon the notion of infinity in mathematics for good. Instead, they continued to this day
with a system containing paradoxes and inconsistencies (er -- I mean "antinomies"). This is not at
all what I would call a "perfectly solid" system.

this is all that mathematicians care about anyway.

I agree. In fact, I think they should care more about it than they do. The inconsistencies don't
seem to bother them nearly as much as I think they should.

But I think my proposal holds more promise for Physicists than it does for Mathematicians. The
infinities that appear in the Physics equations are a real nuisance. In fact, they give Nobel Prizes
to Physicists who figure out ways of getting rid of some of the infinities. I would think that they
would find much more useful a mathematics that doesn't have infinities in it and which is grainy
just as the physical universe seems to be. Not only that, but none of their calculations would be
affected. In reality, all their calculations are already done in a system like the one I propose. (E.g.
no one has ever done a calculation using a value for Pi with an infinite number of digits.)

Paul

 

[Paul Martin]

Hi Matt,

You don't actually give any proper definitions to make it work.

True. As I described it, my essay makes only an intuitive appeal.

Your notion of penumbra is hazy (what is it for heaven's sake?)

That's a good one! LOL Not only my notion, but a penumbra itself is hazy. I'll give you the credit for intending the humor. I'll also assume your question was meant for readers (other than yourself, of course) who may not know what a penumbra is. If you know what a penumbra is, you may skip the next paragraph.

If you have a 1-foot diameter light source 10 feet from a wall and directly between them, 5 feet from the wall you have an opaque disc 1 foot in diameter, the disc will cast a shadow on the wall that has two distinct parts. In the middle will be a 1-foot diameter shadow which will be uniformly dark. If you sketch a diagram, you can see that no ray from the source can directly hit any part of this shadow. That part of the shadow is called the umbra. Surrounding the umbra is a more or less hazy shadow extending 1 foot beyond the umbra with the outside boundary making a 3 foot diameter circle concentric with the umbra. This band of shadow is called the penumbra. It is not uniform but instead is almost as dark as the umbra in regions near the umbra, and it is nearly as light as the unshaded wall near the outer boundary of the penumbra.

The metaphor of a shadow works well for explaining my proposal. If we assume the largest integer is Q, then you can think of a Cartesian plane with two concentric circles centered at the origin, one of radius Q the other of radius P = square root of Q. The outer ring is analogous to the panumbra and the inner circle to the umbra. The (x,y) pairs inside the inner circle represent all possible pairs of the Practical Numbers (those between -P and P), Sums and products are defined for all such pairs although a sum or product may not yield another practical number. But in all cases, the sum or product will be a number between -Q and Q. Numbers within the penumbra are called "impractical numbers" and may or may not have sums or products defined with other numbers.

[Your scheme] requires the arbitrary removal of closure from the additive property of the natural numbers, and cartesian products (why should I be allowed to take the prodcut of two sets if I can't sum two numbers?)

It isn't arbitrary. It is necessary. If all sums and products are defined, then you have a situation where you can't define them all explicitly as I insist. If you assume that you can somehow acquire numbers without explicitly defining them, as is typically done, then you end up with infinite sets. It is that tacit acquisition assumption which I refuse to accept.

So, in my explicitly defined system, all numbers do not end up equal, even in their rights. Some have more or less operations defined on them than others. It's no different, except in scale, from what we have in typical number systems. There is an infinite (your concept, not mine) number of division operations which are not defined, viz. all those with zero divisors. My system simply has some sums, products, differences, and divisions which are also not defined. No problem.

Why does Q even possesses a square root?

Because it is a defined number in my system. It happens to be the largest number, but it is a number just the same. But you are probably getting at the situation where Q has an irrational square root in conventional mathematics. As is obvious, my system contains no irrational numbers, so the definition of 'square root' would have to involve a limiting process and assign the "closest" practical number to the desired root. This is exactly what we do in our computers and our calculations today. Nobody has ever worked with an irrational number. No problem.

nb Q is the 'largest integer' not the rationals.

I'm not sure what you are getting at here since I can't parse your statement.

Point 15 wrong. If every set is closed every set is open too, or would you need to redefine what a topology is?

I disagree with you here, but as I said in Point 15, "I will leave this for others to ponder."

