.999999999999 repeating = 1

[LauricAcid]

A system of ideas in our minds, which you called "the way both you and I think about these numbers in their more everyday sense" ; let me just refer to it as E.

A formal system, F, that is isomorphic with E.

E is based in reality, while F is based on axioms. The axioms are chosen so as to ensure that F is isomorphic with E. We prove theorems using F, since only F has the rigor required for proofs. We engineer our technology (and count our money, etc.) using E, as E is the system that is "plugged into" reality.

Without my necessarily committing to that entirely, I think yours is basically a nice summary. Thank you for it. Except one very important point:

F is not claimed to be isomorphic (and we're using 'isomorphic' in an everyday, not technical sense, here) with E all the way. But rather, the isomorphism starts at a certain point in F. That is, in the first stages of F, it runs along without isomorphism with E, then only at certain later stages do certain parts of F become isomorphic with E. For example, some of the stages in the development of the natural numbers in F is not isomorphic with E, but at a certain point of fulfillment in those stages, the way the natural numbers work in F is isomorphic with the way they work in E. An even better example would be, say, the rational numbers, since they are defined as equivalence classes in F while E doesn't give a damn about equivalence classes, but after the equivalence class construction in F is done, the rational numbers in F work like they do in E.

Also, and I think this is were most detractors of set theory have their biggest objection: Because of what F needs to do in the early stages, it turns out that in some later stages F results in a proliferation of inifinte sets and of even different "sizes" of infinity. This is anathema to some people. Meanwhile, most people who study F actually welcome dealing with these different infinities. On the other hand, even if proponents of F were to concede that F would be better off without these different infinities, then it is not clear how we could still accomplish the early stages of F without the infinities popping up later.

The first axioms of F say nothing about infinity and basically just provide common ways of forming sets (as well as just asserting that sets are equal if they have the same members), and as far as F goes at that point, F does not say one way or another whether there exist infinite sets. But then, especially for real analysis, it seems that we do need to have not just each individual natural number but the whole set of natural numbers taken together, which is an infinite set. Then, since from the first stage there is a power set axiom, once there is at least one infinite set, there is no way to prevent there being sets of ever greater and greater infinite "size".

Now, there are alternatives to this, including theories that reject the existence of infinite sets, but I'm not aware that these alternatives have been successful. To whatever degree they are successful, at least as far as I have seen (I admit that I haven't looked high and low), they are significantly more complicated than Z set theory in classical first order logic. And I could not assure you that the alternatives are any closer to isomorphism with E in the early stages than Z set theory is. And in some cases it seems to me (though I'm not expert) that some of the complications that pop out of the alternatives might not be acceptable to detractors of F anyway, especially in the formal semantics.

I have been proposing to replace this isomorphy with an identity. Instead of having two systems, one in our mind and one on paper, one based in reality and one on a couple of seemingly arbitrary postulates--why not have just one reality-based system whose propositions can be expressed on paper as well as mentally grasped.

Yes, I am all for this. But what I've been saying is that when you understand what this actually requires in terms of formalization and mathematics, you realize that just proposing it is light years away from doing it or even being able to do it. Also, since mathematicians and philosophers have had this aspiration (basically speaking) for a long time, it behooves us to look at the history of mathematical logic and set theory to see what obstacles have been encountered already and how previous mathematicians have dealt with them.

Note: One exception I take with your remark is your use of 'identity' here. I think it would be better to say that what is desired is that F be isomorphic with E from the first axiom and through all stages or that F exactly express E from the start. The reason is that this formal system is what it is, which is a kind of description of reality, while the description is not the reality itself. The description is given in language, which is a part of reality, but is not identical with all of the reality it describes, since for example, the word 'chair' is not a chair.

My motivation for this is that, although the "two-system solution" has been wielded successfully for engineering purposes, it has been causing philosophical confusion.

I quite agree that mathematics, mathematical logic, and set theory give occasion for some profoundly perplexing philosophical problems. However, as to confusion, a great deal of it is not caused by these fields of study but rather by some people's misunderstanding of the studies.

[continued next post]

[LauricAcid]

Given that F, the only one of the two systems that people ever get to see, has a foundation that appears (to a naive person) to be just a set of arbitrary postulates, naive people can easily be led to question the validity of mathematics and, by transitivity, reason.

Yes, I see. Interesting point. However, I think you exaggerate the arbitrariness of the axioms. The axioms are arbitrary in the sense that we could adopt different sets of axioms and, roughly speaking, come up with pretty much the same results. But the axioms are not so arbitrary that they don't reflect common mathematical notions. Let's look at the axioms of Z set theory (given informal translation here):

Extensionality. Sets are equal that have the same members.

Subset. If you have a set x, then you have any subset of x such that the subset has just the members of x that also have a particular property defined by a particular formula. (And this axiom is formulated in a way that disallows a certain kind of self-reference that is exploited to derive Russell's paradox.)

Pairs. If you have sets x and y, then you have the set whose members are just x and y, which is {x y}.

Union. If you have a set x, then you have the set formed by taking all the members of members of x.

Power set. If you have a set x, then you have the set of all subsets of x.

Infinity. There exists a set x that has 0 as a member and if n is a member of x, then n+1 is a member of x. (Note: '0' and '+1' will have been defined by the time this axiom is stated. Notice that all this axiom asserts is pretty much what you alluded to in one of your posts: We can start with 0 and "generate" from it all the natural numbers by just adding 1 to each natural number in the "process" and, finally, arrive at a set that has all of the numbers thus "generated". Note, however, that this set might have as members other things that are not natural numbers. But from this we can prove that there exists a set that has just the natural numbers.)

Aside from first order logic itself and plain old identity theory, that's all there is to it. That's pretty much all you need to axiomatize mathematics for real analysis.

There are additional axioms, especially the axiom of replacement, the axiom of regularity, and the axiom of choice, but these are mainly needed for the parts of Z that you don't care about for E (except, for certain theorems, the axiom of choice is called upon in real analysis too).

So the axioms of set theory are not just arbitrary strings of symbols. But I still do not necessarily concede that one cannot rationally maintain a moderate formalist philosophy. While acknowleding that mathematical endeavors are most basically motivated by man's technological endeavors, and while also preferring an axiomatization for mathematics that is closest to "E", one need not reject that also we there is also intellectual benefit in the study of mere consequentialist systems.

Since E, the only system connected to one's senses and practical decisions, remains unformalized, sense perception and action can easily become divorced from logic in people's minds. Since the isomorphy between E and F has not been explored and validated in a widely publicized way, many people are unaware of it, so the duality between the "theoretical" system (F) and the "practical" one (E) helps promote the deadly mind-body dichotomy. It engenders rationalism in the classroom and irrationalism outside it.

