A. 1 through 9, are numbers, why? — Qwex

Two things there ain't. One, there ain't no sanity clause. Two, there ain't no why.
They're just ideas that have a wide range of uses. For convenience, they're called numbers.

Okay, thanks for your input.

May I ask that you read the 'shadow argument' I edited into the original post?

If you were imagining shapes, a finger is a finger number more than a 1 number, it's that simple.

Mind is, evalutation as well as observation, of shapes in nature amongst other things.

How you formulate a shape is very partially, by drawing a line and at minimal you can formulate dots to help you, that is why we use a rule to draw a straight line, we connect two points. You can imagine two dots and if powerful enough, connect them. Imagining a illusion of a line. This line can take us into a powerful thought network.

The multi-verse may just as well exist in the background of our universe.

It seems to me you're looking for an origin of these things outside of mind, as if delivered by a Moses magically carved in tablets of stone. One matures out these ideas. If you do not, and until and unless you do, you're stuck.

Try indicating quantity without numbers. Or, just try indicating quantity. How many sheep are in your flock? Small, medium, large (indicated by holding hands apart)? Your guess as to how numbers came into use as good as mine. And I think that is it.

Modern physics is finding that the universe seems to be in some weird ways mathematical - but that isn't how they came into use. Stay tuned, maybe the final word on that within fifty years, maybe. Here's a teaser video:
https://www.youtube.com/watch?v=qTx98PUW6lE

Numbers are names for quantities.

I would say all mathematics, non regarding of the basic up, left, right, left, is impartial. I'm not saying it's bad, but it's language holds no bearing over purer symbol.

I need to create symbols, I do, why would I think in symbols that are base 4?

Why not base 8 which would sum up base 4 as 08? Also any symbol or pure symbols ' I am working', per se.

I think our language is bad but lexscribed well, I like more a language like a whisper. Short words, pauses.

I think N is 0.8

If there is a natural number then there's the natural golden number which proves N, which is 0.8, on base 4 you move at 0.8, and on base 8 you move at 64.

A. 1 through 9, are numbers, why? — Qwex

Because "rocks" was already taken.

What I want to know is how N is defined. — Qwex

Two ways: via the Peano axioms, in which $\mathbb N$ is a collection but not a set; and in ZF via the axiom of infinity. In the latter approach 0 is defined as the empty set; 1 is the set containing 0; 2 is the set containing 0 and 1; and in general, n is the set containing 0 through n - 1.

Of course numbers are not literally sets. The set-theoretic definition is just a representation of the abstract idea of the counting numbers.

https://en.wikipedia.org/wiki/Peano_axioms

https://en.wikipedia.org/wiki/Axiom_of_infinity

Hi Qwex, I really liked that you question the concept of numbers.
I believe that when you try to search for the origin of words and the way in which we pour meaning into the world around us - it is most fruitfull to search how those meaning are first discovered, whether it is understanding how a baby generates those concepts or reading the history of how those concepts were created in the early human culture.
A child starts by counting objects in the world - the concept of numbers as something other than an adjective to a group is outside the scope of our first usage of them. This is on par with how we evolve language itself - we use sounds and words that allow us to interact better with the world around us.
In the same way the usage of numbers in the human history evolved as a way to describe actual acts in the world - ways to divide a field of wheat, to make commercial transactions, etc...
It was then used as a way to allow us to make astronomical inquiries - taking part in one of the most early abstract human endeavours: religion. The Pythagorean sect in ancient Greece even evolved mathematics as a religion of its own. It is not weird than that Geometry, so closely related the world of mathematical usage in those days, was the tool by which the Pythagorean advanced their understanding of the world. And an ancient greek would asked then, in the same way you ask now - why do we talk about lines, what are lines?
So my answer to what are numbers - they are signs (verbal/graphic), like words, by which we are able to interact better with the world. The scope in which we use them is sometimes confined to the abstract term - mathematics; But this term, describing a narrower usage of them, isn't all that we do with those signs. We use numbers to describe concepts in every field of human knowledge.
And for a more concrete terminology - they are most widely used to describe a quality of a group of objects, distinguishing that group by the count of object within it. You can basically grow all of the mathematical knowledge from that definition (given a set of logical axioms on such groups).