Obviously all axioms are arbitrary ultimatelym but why must I forgo the fact that Z is a ring in order to satisfy your need to have "no infinities", especially since as we've seen nothing you've said forces there not to be some infinite object

I don't even know what "Z" is so I can't comment on its being a ring. My "need" is to accept terms that are explicitly and consistently defined. I am aware of no explicit and consistent definition of "infinity" or of any "infinite object". It isn't that I reject the notion of infinity. Instead I claim that the notion has never been properly introduced.

You claim that you don't accept the set of integers if finite. Do you accept that there is at least 1 integer, say 1? Do you accept that 1+1 is also an integer, and that it is 2 and proceeding can we say 2+1 is an integer?

If the set is finite there is a maximal element, agreed? Say M, which is gotten by adding 1 M times, so why can't I logically add one more to M and get a bigger integer contradicting the assumption of finiteness.

Which of those steps is wrong in your opinion?

It goes wrong where you say, "contradicting the assumption of finiteness". If, by " logically add[ing] one more to M", you mean defining another number beyond M (your 'M' is my 'Q'), then, yes, you do get a bigger integer. You have defined a bigger system. (Now your M+1 is my
Q). But M+1 is still finite.

You are also wrong in characterizing the notion of "finiteness" as an assumption. I don't ever need to introduce that term into my system. The only reason you do, is to be able to make a distinction between my system and your notion of "infinity" which I claim you can't define in any consistent way. (NB the commonly accepted definition of infinity is Cantor's and he ran into inconsistencies with it right away.)

Note we do not assume that the whole set of integers exists and is infinite. We have 1 'existing' in some sense and we can add to it as we want. Purely a mathematical idea this nothing to do with what can be constructed (construct the number 1?) in any physical sense.

You have to be careful here if you want to be rigorous. Assuming that the whole "infinite" set of integers exists was the common foundation of arithmetic prior to Peano. He was the first to show how the integers could be constructed from an axiomatic base. But that "construction process" has holes in it which are my main objection. Similarly, we don't have 1 'existing' in any sense. To be rigorous, the number 1 must be defined. When I was studying this stuff in college 45 years ago, we spent six weeks defining the number 1 from a set-theoretic starting point. Bertrand Russell and Alfred North Whitehead took (if I remember right) a couple hundred pages at the beginning of their "Principia" to do the same. It's not trivial if you want to be rigorous.

Why is it important to you that in your mathematics you have no infinities?

I think it would work better for Physics as they try to describe our grainy universe.

They cause no problems.

In my view, they cause a host of problems, starting with Cantor's Paradox and followed by many others. They also cause horrendous problems in Physics calculations.

Thanks for all the time you have put in on this, Matt. I enjoy talking to you.

Paul

 

[matt grime]

Firstly, mathematics isn't physical. The physical sciences use mathematics, but that infinite is well defined mathematically (contrary to your opinion) doesn't force infinite things into phyiscal reality. If it does it is either because of a flaw in the model, a flaw in the logic, or becuase there is indeed a singularity in the model there that needs to be physically interpreted. Just because the energy levels of a bound electron are quantized doesn't make the real number system invalid because you could divide the lowest energy by two and still have a real number that isn't the energy of a bound electron.

A set is infinite if it is not finite. A set is finite if it can be put in bijection with any of the inductive elements that arise in the inductive set in the axiom of infinity, or if you prefer, as we all should, with any of the sets {1,..,n} for some n in N.


nb N means the naturals, Q means the rationals informally in texts like this (usually we write $$\mathbb{Q}$$ hence why i qualified what Q is in my post incase anyone who hadn't read you essay thought it meant the rationals).

I'm sorry, what do you mean by Cantor's Paradox?

 

[Hurkyl]

First of all, I'm not sure what you mean by "ZF". I'm guessing you mean the system of arithmetic as axiomatized by Zermelo and Fraenkel, but that's only a guess.

Almost right; the Zermelo-Fraenkel axioms are for set theory, not arithmetic. (But, of course, arithmetic can be modeled in this set theory)

It was evidently exciting enough for mathematicians
to see the consequences of Cantor's work that they were willing to sweep the inconsistencies
under the rug.

That is exactly wrong; much work went into developing a set theory that did not suffer from the inconsistencies. The most popular resulting theory is ZFC: the Zermelo-Fraenkel axioms + the axiom of choice.

All of the paradoxes of naive set theory arise from unrestricted comprehension: that is, saying "Let S be the set of all blah" where "blah" could be anything.