I think you're too alarmist here. I don't think you give students of mathematics enough credit. At least my anecdotal impression is that most people do understand these relations. In fact, the main set theory text I use does include a discussion of the matter, even with grade-school style drawings to illustrate the point. Also, it is true that some mathematicians and philosophers of mathematics have adopted philosophical postions that you would deem to be rationalism, as well as some are avowed realists, but I don't know that use of the mathematics itself tempts one into philosophical views that one were not already inclined toward anyway. As to mind-body dichotomy, I'd have to take another look at some of the debates to recall how mathematical logic has influenced them. At least in the AI debates, results in mathematical logic have been brought in, but there have been withering critiques of this too.

From the facts that E exists and is isomorphic with a formal system, it follows that E itself is formalizable. A formalization of E would include all of F except for the "acrobatics" that F uses to achieve isomorphy with E. Instead of those, the formalization of E would contain a formalized expression of the corresponding relationships within E.

That would be fine, except the "acrobatics" are in deriving the isomorphism with E from just two primitives: identity and membership. If you omit the acrobatics, then it's not clear how you'll be able to have E without obligating yourself to tons and tons of primitives that will be quite difficult to manage. Also, another angle on this is that not only does F provide an isomorphism with E, but F provides the roads from which to travel among various parts of E. For example, E includes not just natural numbers, but also rational numbers and real numbers and systems of greater and greater complexity. F, with its sets, functions, bijections, etc, allows us to move among these systems so that they are thus unified, or, more fundamentally, they are made "commensurate" with one another by F.

Instead of 2 = {{},{{}}}, it would say something like:

a != b -> |{a,b}| = 2

Addition of natural numbers would be defined in terms of a union of disjoint sets:

A intersect B = {} -> |A union B| = |A| + |B|

(The commutativity of addition would thus follow from the commutativity of set union.)

But from where did we get the commutativity of set union? From F! And your formulas are just Z set theory anyway. Your definition of '2' is just a theorem of Z set theory. Your definition of cardinal addition is just Z set theory. Moreover, your definition of '2' might work as a definition in some other axiomatization, but the Z and the von Neumann construction of the natural numbers has an attraction that your construction might not provide. It is not even clear how you would state and prove that there is a set of natural numbers. The von Neumann construction allows each natural number to be defined in terms of the previous natural number. This accomplishes the "generating" of numbers in such a way that all of the properties of natural numbers, including mathematical induction, pour out as theorems as smoothly as honey out of a jar.

the axioms would be consistent with reality, and that, together with the fact that reality is internally consistent, would guarantee E's internal consistency.

Yes, any set of sentences that is a set of sentences true in some model is a consistent set. However, without my committing to some pretty complicated details here, you're going to run up against the second incompleteness theorem here. Granted the second incompleteness theorem is itself formalized in a meta-theory that is just a copy of the set theory you're objecting to. But I strongly suspect that still you're pretty much chasing a pot of gold at the end of a rainbow if you think you can have a sufficiently rich mathematics and also the kind of guarantee of consistency you expect to have for it.

Thanks for your post. I enjoyed some of the ideas you've presented and have enjoyed responding to its challenges.

Edited September 5, 2005 by LauricAcid

[LauricAcid]

Cantor worked on infinities a lot and he ended up as a nutcase.

For crying out, loud, Cantor went insane and struggled for years with it and he was a mystic (pace, those who claim they're the same ) and he may have had many personal flaws. So f-ing what? It has nothing to do with the mathematics, unless, of course, you think ad hominem is a valid form of mathematical argument.

But somehow once you look at infinity magically both groups are of the same magnitude.

We do look at infinty magically. We look at it mathematically, in completely rigorous formal systems, with axioms, primitives, and definitions. And we don't compare magnitude of groups, we compare sets. And the claim is one that is proven. There is a bijection from the set of naturals to set of squares of naturals.

There is a branch of mathematics that took out infinity completely and it could repeat all the mayor proofs even in the field of quantum mechanics.

Please provide a reference to it, preferably on the Internet, or at least to the literature that can be found in libraries.

Both the set of real numbers and the set of rational numbers spread into infinity both ways.

Yes, under a the standard ordering they do. But they can also be ordered to go in just one direction. As to your remarks about density, size, and magnitude (or size), I see what you're driving at, but since by 'equal size', mathematicians just mean the existence of a bijection, I don't see any harm in this or in speaking of size comparisons of infinities. Your concern with density is tied with orderings, since density is defined in terms of a given ordering. But a bijection does not depend on there being ordering that's already been given. I think your notion of density may give insight about a larger perspective and why certain bijections fail, but we don't need to even mention density to define 'equinumerosity', even between infinite sets.

There is no mathematical theorem that will allow you to compare infinities.

That depends on what we mean by 'compare infinities'. As I remarked, we can look for the existence of bijections between infinite sets. I think that's what most people would mean by saying they are comparing infinties (though, I agree that such is not good terminology). Also, upon adoption of the axiom of choice, we have the trichotomy law for cardinals, as well as that every set has a cardinality, so that any two sets x and y, exactly one of three hold: 1. x less than y. 2. y less than x. 3. x and y have a bijection between them. Edited September 5, 2005 by LauricAcid

[LauricAcid]

My exact words were: "However, the longevity of a system based on arbitrary axioms is determined by its practical use - if there is none, it won't survive. It will be dismissed as junk."

Fair enough. But since the von Neumann definition is such an important part of set theory for over eighty years, it follows from your assertion above that this amount of longevity attests to the practical use of the von Neumann definition.

The von Neumann approach became the standard approach because it makes so many theorems because since it is so very easy to grasp and conceptually elegant: each natural number is just the set of all previous natural numbers. As well as, the general idea suggests strategies that can be applied for other situations. Look at the kind of nesting that is used here and how the method applies the concept of inheritance. Talk about fountainheads. Post, Turing, Church, Godel, Curry, Markov, Kleene, Rosser, von Neumann, not to mention so many brilliant mathematicians before and after them. Are you kidding? Mathematical logic and set theory in the early twentieth century are crucial (I wonder what could even be said to be more crucial) in the conceptual development of modern computers. Hey, and until later, they weren't even trying to make computers! Their mathematics is so rich that the foundations of computer science just fall out like corollaries, like conceptual footnotes.

However, I doubt that this particular notation has much meaning in design of digital computers.

It's quite reasonable to doubt it. That's not at issue. Anyway, it's not a matter of notation, but of conception. That von Neumann's conceptions in set theory contributed to his insight into building computers has been remarked upon many times. As for doubts, I have little doubt that there is book that gives the history of computers that doesn't discuss the monumental work of von Neumann and his colleagues, especially in the transition from the ENIAC to the EDVAC. Here are a couple of links just for starters:

http://turnbull.dcs.st-and.ac.uk/~history/...on_Neumann.html

http://www.siam.org/siamnews/06-02/logic.pdf

I haven't had a chance yet to read the Martin Davis book, but I am looking forward to doing so. Davis is a real good source. Notice that even the book review mentions that von Neumann's work as a logician enabled him to provide the basis for these early computers. Integers, of course, being part of that.