via the Peano axioms, in which N is a collection but not a set — fishfry

I tried to look up this concept but the wikipedia pages for peano axioms and natural number do not seem to mention this subtlety. I assume that "a collection but not a set" means that N cannot be an element of another set?

There is also the concept of "proper class":

In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity.

A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems. — Wikipedia on class versus set

So, according to the above, the ordinal numbers are not a set but a proper class.

In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a (possibly infinite) collection of objects in order, one after another. Any finite collection of objects can be put in order just by the process of counting: labeling the objects with distinct natural numbers. Ordinal numbers are thus the "labels" needed to arrange collections of objects in order. — Wikipedia on ordinal numbers

So, according to the above, ordinal numbers are not a set in set theory. I couldn't find a reference to the idea of distinguishing between natural numbers and ordinal numbers in Peano arithmetic (PA). It even looks like expressing this distinction requires the full power of the machinery in set theory, such as, for example, by defining Von Neumann ordinals.

Therefore, I am a bit surprised that PA would even be able to introduce this type of subtlety through its axioms.

(Or maybe it actually does, but then implicitly/unexpectedly.)

By the way, I also found this remark on the subject:

Sets are those things given by the axioms you use, and it results that the notion of set becomes relative to the theory being considered. Something may be a set in one theory, but not in others. “Collection” is, as far as I can imagine, an informal word for aggregate or amount of things, standard things like pebbles and cats, which of course can be represented by sets, but have nothing to do with abstract mathematics. — Quora answer on sets versus collections

The answer above even seems to object to using the term "collection" in mathematics, because the term does not naturally emerge from any theory's axioms.

I understand you can count your fingers, 1 - 4, but what says a finger is a 1 and not crossed fingers? The 'whole' of the finger? — Qwex

The finger is a representation or symbol representing a mathematical concept.

Like in programming when i name a variable x that represents the number of apples being bought at a store.

Obviously you wouldn't name the variable X using correct coding procedures but a variable represents a number and a number is typically a Asian Indian/Arabic symbol that represents a mathematical concept.

I understand:

Natural: 1, 2, 3...
Whole: 0, 1, 2...
Integer: -1, 0, 1...
Rational: m/n
Irrational: x - m/n
Real: applicable to number line.

What I want to know is how N is defined.

Is there special use of the word 'is'? Natural numbers are N, is incomplete.

A. 1 through 9, are numbers, why?

B. Why does the number system progress, beginning from the left, proceding to the right?

C. Is human number just a tool?

I'm just getting into mathematics...

Sorry for having an intricate view - I'm not trying to distract. My primary question is (A).

Further Edits:

A shadow-argument:

I understand you can count your fingers, 1 - 4, but what says a finger is a 1 and not crossed fingers? The 'whole' of the finger?

In which case it's not a single, there's an organism involved(such as under the skin of the finger), and thus, a finger is not a 1.

I understand 1 is a concept but mathmatically, 1 is a point.

Perhaps, to point at your finger you'll use the number 1 but to define it numerically it's a different number. — Qwex

You are really good at this. Are you a team or do you come up with this stuff yourself?

I tried to look up this concept but the wikipedia pages for peano axioms and natural number do not seem to mention this subtlety. I assume that "a collection but not a set" means that N cannot be an element of another set? — alcontali

Yes. In PA (Peano arithmetic) each of the numbers 0, 1, 2, 3, ... exists, but not a completed set of them.