The usual solution is to only allow "safe" ways of making big sets: unions and power sets. Then use restriction (aka the axiom of subsets) to prove the existence of the set you really wanted.

The part of it which is of interest to me is the ellipsis at the end. I would like you to give me a rigorous definition of what you mean by the ellipsis.

First, allow me to define "transitive set":

A set S is transitive if and only if whenever x is an element of y and y is an element of S, then x is an element of S.

Then, the class Ord is defined to be all transitive sets that contain only transitive sets.


An example of a transitive set is:
{ {{0}}, {0}, 0 }
(where 0 is the empty set, as before). This is not an ordinal number, though, because it contains {{0}} which is not transitive (because it does not contain 0).

Kurt Goedel served up the fatal
blow. Goedel's theorem says that any system of math that includes the infinite set of integers
cannot be consistent and complete.

To be slightly more precise: the theory must be able to define a model of the integers, and the operations of multiplication and addition.

Whether they fit in a set or not is irrelevant to Godel's theorem.

And, it's not enough to simply contain the integers: the theory has to be able to define them too. See Tarski's completeness theorem -- Godel's theorem is not applicable to the first-order theory of real closed fields, even though every real closed field contains the integers.

(and, as a technical point, it also requires you to be working from a finite set of axioms)


Anyways, I don't see why you think this is a "fatal blow": while it is nice for a theory to be complete, it isn't required to do math. All this property means is that every statement either has a proof or disproof.

 

[StatusX]

how can you define a largest number? I just don't understand that. if addition by one is defined for the number below it, how can it not be defined for it? what if one day we need a higher number for a practical application? can we extend the range of numbers? is that allowed, or will we have to pay some kind of fine to your descendants first? if you define the number 1, define addition by 1, then intuitively, there is no upper limit to the integers. this is mathematically well defined, and necessary for countless applications.

is it that you don't understand the concept of infinity? if so, just study calculus: understanding infinity is necessary in grasping the idea of a limit. but you probably do understand it, and if so, then YOU CANNOT DENY THAT INFINITY EXISTS, AS AN IDEA. all math is is ideas. sometimes people say that 0.999... is not exactly one, but for any practical purpose it is. what practical purpose? it is nothing but a completely abstract idea. there is no physical analogue to the mathematical question of 0.999... = 1. math is made to describe the real world, but in doing so, it has generated so many rich new ideas that it is worth studying for its own right, and it can generate questions within its own abstract system that need to be answered. if there's anywhere you want to place a limit on numbers to reflect the "graininess" of space, its in a physical theory, and that is exactly what physicists are doing now to resolve infinities in QM, ie., by quantizing space and time.

 

[master_coda]

Since our world is grainy, it seems likely to me that a grainy math would be better suited to
describe the world than a smooth, continuous math would.

This is a good reason for wanting to explore certain fields of math in more depth. This isn't a good reason for discarding other fields.

But the "facts" are otherwise. To be "perfectly solid", math must be consistent. Currently
accepted math and logic failed this test as soon as Cantor encountered his first paradox. I stand
with Kronecker and Brouwer on this issue. It was evidently exciting enough for mathematicians
to see the consequences of Cantor's work that they were willing to sweep the inconsistencies
under the rug. They euphemistically called the paradoxes and inconsistencies "antinomies".

You seem to have a very poor understanding of the history of set theory. All of the paradoxes Cantor found were paradoxes in naive set theory. Modern set theory did not sweep these paradoxes under the rug; it was an entirely new theory constructed specifically to avoid Cantor's paradoxes.

Russell tried systematically to eliminate specific sets and propositions, with his Theory of Types,
in an attempt to isolate the inconsistencies. But, the way I see it, Kurt Goedel served up the fatal
blow. Goedel's theorem says that any system of math that includes the infinite set of integers
cannot be consistent and complete. In my opinion, that news should have caused mathematicians
to abandon the notion of infinity in mathematics for good. Instead, they continued to this day
with a system containing paradoxes and inconsistencies (er -- I mean "antinomies"). This is not at
all what I would call a "perfectly solid" system.

The standard modern set theories (ZF, ZFC, NF, NBG, etc.) do not contain any known antinomies. If any of them did they would immediately be discarded by mathematicians.

Demanding that all theories in math be consistent and complete is a ridiculously strict requirement. Sure, it's great when a theory turns out to be consistent and complete, but that doesn't happen because the theory is more "solid"; it happens because the theory is weak. And often that lack of power is a bigger problem than a lack of completeness.