At least at this moment, I don't propose to tell you the exact piece of hardware or the exact step in a flow chart that the definition of 2 comes into play. The definition is part of a larger theory, larger conception, and larger context. It would be ridiculous for me to isolate a passage from a book and say, "This sentence has no practical value" or to pick a number from a phone directory and then present to someone that it is just a random number without telling them it's the number to the best Thai food delivery restaurant in town. It's like the old George Carlin bit where he's the sports news anchor and he says, "And now for the baseball scores: 5 to 2, 8 to 4, 2 to 1, still in extra innings tied at 4 to 4, 7 to 4...".

I am curious as to how you would have someone prove to you a nagative assertion? If I was still convinced that 2 = {{}, {{}}} has no practical value, how do you suggest I would go about proving it doesn't? It is impossible - which is why only positive assertions need proof. That is the way I understand it.

Let's take it from the top: First you claimed there's no practical value. Then I asked you how you know that. Then you said you can't prove a negative. Then I said you can give support for a negative assertion, such as I can give support for the negative assertion that no oranges have pits by showing oranges that don't have pits (and it would help if they were from all over the world, by the way), or I can support the negative assertion that there are no pink unicorns by pointing to the fact that zoologists have looked virtually everywhere on the planet for different creatures and have not come across unicorns or even any trace of them. Now, around this time (a little before or after?) you mentioned that your own experiences in math and computing have not led you to contact with the formula let alone its use. Fair enough. Your experiences are some inductive evidence for your claim. And if that had been your first answer, then I would have responded that your experiences then have been too limited, especially since they have not included exposure to the standard set theoretic development of the naturals nor to some of the most famous history of computing such as the von Neumann architecture. Then you might have said, fair enough, and asked for some links, and I'd have given them to you, and this whole lousy argument would have been avoided (though I think some good things have come from the argument).

only positive assertions need proof.

Negative assertion: This house has no termites.

If you think that requires no proof, then here's my little radio skit:

[sound effect: Ringing phone]

"Hello, Sourcerer Termite Control."

"Hi, I need termite inspection for my house. How much is that?"

"That's three hundred and thirty five dollars."

"Fine, because I really need to find out if my house has termites."

"You don't have termites."

"What do you mean? How do you know my house doesn't have termites?"

"What do you mean, how do I know? I can't prove a negative, you know."

"What?! I just want to find out if I have termites. I don't understand how you know that the house doesn't have them."

"Look, pal, I can't prove a negative, and it's as simple as that."

"But I might have termites. You don't know that I don't have them."

"Oh, when you say you might, then you open the door for anything goes. When you say might you can then say any false thing you want. There's no might. Might is the arbitrary. There is only certainty."

"But how are you certain that I don't have termites in my house?"

"Like I told you, I can't prove a negative. Besides, I've looked at a few houses before, and they didn't have termites. So, by induction, your house does not have termites."

"Never mind, I'll call someone else."

"Okay, suit yourself. Meanwhile, I'll send you my bill."

"Your bill?! You didn't do anything!"

"Sure I did. I told you that you don't have termites. What's your address?"

"I'm not giving you my address for you to send a bill! That's ridiculous."

"That's okay. I have your address right here. It's John Doe, 100 Main St. Anytown, USA."

"That's not my address."

"Of course you say that since you don't want me to know your address, so if I tell you your address, you'll deny it, so it must be address. Hey, listen, if I'm wrong, prove it."

"I can't prove to you that that's not my address."

"See what I was talking about? You can't prove a negative."

"This is nuts. Good bye."

[sound effect: Phone hanging up]

[sound effect: Ringing phone]

"Sourcerer Termite."

"Yes, thanks, I need to find out if there are termites in a house I'm selling."

"You don't have any termites."

[sound effect: 'Twilight Zone' theme music]

The End.

I was speaking about formal systems, but not in the context of formal systems.

In the context of formal systems, you challenged whether I know what consistency is. That was silly. That's all. Anyway, I hope that our conversations can turn more profitable now, which is not to say that I think they've been entirely unprofitable so far. Edited September 6, 2005 by LauricAcid

[LauricAcid]

Self-correction: 'We do look at infinty magically' should be 'We do not look at infinity magically.' What a goof I made there!

[Hal]

What you can ask is how dense they are. It is because of density that the cardinality of real numbers (which is more dense) is greater. For set A to have greater density than B means that you can't form a bijection (transformation one-to-one and onto) from B to A. So a set with greater density has greater cardinality.

I dont know what you mean by 'dense'. I know what that word means within real analysis, and both the real numbers and rational numbers are dense in the sense that between any 2 members of either set lies a third. We can also show that the rational numberes are dense in the reals, meaning that between any 2 real numbers lies a rational (and vice versa). So saying that one set is more 'dense' than the others is meaningless unless you explain what you are trying to say.

Rather than considering the set of rational numbers versus the set of real numbers, we can compare the set of real numbers lying between zero and 1 (a finite interval) with the set of all rational numbers (going off to infinity in both directions). And the set of real numbers in this finite interval will still have a higher cardinality than the set of all rational numbers. I'm not sure how this relates to density, or your statement that "the finite is neglible in comparasion to the infinite".

Edited September 6, 2005 by Hal

[jrs]

LauricAcid already gave the proper proof that 0.999... = 1 .

But for those who are still unconvinced, let me offer this "proof":

0.999... + 0.999... = 1.999... because 9 + 9 = 18 and 1 + 8 = 9

1 + 0.999... = 1.999...

thus 0.999... + 0.999... = 1 + 0.999... because both equal 1.999...

subtracting 0.999... from both sides gives:

0.999... = 1

which is what was to be proved.

Forgive me if someone has already given this proof. I did not want to re-read the entire thread to check.

[source]

But they can also be ordered to go in just one direction.

Yes, but there is no need here to go beyond standard ordering.

I don't see any harm in this or in speaking of size comparisons of infinities.

There really isn't (except that the terminology isn't what a mathematician would use), if you are familiar with mathematical theorems concerning this. However, the context of my response to Felix (which I kept in my response to Hal) also included that he is misunderstanding the concept of infinity. Therefore, I was trying to explain to him that there isn't anything "magical" in it, by trying to explain his confusion (remember the sets he suggested) without much involving infinities. The problem with infinity is that when many people hear it, they tend to blank-out on it, and/or they drown in mysticism.

That depends on what we mean by 'compare infinities'.

Exactly what I said - compare infinities. If you have two infinite sets, you can't count how many elements each one has (like you would do with finite sets) and then compare the result. You must instead prove that there is/isn't a bijection from one infinite set to another to prove that one has/hasn't a greater cardinality than the other.

As I remarked, we can look for the existence of bijections between infinite sets. I think that's what most people would mean by saying they are comparing infinties (though, I agree that such is not good terminology).

You mean most people who studied higher mathematics?