There is also the concept of "proper class": — alcontali

Yes in fact in PA, the natural numbers are a proper class. "Too big to be a set," as they say. It's a great example of how you can visualize what that means. It's the axiom of infinity in ZF that bestows set-hood on $\mathbb N$.

In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity. — alcontali

Right. There are no proper classes in ZF even though we use the term informally, but in other set theories the notion can be formalized.

A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems.
— Wikipedia on class versus set — alcontali

Yes.

So, according to the above, the ordinal numbers are not a set but a proper class. — alcontali

Yes. That's the famous Burali-Forti paradox, that the collection of ordinals can't be a set.

In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a (possibly infinite) collection of objects in order, one after another. Any finite collection of objects can be put in order just by the process of counting: labeling the objects with distinct natural numbers. Ordinal numbers are thus the "labels" needed to arrange collections of objects in order.
— Wikipedia on ordinal numbers — alcontali

Yes. The ordinals are awesomely cool, sadly not as well known as the cardinals.

So, according to the above, ordinal numbers are not a set in set theory. I couldn't find a reference to the idea of distinguishing between natural numbers and ordinal numbers in Peano arithmetic (PA). It even looks like expressing this distinction requires the full power of the machinery in set theory, such as, for example, by defining Von Neumann ordinals. — alcontali

Yes, you can't define the ordinals in PA because you can't get to the first transfinite ordinal $\omega$ by successors. You have to take a limit; or what amounts to the same thing, you have to consider the completed set of natural numbers. That in fact is the definition of $\omega$.

Therefore, I am a bit surprised that PA would even be able to introduce this type of subtlety through its axioms. — alcontali

Not sure what you mean. PA can't introduce ordinals or transfinite numbers. You can do a lot of number theory in it but not enough to get the real numbers off the ground. Unless you mean that it introduces subtleties in terms of what it can't do.

(Or maybe it actually does, but then implicitly/unexpectedly.) — alcontali

Don't know what you mean here.

By the way, I also found this remark on the subject:

Sets are those things given by the axioms you use, and it results that the notion of set becomes relative to the theory being considered. Something may be a set in one theory, but not in others. “Collection” is, as far as I can imagine, an informal word for aggregate or amount of things, standard things like pebbles and cats, which of course can be represented by sets, but have nothing to do with abstract mathematics.
— Quora answer on sets versus collections — alcontali

Yes, collection is what you say when you don't want to imply that something's a set.

The answer above even seems to object to using the term "collection" in mathematics, because the term does not naturally emerge from any theory's axioms. — alcontali

Quora giveth and Quora taketh away. I'd like to know who the author is before passing judgment on the sanity of anything math-related on Quora. There are a small handful of math experts there and a lot of people who don't know much. But in general, collection doesn't have a formal meaning. But then again neither does proper class.

But we can define collection. A collection is the extension of a predicate. That is, a collection is all the things in the universe that satisfy some predicate, like the collection of sets. Collections may or may not be sets. I tend to use the word collection synonymously with class. Every predicate defines a class, which may be a set or not. If it's a class but not a set, it's a proper class. But a lot of this usage is informal.

My ability to create symbols is damaged by the movement that base 4 math is good.

Imagine that sound visualization that creates a zig zag - you probably know what I'm on about.

Imagine a pure symbol that can be anything; much like the sound visualization, picture the symbol fluctuating.

Do I want to restrict my minds capacity to create symbols?

I know that base 4 isn't going to restrict that, but understood in a manner contrary to the way that I understand it, it does.

When I think mathmatically, I use line-form, icons squares, projected motion, mapping, etc.

I said in a previous thread that thought is partially mathematical in nature. This side of thought is not restricted to base 4. That would be stupid.

That's the famous Burali-Forti paradox, that the collection of ordinals can't be a set. — fishfry

Wow. That is very interesting!