Incidentally, there is a consistent and complete theory known as the Presburger arithmetic which is a theory of the integers without multiplication. The integers forumlated in that theory are not finite. Infinite sets don't automatically make a theory inconsistent or even incomplete; getting rid of them won't magically make a theory easier to work with.

But I think my proposal holds more promise for Physicists than it does for Mathematicians. The
infinities that appear in the Physics equations are a real nuisance. In fact, they give Nobel Prizes
to Physicists who figure out ways of getting rid of some of the infinities. I would think that they
would find much more useful a mathematics that doesn't have infinities in it and which is grainy
just as the physical universe seems to be. Not only that, but none of their calculations would be
affected. In reality, all their calculations are already done in a system like the one I propose. (E.g.
no one has ever done a calculation using a value for Pi with an infinite number of digits.)

Paul

The physicists who worked around the problems that infinity can introduce didn't do it by just pretending infinity doesn't exist. That would leave them with a system to crippled to do anything usefull. Saying "let's never use division" is not a helpful workaround for the division-by-zero problem. A great deal of physics is based on math that relies on infinities, and it may not even be possible to produce a more accurate theory of physics without such theories.

It's also unlikely that physicists or people in any more applied field would care about the incompleteness of a theory either. Scientists used calculus long before mathematicians came up with a rigorous foundation for it. If they didn't care when the theory wasn't even properly defined, why would they care if that definition isn't complete?

Of course theories can be useful in very specific fields such as numerical computation, but even then it's often useful to use an exact number system to work out exact results as much as possible. The difficulty of doing exact computations on a computer is a limitation to be worked around, not something we should base all of mathematics upon.

And your remark that nobody has ever done a calculation using a value of pi with an infinite number of digits is also wrong. The fact that the native format on a computer is numbers with a finite binary representation does not mean that all computation has to be done using only those numbers; that's just the easiest and fastest way. If we can produce a finite representation of pi on a piece of paper, then we can certanly do it on a computer.

 

[Paul Martin]

Hi Matt,

I agree with what you said in your first paragraph. Your second paragraph was

A set is infinite if it is not finite. A set is finite if it can be put in bijection with any of the inductive elements that arise in the inductive set in the axiom of infinity, or if you prefer, as we all should, with any of the sets {1,..,n} for some n in N.

I agree you could define 'infinite' this way, but it would leave the burden of an existence proof on you to prove that some infinite set exists. That is where my doubt lies. I doubt that you can demonstrate the existence of an infinite set without introducing inconsistency.

nb N means the naturals, Q means the rationals

Sorry about that. I'm not up on the currently used symbols. I think you understood me that my P and Q were simply positive integers. I chose 'P' because I call them Practical Numbers and I chose 'Q' simply because it sort of goes with 'P'. I hope I didn't confuse anyone with my choices.

I'm sorry, what do you mean by Cantor's Paradox?

First, Cantor's Theorem: "The cardinal number of the set of the subsets of a set S is greater than the cardinal number of s."

Second, Cantor's Paradox: "Let S be the set of all sets, and T the set of the subsets of S. Since T corresponds one-to-one to itself as a subset of S, it cannot have a greater cardinal than S. Yet by Cantor's theorem it must; this is Cantor's paradox (1899)."

I believe Cantor's Paradox was first published by Bertrand Russell although maybe under a different name.

Good talking to you, Matt. Thanks for writing.

Paul

 

[Paul Martin]

Hi Hurkyl,

The usual solution is to only allow "safe" ways of making big sets: unions and power sets. Then use restriction (aka the axiom of subsets) to prove the existence of the set you really wanted.

I call that kind of "solution" and "restriction" sweeping inconsistencies under the rug. Sorry.

First, allow me to define "transitive set":

OK. I got it. But where is the "second" part? You didn't even mention the ellipsis much less give a rigorous definition for it.

I don't see why you think this is a "fatal blow": while it is nice for a theory to be complete, it isn't required to do math. All this property means is that every statement either has a proof or disproof.

I agree that there isn't much of a problem if a system in incomplete, but I forgot to add that in addition to the system not being both complete and consistent, the killer is that you can't know which it is. It is this possibility for inconsistency which I claim is the fatal blow. The fact that the inconsistencies (paradoxes and antinomies) popped out immediately tells me that the notion of infinity should not be entertained. While you can take the approach you are talking about, and restrict your system sufficiently in order to avoid inconsistencies, I don't think what you gain is worth it. What you gain is a bunch of infinite sets that are not very useful in applying to real problems.