Are you kidding? Mathematical logic and set theory in the early twentieth century are crucial (I wonder what could even be said to be more crucial) in the conceptual development of modern computers.

I didn't say that mathematical logic and set theory weren't important. I only said, and apparently I was wrong, that 2 = {{}, {{}}} has no practical value. I have never seen it, and I studied both - logic and set theory - and I applied both. When I saw it in your post, I thought it was some kind of a joke. Still, I may encounter it in my further studies.

[source]

Now, around this time (a little before or after?) you mentioned that your own experiences in math and computing have not led you to contact with the formula let alone its use. [...] And if that had been your first answer, then I would have responded that your experiences then have been too limited, especially since they have not included exposure to the standard set theoretic development of the naturals nor to some of the most famous history of computing such as the von Neumann architecture.

It would be nice if my first response could have been that - but it couldn't. For starters, I thought the formula is come from your imagination. I've never seen it or heard about it and I couldn't have concluded that it has to do something with computing, especially because my response was to Capitalism Forever and I haven't read your previous posts.

If you think that requires no proof, then here's my little radio skit:

Haha! That was great, thanks. Indeed, even a negative assertion must have some basis. However, this doesn't mean you can prove it beyond all doubt. A positive assertion, however, can be proven - beyond all doubt.

So saying that one set is more 'dense' than the others is meaningless unless you explain what you are trying to say.

I have given you the exact definition of this. If you don't understand, consider this: square root of 3 is a member of real numbers, but not a member of rational numbers. Yet, it is larger than 17/10 and smaller than 18/10, larger than 173/100 and smaller than 174/100, etc. It is here somewhere, in between of all the rational numbers.

For a set to be dense means that there are infinite elements of that set between any two given elements of that set (so when I was speaking of natural numbers and their squares, I should have said that they are equally rare, although not even this is the correct mathematical terminology - I should in fact be speaking of cardinality - so consider this a correction. Note that the two sets are both discrete).

[Capitalism Forever]

Yes, I am all for this. But what I've been saying is that when you understand what this actually requires in terms of formalization and mathematics, you realize that just proposing it is light years away from doing it or even being able to do it.

That's correct of course. I'm aware that this is not a one-day job. If it were, I would have done it already.

As for being able to do it, I am confident that it is possible in principle, i.e. that a formal language can be powerful enough to express the facts that anchor E to reality. I believe what has kept mathematicians from doing so is that their approach has been rationalistic: They've been striving to minimize the number of axioms, motivated by the consistency concerns you described earlier. In this "top down" approach, you start with a small number of basic truths, derive general theorems from them, and finally you apply the general theorems to specific cases. The integration of the various sub-disciplines is ensured, as you said, by having a common starting point for all of them.

Objectivist epistemology goes "bottom up" first, and then "top down." Any correct observation of a fact of reality can serve as an input for a theory (an "axiom" in a very broad sense). From observations of concretes, general principles are derived by means of abstraction and induction; then the general principles are applied to other concrete cases. Integration of various areas of thought means finding commonalities among them and forming further abstractions and general principles.

A good example for the latter would be the way the concept ring has been abstracted from the sets of integers, matrices, polynomials, etc.

(For a formal system, the inputs would not be limited observations of concretes, as such a limitation would indeed make formalization impossible. A mathematics constructed this way would mostly be based on "observations" of concepts like "set" and "natural number," and relationships between them, such as "if a set has the property finite, then it also has a property size whose value is a natural number.")

However, I think you exaggerate the arbitrariness of the axioms.

I wouldn't call the axioms arbitrary myself, but they may seem arbitrary to people who are not sufficiently familiar with the isomorphism between E and F, and a philosopher wishing to undercut reason can exploit that apparent arbitrariness.

While acknowleding that mathematical endeavors are most basically motivated by man's technological endeavors, and while also preferring an axiomatization for mathematics that is closest to "E", one need not reject that also we there is also intellectual benefit in the study of mere consequentialist systems.

I have found it fun to study the set theoretic derivation myself--and it must have been even more fun to author it!--so I have to concede that. It's OK as long as it is treated what it is: a hypothetical universe where everything is a set, created in one's imagination for the purpose of exercising one's brain--but not meant as a statement of fact.

"If each natural number were a set of the previous natural numbers, ..." -- fine;

"Each natural number is a set of the previous natural numbers, therefore..." -- no please.

Thanks for your post. I enjoyed some of the ideas you've presented and have enjoyed responding to its challenges.

I would like to thank you for your input also, and for prompting me to think about the issue in the first place.

[LauricAcid]

Thanks for your comments, source. (I'm glad you didn't take offense at the skit.)

Indeed, even a negative assertion must have some basis. However, this doesn't mean you can prove it beyond all doubt.

There's a couple of problems with that. 1. We need to define 'negative statement'. 2. Some statements of the form 'No F are G', which is the basic form of the negative statements we were talking about, can be proven. For example, though it's trivial, we can prove that no circles are squares. Granted, this is a mathematical, not empirical (oops! I just expressed the forbidden schism) statement. But I wonder, if we're given a finite domain for empirical questions (I take it that in this context it is agreed that the physical universe is finite), then by examining a finite number of existents we do have a decisive proof. If every physical location in the universe were examined and no unicorn were found in any physical location, then we conclude, decisively, that there are no unicorns. Granted, this doesn't account for time, since we would have to examine all future space-time points and all past space-time points too, which I don't know to be possible. And even if one allowed that the physics of the universe might be otherwise, I can't imagine what would even be the meaning of speaking of something being empricially possible to prove in principle in some other possible physical universe but with its physics different from the real universe. So, yeah, I guess my admittedly amateur and convoluted ruminations here do end up agreeing that certain empirical negative assertions cannot be proven decisively.

I thought the formula is come from your imagination.

That would be nice, wouldn't it?

For a set to be dense means that there are infinite elements of that set between any two given elements of that set

That's a consequence of this kind of density. But not only are there an infinite number of reals between any two rationals, there are an infinite number of rationals between any two reals. So I don't see how to use density to explain the difference in cardinality. (But, I think you've now dispensed the density argument anyway?)

Note that the two sets are both discrete

What do you mean by 'discrete'? Do you mean disjoint? If so, it's not required for sets to be disjoint to compare their cardinality. Edited September 6, 2005 by LauricAcid

[LauricAcid]

if a set has the property finite, then it also has a property size whose value is a natural number.

That's pretty good. But, if you have the power set axiom, and you allow for the existence of the set of natural numbers, then, whether you call it 'size' or not, you're going to have uncountable sets. Is that okay with you? If not, then something's gotta give. You have to either trash the power set axiom or trash the existence of the set of natural numbers. If, after having made that choice, you come up with alternative axioms to derive mathematics, then more power to you.

the isomorphism between E and F

Remember that I don't want to be misunderstood to be overstating that F is isomorphic with E all the way. In the early stages F is not isomorphic with E, and in later stages, with transfinites, F goes way outside the isomorphism. What's isomorphic are certain sections, let's say modules, or less charitably, pockets.

a hypothetical universe where everything is a set, created in one's imagination for the purpose of exercising one's brain--but not meant as a statement of fact.