It is named after Cesare Burali-Forti, who in 1897 published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Cantor. — Wikipedia on Burali-Forti

The proof based on Von Neumann ordinals is also straightforward. In that definition, (some interpretation of) $\mathbb{N}$ itself is also an ordinal. However, with "each ordinal is the well-ordered set of all smaller ordinals", it leads to $\mathbb{N} \lt \mathbb{N}$, i.e. the ordinal $\mathbb{N}$ being smaller than itself. Apparently, Cesare Burali-Forti already managed to prove it without using set-theoretical constructions (but Wikipedia does not explain the older proof strategy).

Yes, you can't define the ordinals in PA because you can't get to the first transfinite ordinal ω by successors. You have to take a limit; or what amounts to the same thing, you have to consider the completed set of natural numbers. That in fact is the definition of ω. — fishfry

Yes, I should probably not have used the symbol $\mathbb{N}$ (above) to designate the ordinal ω. I already sensed this because Wikipedia explicitly stays clear of doing that:

"Let Ω be a set that contains all ordinal numbers."

So, in this context, the subtlety is that $\mathbb{N}$ = { 0, 1, 2, ... } is not complete, while ω = { 0, 1, 2, ..., ω } is complete but then rests on something ultimately contradictory, i.e. a non-well-founded set expression of the type A = { B, A } which is then equivalent with A = { B, { B, A } } = { B, { B, { B, { B, A } } } } and so on, ad nauseam.

It looks like there is a strong (but unexpected) link between and Cesare Burali-Forti's work and what Dmitry Mirimanoff pointed out:

The study of non-well-founded sets was initiated by Dmitry Mirimanoff in a series of papers between 1917 and 1920, in which he formulated the distinction between well-founded and non-well-founded sets. — Wikipedia on non-well-foundedness

So, this result is understandable if we keep the subtlety in mind that $\mathbb{N}$ is not materialized while Ω is materialized. The act of materializing $\mathbb{N}$ substantially changes its nature. The work of Burali-Forti is quite interesting. In my opinion, it is surprising and even intriguing!

Wow. That is very interesting! — alcontali

Reading ahead, I surmise that you may making a little more of it than it deserves. Towards the end of your post you seem to draw some metaphysical or philosophical conclusions or associations, and I don't think those are justified. But as it ended up I didn't have time to get to that part of your post tonight. Instead I wrote a little disquisition on $\omega$ to clarify some technical points.

It is named after Cesare Burali-Forti, who in 1897 published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Cantor.
— Wikipedia on Burali-Forti — alcontali

Yes.

Yes, I should probably not have used the symbol ℕN (above) to designate the ordinal ω. I already sensed this because Wikipedia explicitly stays clear of doing that: — alcontali

Ah then I boldly go where Wikipedia dares not. Because in fact $\mathbb N = \omega = \aleph_0$ as sets. These are three different names for the exact same set.

Permit me to explain in order to make this point perfectly clear.

* The axiom of infinity gives us the existence of a set which we commonly call $\mathbb N$. Specifically it is the set of the finite von Neumann ordinals, also known as the natural numbers. We have in fact

* $0 = \emptyset$

* $1 = \{0\}$

* $2 = \{0,1\}$

* $3 = \{0,1,2\}$

and in general if we denote the successor of $n$ as $n +1$, then

$n + 1 = n \cup \{n \}$.

When you grok that last little bit of notation you will be enlightened! And now finally we define

$\mathbb N = \{0, 1, 2, 3, \dots \}$

That last definition relies on the axiom of infinity. In PA there's no rule that lets us form infinite sets. At best we could define set theory for PA, the set theory of finite sets. Everything would be the same as in full set theory except there are only finite sets. In fact PA is equivalent, as a theory, to ZF minus infinity; that is, ZF with the negation of the axiom of infinity.

The axiom of infinity allows us to take the "output of the completed induction," if you think of it that way, and put it all together into a set; which can then be operated on by the rules of set theory.

So now we have defined $\mathbb N$, the set explicitly given to us by the axiom of infinity.