Good talking with you, Hurkyl. Thanks for writing and have a happy Halloween.

Paul

 

[Paul Martin]

Hi Statusx,

Thanks for writing.

how can you define a largest number?

There are several ways. In mathematics, it is fair game to define anything you want as long as you remain consistent within your system. So, for example, you could define a finite set of numbers, among which one of them would be the largest number. Then, within the context of discussions about that set, you have defined a largest number.

More practically, you can define a largest number which can be used within a calculating machine. Take your calculator for example. If you start with some number and then successively add one to it, you cannot continue indefinitely. As soon as the sum exceeds the maximum capacity of the display (or the buffers in the circuitry) the addition won't work. For every calculating machine, or algorithm, the designers must define the largest number that can be represented in their machine.

In my proposal, I don't specifically define a largest number. I suppose that a set of integers has been constructed up to some maximum number which I called Q. (I have since realized that that was a bad choice, so I don't mind if someone changes the symbol to something else.) Since Q is a variable, anything that can be proved in my system would hold for any finite Q. So, yes, if you wanted a bigger largest number, nothing would prevent you from defining all the numbers up to and including any larger number you like.

I just don't understand that. if addition by one is defined for the number below it, how can it not be defined for it?

As I pointed out in an earlier post, the definition of an operation like addition is done after the numbers are defined. We therefore can't use the operation to define the numbers. So, in my Practical Number system, if R = Q - 1, then the sum R + 1 is defined, and the sum happens to be Q, but the sum Q + 1 is simply not defined. This is no different from traditional arithmetic where the quotient 5/1 is defined but the quotient 5/0 is not defined.

what if one day we need a higher number for a practical application? can we extend the range of numbers?

Yes, absolutely. No problem. In fact we do that all the time as we build calculators, computers, and algorithms with ever larger number capacity. In my opinion, if we were to actually set the number Q to some large value for some reason, I would say that it would be so large that it wouldn't limit any calculations we would ever want to make. So, for example, the smallest interval, which would be of length 1/Q, would want to be much less than the Planck length if the system were to be used for physics for example.

will we have to pay some kind of fine to your descendants first?

Sort of, but not to your descendants. First of all, there is no mathematical reason to pick a specific value for Q, the largest number. It will be sufficient simply to assume that such a number exists. But, if you are designing and building computing machines or algorithms, then there is a penalty to pay for designing it too small to start with and then having to extend it later. But you pay, not your kids.

if you define the number 1, define addition by 1, then intuitively, there is no upper limit to the integers. this is mathematically well defined, and necessary for countless applications.

That is roughly the position and approach of current mathematics. I challenge it in a couple ways.

First, How do you propose that all these successive additions are to be done? Or don't we have to do them at all? Either way, what makes you think such a process could ever produce anything but a finite number?

Second, I disagree that it is "necessary for countless applications." If you counted up all the applications of anything whatsoever ever made by man or his hominid ancestors, you would have a finite number of them which is countable. There have not yet been "countless applications", nor in my view could there ever be.

is it that you don't understand the concept of infinity?

No. I don't think that's it. I got good grades when I studied the subject a long time ago and I feel that I understand the concept of infinity as Cantor defined it.

YOU CANNOT DENY THAT INFINITY EXISTS, AS AN IDEA.

You are exactly right. I do not deny that infinity exists as an idea. I am well aware of the concept as developed by Cantor and those who followed him. I also am well aware that the concept of unicorns exists.

if there's anywhere you want to place a limit on numbers to reflect the "graininess" of space, its in a physical theory, and that is exactly what physicists are doing now to resolve infinities in QM, ie., by quantizing space and time.

Yes, I agree. The potential usefulness in physics is my main motivation. BTW, in this regard I am wondering why nobody has commented on the implications of my proposal for the Dirac Delta function.

Thanks for writing, and have a happy Halloween.

Paul

 

[Hurkyl]

I doubt that you can demonstrate the existence of an infinite set without introducing inconsistency.

He can't, but not for the reason you think. The reason is:

Consistency can never be proven

The best you can do is say things like: "According to theory A, theory B is consistent." Mathematicians phrase this by saying "Theory B is relatively consistent to theory A".