This is not only an exercise, but a means to organize concepts, and for various ends ranging from pure abstract interest to eventual practical application. And I appreciate that your criticism is focused so that it is against realism and not just mathematics in general. And for a better understanding of your philosophical adversaries here, I highly recommend Hilary Putnam's little book (I forgot the title) and the writings of W.V.O. Quine. These are a couple of brilliant philosophers and mathematicians, with whom you are in fundamental disagreement, so that reading them will give you chance, as it were, to sharpen your ax against, or to spar with, the very best. Another realist is Godel, but I don't think he'd be a good foil since he's so idiosyncratic and doesn't present much as a polemicist.

"If each natural number were a set of the previous natural numbers, ..." -- fine;

"Each natural number is a set of the previous natural numbers, therefore..." -- no please.

You've summarized the difference between realism and consequentialism. And I do think that consequentialism is a reasonable view. An interesting book (and a rather bold broadside) is Chiara's 'Ontology And The Vicious Circle Principle'. It includes some lighthearted thinking about consequentialism. And you might appreciate the vigilance against impredicativity (here, backed up by a system by Hao Wang). Also, the great mathematician Haskell Curry's little book (I forgot the name) gives a beautiful explanation of his philosophy of moderate formalism.

[source]

So, yeah, I guess my admittedly amateur and convoluted ruminations here do end up agreeing that certain empirical negative assertions cannot be proven decisively.

That would be nice, wouldn't it?

No! If it was, who knows what computers would be like today. Possibly several decades behind in technology. I couldn't stand that!

That's a consequence of this kind of density. But not only are there an infinite number of reals between any two rationals, there are an infinite number of rationals between any two reals. So I don't see how to use density to explain the difference in cardinality. (But, I think you've now dispensed the density argument anyway?)

Yes. You can see my original definition of what it means for a set to be denser than another set, is in fact the same as the definition which tells which set has greater cardinality. I made a loophole somewhere as I tried to explain infinite sets without using infinity, and I should have used cardinality instead.

What do you mean by 'discrete'? Do you mean disjoint? If so, it's not required for sets to be disjoint to compare their cardinality.

Set S proper subset of X is descrete in X if every x element of S has a neighborhood U subset of X, such that S interjection U == {x}.

So, both of Felix' sets are discrete in R.

If set S is discrete, then it is certainly not dense, so I misapplied the term "density" in my previous posts, which is what I was trying to get across.

Edited September 7, 2005 by source

[Capitalism Forever]

That's pretty good. But, if you have the power set axiom, and you allow for the existence of the set of natural numbers, then, whether you call it 'size' or not, you're going to have uncountable sets. Is that okay with you?

Yes, it is okay with me. I require finite sets to have natural numbers as their sizes, but I do not require every set to be finite. The sizes (or cardinalities) of infinite sets are expressed by cardinal numbers, which are a superset of natural numbers.

Remember that I don't want to be misunderstood to be overstating that F is isomorphic with E all the way. In the early stages F is not isomorphic with E

That's pretty clear.

and in later stages, with transfinites, F goes way outside the isomorphism.

Well, it's certainly true that transfinites do not play much of a role in our everyday applications of mathematics. But a formalization of E would be capable of expressing them, in terms of their relationships with concepts like set, bijection, and order.

You've summarized the difference between realism and consequentialism. And I do think that consequentialism is a reasonable view.

Remember, though, that what I am really for is objectivism, which means that I think that mathematical objects should refer to facts of reality (more specifically, to the values of the attributes of and relationships among physical objects as well as man-made institutions) but, unlike realists, I do not believe that they exist in their own right, apart from the physical objects and man-made institutions that give rise to them. Nor do I think that all mathematical theories automatically will refer to facts of reality--as exemplified by the ominous 2 = {{},{{}}}.

Now, a logical argument, including a formalized theory, does tell us what the consequences of its premises are. If natural numbers are taken to be sets of the previous numbers, then you can derive the consequences of such a hypothesis. The derivation itself will be a fine piece of logic--but, since the premises do not correspond to reality, it cannot be said to be an objective theory.

Thanks for the book recommendations!

[LauricAcid]

You can see my original definition of what it means for a set to be denser than another set, is in fact the same as the definition which tells which set has greater cardinality. I made a loophole somewhere as I tried to explain infinite sets without using infinity, and I should have used cardinality instead.

I'm lost as to what you're claiming. Are you saying that the reals are denser in the reals with respect to the standard ordering than the rationals are dense in the reals with respect to the standard ordering? What does 'denser' mean here? If you mean that between any two reals there are more reals than there are rationals, then that's true and one might use that fact in connection with comparing the cardinality of the reals with the cardinality of the rationals. But first, you have to prove the fact, which is a cardinality proof anyway. Second, there are cardinality comparisons one can make without mentioning density or even any ordering at all.

So what is your definition of 'denser' and how do you define 'greater cardinality' in terms of it?

Set S proper subset of X is descrete in X if every x element of S has a neighborhood U subset of X, such that S interjection U == {x}.

Hold on a minute, pardner. We're gonna get real messed up if we're not clear what we're talking about here: topological spaces in general, metric spaces, or what. And you've got stipulations about 'proper subset' and 'U subset of x' that I don't know that we need. Also, 'interjection' should be 'intersection'. Anyway, I suggest we make things easy on ourselves by first talking specifically of the set of real numbers. Then, after we see how your ideas work out in the specific case before we generalize to other topological spaces. Here's a definition that should work for our present purposes:

Let 'e' stand for 'is an element of'

Let 'A' stand for the universal quantifier

Let 'E' stand for the existential quantifier

Let R = the set of real numbers.

Definition: i is an R-interval around x iff

(i is an interval in R and

x e i and

Ey(y e i and y < x) and

Ez(z e i and z > x))

Definition: S is discrete in R iff

(S subset of R and

Ax(x element of S -> Ei(i is an R-interval around x and i intersect S = {x}).

Now, the set of natural numbers is discrete in R. The set of primes is discrete in R. The set of square natural numbers is discrete in R. But the set of rational numbers is NOT discrete in R. Yet the set of natural numbers, the set of primes, the set of square natural numbers, AND the set of rational numbers share the same cardinality. So I don't see how discreteness contributes to inferring anything about cardinality here.

Edited September 8, 2005 by LauricAcid

[Unconquered]

I'm not a mathematician but I have an eclectic self-education in various mathematical areas including one class in point-set topology. With that said ...

I'm curious, primarily as a question to LauricAcid but also to anyone familiar with "set theory": If the (mathematically infinite) set of numbers in any real number range is claimed to be equivalent to any other real number range, how does set theory distinguish such mundanities as the actual difference between - say - the real numbers between 1.0 and 2.0 (i.e. [1.0,2.0] ), and, say, 1.0 and 1000.0 (i.e., [1.0,1000.0]), since both are ranges between two different real numbers and there is clearly, out in the real world, a difference between these two ranges?

I've always found it a bit disturbing that it's claimed that both ranges contain an "equivalent infinite set" of real numbers, when they are clearly, actually, different in some measurable way. I tend to think that the answer lies in the metaphysics: in reality you cannot actually subdivide forever.

But, I am still interested in reading thoughts about this.

[LauricAcid]

a formalization of E would be capable of expressing [transfinites], in terms of their relationships with concepts like set, bijection, and order.

But if you allow transfinites, then aren't you out of E? If you say, well, transfinites are a part of E but they're just abstractions, then I don't see how your system is significantly different from F. The distinction between regarding these mathematical objects as real, or abstractions, or methods of concept, or as just meaningless symbols is not a distinction found in the system itself but rather in how we decide to regard the system. If your system can't suppress transfinites, and you explain them away as being abstract offshoots of an empirically based foundation or as methods of concept or whatever, then subscribers to F can just say the same thing about F.

I'll reinforce that. If you allow the axiom of infinity, then F may as be as empirical as any alternative you devise.

First start with identity theory, which has two axioms: (1) x = x. (2) If x = y, then all properties of x are properties of y.

Then, pick some physical entity - whatever entity you like. Call it not 'the empty set' but instead call it 'the base object', and as such let's call it not '0' but rather 'O'. Then we form sets such as {O} which corresponds to counting the base object once, which corresponds to the number one, then {{O} O}, which corresponds to counting one plus the base object, which corresponds to the number two, then {{{O} O} {O} O}, which corresponds to counting two plus the base object, which corresponds to the number three, etc.

So we notice that we're making sets by gathering finite numbers of already formed things (first a physical object, then sets formed with that physical object). So we state an axiom that says right up front that this is what we're doing. But we see that we don't need an axiom for gathering any number of finite things, but only for gathering just two things as long as we have a union axiom (which we'll state below). So we have a pairing axiom.

And, by the way, if you want any other physical entities represented in this system, then you're welcome to have them. They won't hurt a bit, even though we won't need them to build the mathematics.

And we empirically observe that for any set we form, we can form another set by gathering all the members of members of the original set. So we have a union axiom.

And we empirically observe that we can form the set of all subsets of any set So we have a power set axiom.

And we empirically observe that for any set, we can take subsets of that set defined by different properties. So we have an axiom for this. And we make sure that this axiom doesn't allow us circularity or self-reference in the form of defining a property that refers to the set that is being defined by that property. So we have the axiom of separation, so called since we're forming sets by separating them out, on the criteria of a defined property, from an already formed set.

And we empirically observe that any two sets are the same if they have the same members. So we recognize this with an axiom. So we have the axiom of extensionality.

Now, except for the axiom of infinity, that is ALL of Z set theory. That's all there is to it. Just these simple, and if you will allow, empirically arrived upon axioms.

But now, empirically, we see that we can keep increasing the set we started with the base object: Take the previously formed set, then make a new set that has as members all the members of the previously formed set as well as the previously formed set itself. Each of these sets has exactly the number of objects that are in the count of how many times we applied this process. And this is just an abstraction of the very process of counting. Counting physical objects is a process, just as each time we count something, we move our eye to the next object, or we move the already counted object out of the pile of objects that remain to be counted. So, just as we have a physical process, we have an abstract, or set theoretical process, and the process gives us counts of the number of times we apply the process, just as moving coins out of a pile gives us a count of how many times we move a coin from the pile, which is just what it means to count coins. So not only do the sets thus constructed have exactly the number of elements as the cardinalities of the numbers the sets represent, but the number of times the construction process is applied is exactly the cardinality of the number represented.

(So, now that I think of it, I may have conceded much too much by saying that F is not "isomorphic" with E from the start.)

So, as we empirically see that we can keep increasing the set of natural numbers we get with the process of forming them, we state an axiom that there is a set that has as among its members all the natural numbers, which is any number that can be formed by the process we started by counting that one base object. So we have an axiom of infinity. (Then, with our other axioms we can prove that there is a set having all and only the natural numbers.)

That's all. That's all the axioms of Z set theory. Why in the world certain Objectivists denounce this I really do not understand. I can understand the Objectivist opposition to realism. Lots of mathematicians and philosophers of mathematics reject realism. But Objectivism does embrace that we form concepts, and abstractions, and what Objectivism calls 'concepts of method'. So it makes no sense to me that the abstractions of set theory are such a bugbear for so many Objectivists.

And now that I've thought about this a little more, I realize that the reason I conceded too much before and the reason mathematicians take all this for granted, is indeed the fact that it is so very true that our mathematical intuitions are formed and guided by empirical observation. This is to say, the axioms of set theory are just so simple and so obviously parallel to everyday thinking that there's no purpose for a mathematician to belabor the point. I guess I just never had a chance to worry about accepting these kinds (though not necessarily the particular ones now in discussion) of abstractions since they made immediate sense to me even as a very young child. And what's ironic is that I wasn't even interested in mathematics as a child, and I don't have any special aptitude for mathematics, and I'm not even any kind of mathematician, not even an amateur one, as an adult. To me, though, the kinds of very simple abstractions such as seen in the axioms of set theory hardly require empirical justification since I had already pretty much digested their empirical basis even before I heard about the formulas. Someone tells you, okay, there's a set that has members that themselves have members, and so you can form the set of all these members of members. You empirical experience with things like boxes within boxes doesn't even have to be consulted explicitly. You just get it immediately: Yeah, all the things that are in the small boxes that are inside the big box. What is there to fuss about? Nothing; that's what.

Okay, so there is a fuss about the axiom of infinity. Fine. But it's not such a big deal really. Again, when you're in grammar school, even if the teacher doesn't mention, you start to become aware that there is not just 1, 2, 3, and whatever other numbers you've had in your homework, but that these numbers go on without end and that you can talk about them at once referring to the set of all of them. You can form the concept. It makes fine sense. You realize that to say that 3 is a member of the set of natural numbers is not much more than a fancy way of saying that 3 is a counting number.

But then you do learn about Russell's paradox. And this does tell us that making sets based on just any definable property leads to contradiction. But then we see that by restricting the formation of subsets with the demand that the defining property not refer to the set being formed, we see that, as far as has ever been tested in nearly a hundred years of unimaginably intense scrutiny, no contradiction is derived. So we keep our fingers crossed that no contradiction will be derived, meanwhile developing rich, beautiful, fascinating, and useful mathematics.