This is actually the subject of Godel's second incompleteness theorem: no consistent theory can prove itself consistent!


Now, I've already explained the approach modern mathematics takes to dealing with the paradoxes. All of the paraodoxes of Cantor's set theory involved unrestricted comprehension. The ZF axioms don't provide a way to carry out unrestricted comprehension, so none of the paradoxes of set theory apply to ZF set theory.

You seem to think the answer is striking out a different aspect of Cantor's set theory; the existence of an infinite set. But the problem is that none of the famous paradoxes care about the size of the sets involved! As long as you still permit unrestricted comprehension, you get paradoxes, whether you permit infinite sets or not!

Since you have to throw out unrestricted comprehension for your set theory to work, I don't think your condemnation:

I call that kind of "solution" and "restriction" sweeping inconsistencies under the rug. Sorry.

carries any weight, unless you think it applies to your approach as well.

OK. I got it. But where is the "second" part? You didn't even mention the ellipsis much less give a rigorous definition for it.

I thought you were more interested in the definition of "ordinal number", so that's what I provided. 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}.

What you gain is a bunch of infinite sets that are not very useful in applying to real problems.

And that is entirely inaccurate, because infinite sets have proven themselves useful for solving "real problems".

 

[master_coda]

I call that kind of "solution" and "restriction" sweeping inconsistencies under the rug. Sorry.

So you think abandoning unrestricted comprehension and replacing it with weaker operations is "sweeping inconsistencies under the rug". But throwing away unrestricted comprehension and all infinite sets is not.

 

[matt grime]

I agree you could define 'infinite' this way, but it would leave the burden of an existence proof on you to prove that some infinite set exists. That is where my doubt lies. I doubt that you can demonstrate the existence of an infinite set without introducing inconsistency.

That is how you define finite and infinite unless you want to use the axiom of choice.

The set of integers is infinite. The underlying set of the torsion free group generated by 1 non-identity element is infinite.

Of course this begs the question "what do you mean by existence?"

Do you understand the difference between the axioms of a set theory and a model of the axioms?

First, Cantor's Theorem: "The cardinal number of the set of the subsets of a set S is greater than the cardinal number of s."

Second, Cantor's Paradox: "Let S be the set of all sets, and T the set of the subsets of S. Since T corresponds one-to-one to itself as a subset of S, it cannot have a greater cardinal than S. Yet by Cantor's theorem it must; this is Cantor's paradox (1899)."

These 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?


Others have adequately answered this question, but there is stil the question of what inconsistency you mean since none of what you have written applies to those theories that contain the integers and not to your alleged finite set theory.

 

[CrankFan]

I'm following this only very loosely, Initially Paul wanted to disallow infinite sets by removing the axiom of choice -- which makes no sense... He gets corrected and then stumbles forward with new suggestions which aren't all that coherent either, but ignoring all of those details :) and sticking with just the theme of only working with finite sets I have a question for Paul.

Traditionally, Z is constructed in terms of N, and Q is constructed in terms of Z, etc...

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?

A lot guys who propose "new foundations" for mathematics on net forums know very little about mathematics; yet to be competent in the endeavor of developing a formal system for which all of mathematics can be embeded, one needs to know a lot about mathematics and how it's practiced.

At a glace, your "foundations" seem to invalidate some of the richest and most fruitful areas of mathematics. I know very little about mathematics (and that's an understatement!) -- but at least I know that if we can't embed bread and butter mathematics, like analysis, into your "foundation" then it's totally useless (as a FOUNDATION).

 

[matt grime]

I think I can guess why he thinks/thought that the axiom of choice had something to do with making "infinity exist"

Basically, we can restate the axiom, unmathematically, as:

it is possible to make an infinite number of consistent unspecified choices one after the other.

So, in this case, it is the axiom of choice that makes us say if 1 exists, so does 2, so does 3 and so on ad infinitum, for an infinite number of steps.

Of course it doesn't actually mean that at all. (naively, there is no 'undefined specification': we don't pick some next integer, we pick *the* next integer. Compare this to picking a basis for a vector space: we pick some vector, then some vector not in its span, then some vector not in the span of the first two and so on, where, if this doesn't terminate in a finite number of steps, we are making an infinite number of choices that need to be consistent and are in effect 'unspecified')

I quite like an example Hurkyl (I think) once gave, that if we have an infinite number of drawers which all contain one red pair of socks and one green pair of socks, then we can consistently pick out a green pair from each. If however each drawer contains two pairs of green socks then we need the axiom of choice in order to pick a pair of green socks from each. at least that I think was the gist of what he was getting at.