If I recall correctly, you remarked that consistency is ensured if the mathematics is built from the bottom up as based on reality. This is true, and the principle can be generalized. Any theory is consistent if it has a model. The model can be the real world or the model can even be a fictional, abstract, or "hypothetical" world. Now, if mathematicians knew of a theory for all of mathematics built on a model such that there is an effective method to check whether a formula is an axiom, then that would be THE mathematical theory. No question about it. Mathematicians would accept that theory and dump set theory in a New York second. If mathematicians knew of such a theory, then set theory would be nothing but history.

But there is no such theory that anybody knows. Not only is there not a known theory based on a model of physical reality or based on a model of physical reality plus Objectivist concepts formed on physical reality, but there is no known theory (for mathematics rich enough to express analysis) based on any model - phsycial, physcial plus concepts, fictional, abstract, or hypothetical - whatsoever. The best we have are models that we can show only if we assume a stronger theory than the one we're using. For example, we can show a model of number theory by constructing the model in the bigger theory that is Z set theory. And we can construct a model of Z set theory by constructing it in set theories that assume even more than Z set theory does. But we don't know a way to first have the model, then form a theory that is the set of sentences true in that model, as the theory is rich enough for real analysis and we have an effective method to check whether a formula is an axiom. And, roughly speaking, the second incompleteness theorem tells us that to have such a theory is MATHEMATICALLY IMPOSSIBLE.

Edited September 8, 2005 by LauricAcid

[LauricAcid]

If the (mathematically infinite) set of numbers in any real number range is claimed to be equivalent to any other real number range, how does set theory distinguish such mundanities as the actual difference between - say - the real numbers between 1.0 and 2.0 (i.e. [1.0,2.0] ), and, say, 1.0 and 1000.0 (i.e., [1.0,1000.0])

The expanation is that we must not conflate cardinality with distance. Instead of 'range' (which has another meaning), let's talk about the distance between 1 and 2 as opposed to the distance between 1 and 1000. Distance on the reals is given by a simple function: The distance between two reals is the absolute value of subtracting one from the other. For example, let d = the distance function. So d(1, 2) = |1 - 2| = 1 and d(1, 1000) = |1 - 1000| = 999. You are correct that the cardinality of the interval [1 2] = the cardinality of the interval [1 1000], but cardinality of a set is not the same as the distance between the endpoints of intervals. The distance doesn't count the number of reals in the interval (obviously, there are an infinite number of reals in the interval that has a distance of 999), so it would be a non sequitor to infer that the number of reals in any interval must correspond with the distance of that interval or even that the number of reals in two intervals of different distance may not be the same cardinality. Edited September 8, 2005 by LauricAcid

[source]

I'm lost as to what you're claiming. Are you saying that the reals are denser in the reals with respect to the standard ordering than the rationals are dense in the reals with respect to the standard ordering? What does 'denser' mean here? If you mean that between any two reals there are more reals than there are rationals, then that's true and one might use that fact in connection with comparing the cardinality of the reals with the cardinality of the rationals. But first, you have to prove the fact, which is a cardinality proof anyway. Second, there are cardinality comparisons one can make without mentioning density or even any ordering at all.

It was an error! I don't know how much more explanation you need to understand that.

Also, 'interjection' should be 'intersection'.

Thanks for the correction. I don't know if you read that post in particular, but I mentioned that I need to translate all mathematical terms from my language into english before I can use them. This is very delicate, especially because dictionaries usually don't translate mathematical terms into mathematical terms. As for this particular term, I must have misread it on Mathworld.

So I don't see how discreteness contributes to inferring anything about cardinality here.

You are constantly losing context. Look back! I mentioned that Felix' sets are both discrete because I was pointing to my error in the usage of the term density! Discrete sets aren't dense. Instead of speaking of density, I should have been speaking of cardinality, for the term is valid for both - dense and discrete sets - and if I did that, then there would have been no confusion.

As for my original idea, of what exactly I meant about R being denser than Q.

Q is a proper subsed of R. This means that R has all the members of Q, and some more. Square root of 3 is an example. Moreover, between each two members of Q there are infinite members of R. Remembering these two premises, it is a purely empirical (not mathematical) conclusion that R is denser than Q.

[Capitalism Forever]

But if you allow transfinites, then aren't you out of E? If you say, well, transfinites are a part of E but they're just abstractions

Just to clarify what we mean by "abstraction": The concept "transfinite number" is an abstraction. A transfinite number is a concrete--i.e., not an abstraction. (Similarly, the concept "natural number" is an abstraction but each natural number is a concrete.)

then I don't see how your system is significantly different from F.

It is different at the foundation, in two ways as far as natural numbers are concerned:

• In F, you form natural numbers by counting specific things: a "base object" and zero or more of your previous counts. In E, natural numbers are formed by recognizing that you can count any discrete objects: 2 = |{a, b}| where a and b can be anything as long as they're not the same thing. In F, the definition of 2 "binds" it to the empty set and to the singleton set containing the empty set. In E, the variables are unbound in the definition and can be bound when applying the definition to whatever objects you wish to apply it to.
  
  
• In E, it is a theorem that 2 != {{} {{}}}. 2 is what you think of when you count any objects a and b such that a != b. {{} {{}}} is what you think of when you think of {} and {{}} together. The two are not the same, just like the sky and the color blue are not the same.
  

It is true that you can express F within E if you wish to, except that you will have to use different symbols, such as:

• 2' = {{} {{}}} or
  
  
• 2 ~ {{} {{}}} or
  
  
• 2 = |{{} {{}}}| or
  
  
• F(2 = {{} {{}}})
  

The reason you can express F within E is that F is a formal language, and E is capable of expressing formal languages, just like it is capable of expressing natural numbers &c.

If your system can't suppress transfinites, and you explain them away as being abstract offshoots of an empirically based foundation or as methods of concept or whatever

In my system, transfinites are just what natural numbers are: cardinalities of sets. The only way in which natural numbers are closer to "empirical" is that a natural number that is relatively small can be the cardinality of a set made up exclusively of physical objects, while a transfinite number cannot--but then, neither can an ultra-gigantic natural number, so the line isn't even drawn at the boundary between naturals and transfinites.

then subscribers to F can just say the same thing about F.

What I can say about my system but they cannot say about F is: In my system, all mathematical objects that also play a role in our everyday thinking are identical to what they are in our everyday thinking. In my system, the 2 of mathematics is the same thing as the 2 of my accountant and the 2 of the software I write and the 2 of the Japanese gentlemen who engineered my car and 2 of the pilot who will fly me to London tomorrow. The 2 of F is the isomorphic image of all these, which is just as good for most purposes, but still it's something different--which can lead to unnecessary confusion, especially in the field of philosophy.

If I recall correctly, you remarked that consistency is ensured if the mathematics is built from the bottom up as based on reality. This is true, and the principle can be generalized. Any theory is consistent if it has a model. The model can be the real world or the model can even be a fictional, abstract, or "hypothetical" world.

Not so fast! Reality is internally consistent, but a fictional world is not necessarily so. The guarantee of consistency applies only when you have a real model, or an imaginary model you already know to be consistent.

[LauricAcid]

source, as I suggested, I admit that it was not clear to me what you were asserting and what you were retracting.

It is true that between any two members of Q there is an uncountable number of members of R, and between any two members of R there is a countably infinite number of members of Q. If that's what you mean by the density comparison, then okay. But I have no idea what you mean by saying that there is an empirical but not mathematical conclusion to draw. All of these things about density, cardinality, etc. are defined and proved in an axiomatic deductive system. Whatever empirical inferences you feel there are to be made do not affect that the theorems themselves are proven mathematically/deductively.

[LauricAcid]

Capitalism Forever,

There is no such thing as an inconsistent model. Consistency is not even a property of models. Speaking of a 'consistent model' or an 'inconsistent model' is like speaking of a 'dialectical tuba' - the adjective just doesn't apply. So the point I made is that since it is not mathematically possible to form a sufficiently rich set of decidable axioms from a model, a fortiori, it is not mathematically possible to form a sufficiently rich set of decidable axioms from a model that meets your philosophical requirements.

Just to clarify what we mean by "abstraction": The concept "transfinite number" is an abstraction. A transfinite number is a concrete--i.e., not an abstraction. (Similarly, the concept "natural number" is an abstraction but each natural number is a concrete.)

I'll keep in mind that that is your useage. Logicians would put this differently. In this instance, 'is a transfinite cardinal' is English for a certain formal predicate symbol, and one then may attempt to prove for given x that x does or does not have the property of being a transfinite cardinal. If a given x can be proven to have properties that make it unique so that it can be distinguished from all other y, then this x can be named as an object, and if it is proven unique, named as an object, and has the property of being a transfinite cardinal, then we've named a particular transfinite cardinal.

In F, you form natural numbers by counting specific things: a "base object" and zero or more of your previous counts. In E, natural numbers are formed by recognizing that you can count any discrete objects: 2 = |{a, b}| where a and b can be anything as long as they're not the same thing.

You keep missing my point here. Z also has the representation of cardinality you've mentioned, but you're method in E of defining each natural number via cardinality requires an infinite number of definitions, one for each natural number, and you haven't shown how that can be done in a formal system. What you're missing is the successor function. With the successor function you only need to define one natural number, then the rest of them are defined as successive successors. Related to this, you haven't shown how you define the set of natural numbers by giving a finite statement. All of the statements of Z are finite statements (i.e., the linguistic expression itself is a finite expression) or they are finite statements of expression forms (axiom schemata). But you've not given a clue how you would define 'is a natural number' in a finite expression. You can't just say, "Well, see how I made 1, 2, and 3. Just make the others in the same manner." That expression - "in the same manner" - must be formalized. That's what Z does with the von Neumann construction but what you haven't hinted at with E.

In F, the definition of 2 "binds" it to the empty set and to the singleton set containing the empty set. In E, the variables are unbound in the definition and can be bound when applying the definition to whatever objects you wish to apply it to.

I know you're using 'bound' and 'unbound' informally here, but you should find different words, since 'bound' and 'free' have a very definite technical meaning that should not be confused with your informal sense (in the technical sense, your variables in the definition of 2 will be, must be, bound).

It is true that you can express F within E if you wish to, except that you will have to use different symbols, such as:

2' = {{} {{}}}

Fine. Consider it done. From now on, in every set theory or math text you read, cross out '2' and replace it with '2''. Or, to save eraser and lead, just posit a general rule that every numeral for a natural number is replaced by a numeral from a different character set or in a different font. At this point your objection reduces to an objection about set theory using the standard numerals. But whether set theory uses standard numerals or replaces them with numerals from another character set doesn't make a bit of difference to the mathematics.

In my system, transfinites are just what natural numbers are: cardinalities of sets.

In your contemplated system. Anyway, Z has transfinites that are cardinalities and also transfinites that are ordinals. Just as I mentioned that you haven't shown a way to define 'is a natural number', you should see that it is going to be even more difficult for you to do the same at the infinite level, which would be to definite transfinite cardinals without first defining the ordinals. Edited September 8, 2005 by LauricAcid

[source]

All of these things about density, cardinality, etc. are defined and proved in an axiomatic deductive system.

Yes and I know that. However, it is empirical (or better yet, intuitive) to draw a conclusion from only the premises I mentioned in my last paragraph (using of course, no formal method). I don't know how much of a mathematician Felix is, so I thought it better not to get too deep into mathematics in my reply to him. Heh, but then I had to answer for it to someone who knows mathematics, and you know the rest of the story.

[LauricAcid]

Got it. Thanks.

[Capitalism Forever]

There is no such thing as an inconsistent model.

You realize, though, that there is such a thing as an inconsistent fiction. If your axioms are based on a fictional model, you don't have the guarantee of consistency among them that using reality as a model offers.

Z also has the representation of cardinality you've mentioned, but you're method in E of defining each natural number via cardinality requires an infinite number of definitions, one for each natural number

It does not. You can define the natural numbers from zero to ten and then define the concatenation of a natural number and a digit (0-9) as ten times the number plus the digit. (It would suffice to define them up to 2 and use binary notation, but what I am after is not a minimalistic system but a formalization of E.)

Related to this, you haven't shown how you define the set of natural numbers by giving a finite statement.

I don't define sets. I define concepts, such as the concept "natural number." And I needn't enumerate all units of a concept in order to have a definition. Nor do I need an enumeration to specify a set; once I have defined "natural number," I can use set notation (say, something like: { natural number | all }) to refer to the set of all natural numbers.

But you've not given a clue how you would define 'is a natural number' in a finite expression.

Read my posts more carefully.

Fine. Consider it done. From now on, in every set theory or math text you read, cross out '2' and replace it with '2''. Or, to save eraser and lead, just posit a general rule that every numeral for a natural number is replaced by a numeral from a different character set or in a different font.

Well I don't care how you distinguish the two symbols, but the point is that the 2 of E and the 2 of F refer to two different objects.

At this point your objection reduces to an objection about set theory using the standard numerals.

Have a cup of coffee and review what I wrote. I said that a formalization of E was capable of expressing F (that is, capable of expressing the fact of reality that there is a formal language with these-and-these axioms) but that when expressing F as a part of E, you have to distinguish the symbol 2 of F from the symbol 2 that is already used in E to refer to the size of any set {a, b} where a != b. So my statement presupposes that mathematics has been formalized my way and now I'm formalizing the old formalization within the new system just to show that I can.

But whether set theory uses standard numerals or replaces them with numerals from another character set doesn't make a bit of difference to the mathematics.

The difference is that what I'm talking about here is a formalization^2, i.e. a formalization of a formal system within another formal system, not the formal system itself.

Z has transfinites that are cardinalities and also transfinites that are ordinals.

Correct; I should have written: "Cardinals are just what natural numbers are: cardinalities of sets."