 

[StatusX]

You are exactly right. I do not deny that infinity exists as an idea. I am well aware of the concept as developed by Cantor and those who followed him. I also am well aware that the concept of unicorns exists.

Well, would you deny fantasy writers the oppurtunity to write stories about unicorns because they haven't been observed? they arent inherently contradictory creatures, theyre just horses with horns that don't happen to exist.

infinity is just a mathematical concept which doesn't have an observable physicial analogue in the real world. the same goes for imaginary numbers, perfect spheres, planes, and so on. would you deny us the chance to study any mathematical concept that hasnt been directly observed?

the fact is infinity CAN be studied. so why shouldn't it be? i must be missing something.

 

[master_coda]

Well, would you deny fantasy writers the oppurtunity to write stories about unicorns because they haven't been observed? they arent inherently contradictory creatures, theyre just horses with horns that don't happen to exist.

infinity is just a mathematical concept which doesn't have an observable physicial analogue in the real world. the same goes for imaginary numbers, perfect spheres, planes, and so on. would you deny us the chance to study any mathematical concept that hasnt been directly observed?

the fact is infinity CAN be studied. so why shouldn't it be? i must be missing something.

This kind of thing happens a lot. Someone has an abstraction that's very familiar or natural to them, and they get really attached to it and start to think that their abstraction is somehow more real than other abstractions.

 

[Paul Martin]

Hi Master_Coda,

Thanks, again for writing.

This is a good reason for wanting to explore certain fields of math in more depth. This isn't a good reason for discarding other fields

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.

You seem to have a very poor understanding of the history of set theory.

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.

All of the paradoxes Cantor found were paradoxes in naive set theory.

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.

The standard modern set theories (ZF, ZFC, NF, NBG, etc.) do not contain any known antinomies. If any of them did they would immediately be discarded by mathematicians.

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.

Demanding that all theories in math be consistent and complete is a ridiculously strict requirement.
...
It's also unlikely that physicists or people in any more applied field would care about the incompleteness of a theory either.

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

The physicists who worked around the problems that infinity can introduce didn't do it by just pretending infinity doesn't exist.

...And, I don't propose that approach. I propose that a rigorous, consistent theory be developed assuming a largest integer.

(Incidentally, I just read an article by A.A. Fraenkel in which he says "In abstract algebra, the well-ordering theorem and transfinite induction (or an equivalent maximum principle) solve the problems such as algebraically closed extensions of a field." It caught my attention that the parenthetical reference to "an equivalent maximum principle" seems to imply that an equivalent maximum principle could somehow replace transfinite induction. 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.)

A great deal of physics is based on math that relies on infinities, and it may not even be possible to produce a more accurate theory of physics without such theories.

On the other hand, the opposite may be possible. For example you might be referring to the theory of probability, which of course deals with infinite sets. If a similar theory were developed with a "maximum principle" such as I propose, it may, in fact, lead to more accuracy. I have read a proposal ("Symmetry and the End of Probability", DangSon, TD Printing, San Jose, CA 95123) which I think is consistent with that approach and which I think deserves to be developed.

And your remark that nobody has ever done a calculation using a value of pi with an infinite number of digits is also wrong.

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

 

[matt grime]

"I know that ZFC defines infinite sets. Does ZF also? "


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.


Probability doesn't only deal with infinite sets: it deals with measurable sets.

Re point 15: in topology a subset is open iff its complement is closed. If in your world every subset is closed, let U be an arbitrary set, U^c is closed, hence U is open. Thus every set is open, the topology is the discrete topology

 

[Paul Martin]

So you think abandoning unrestricted comprehension and replacing it with weaker operations is "sweeping inconsistencies under the rug". But throwing away unrestricted comprehension and all infinite sets is not.

I can't confirm or deny what you say without knowing more about "unrestricted comprehension". As for infinite sets, I think throwing them away would result in the loss of some interesting ideas. I wouldn't call it "sweeping inconsistencies under the rug"; I'd maybe call it throwing out the trash.

But, please explain "unrestricted comprehension" and I may change my mind. I certainly will if my approach guarantees its loss and it is something important.

Paul