* Next, we make the observation that since the elements of $\mathbb N$ are themselves all sets, we can ask whether two given members of $\mathbb N$ happen to stand in the relation $\in$ (set membership) to one another.

And now voilà, $\mathbb N$ is a transitive set well-ordered by $\in$ so that in fact the pair $(\mathbb N, \in )$ is exactly the first transfinite ordinal $\omega$.

So $\mathbb N$ and $\omega$ are two different names for the exact same set. We use one notation or the other when we want to emphasize the arithmetic properties or order properties of $\mathbb N$.

* Finally, what about $\aleph_0$? I seem to remember I talked about this a while back in some thread or other. The modern definition is that a cardinal is the least ordinal with a given cardinality. That definition has the virtue of making each cardinal a set. In particular, what is the least ordinal cardinally equivalent to $\aleph_0$? It's $\omega$.

So as sets, $\aleph_0 = \omega$.

Conclusion: The notations $\mathbb N$, $\omega$, and $\aleph_0$ each refer to the exact same set. We use one notation or another when we want to emphasize the arithmetic, order, and cardinality aspects of that set, respectively.

[And I probably should note that NONE of these have anything to do with the $\pm \infty$ of the extended real number system as used in analysis!]

SO: Not only is it mathematically correct to say these three distinct symbols all represent the same set; it is in fact illuminating to do so. If Wikipedia failed to bring out these connections, I would put that down to Wiki being unilluminating in this instance.

I hope this was helpful. This got long so I'll reply to the rest of your post later.

So, in this context, the subtlety is that ℕN = { 0, 1, 2, ... } is not complete, while ω = { 0, 1, 2, ..., ω } is complete but then rests on something ultimately contradictory, i.e. a non-well-founded set expression of the type A = { B, A } which is then equivalent with A = { B, { B, A } } = { B, { B, { B, { B, A } } } } and so on, ad nauseam. — alcontali

Your use of complete is nonstandard and I don't know what you mean. Are you trying to go through the proof of Burali-Forti?

It looks like there is a strong (but unexpected) link between and Cesare Burali-Forti's work and what Dmitry Mirimanoff pointed out:

The study of non-well-founded sets was initiated by Dmitry Mirimanoff in a series of papers between 1917 and 1920, in which he formulated the distinction between well-founded and non-well-founded sets.
— Wikipedia on non-well-foundedness — alcontali

No this is the part where you're reading too much into it.

The axiom of regularity, also known as the axiom of foundation, says that all sets are well-founded. There are no infinite downward chains of membership. No infinite regress, if you like.

This is a standard axiom of ZF. It's never mentioned because it's not used for anything. It's simply a given that essentially set theory is the study of well-founded sets.

Now it happens to be the case that the negation of the axiom of regularity is consistent with the rest of the axioms. And therefore it is possible, though quite obscure, to study non well-founded set theory.

So Burali-Forti is a theorem that follows from the axioms of ZF: that the class of ordinals can not be a set. And non well-founded set theory is a thing, but an obscure thing. These two ideas are NOT at some opposite ends of a pendulum or related to one another at all. You are wrong about any important connection or insight here. I'm being dogmatic to emphasize this point.

So, this result is understandable if we keep the subtlety in mind that ℕN is not materialized while Ω is materialized. The act of materializing ℕN substantially changes its nature. The work of Burali-Forti is quite interesting. In my opinion, it is surprising and even intriguing! — alcontali

I don't have any idea what you mean by materialized, nor how $\mathbb N$ substantially changes its nature. You are not speaking mathematical sense here in my opinion.

I don't want to take away from you the pleasure of discovering Burali-Forti's result, and the nice proof sketch on Wikipedia. But it's not a very important theorem nor does it lead to anything of interest. It's not really surprising once you know that the union of any collection of ordinals is an ordinal. Therefore if the class of ordinals were a set we could take its union to get another ordinal that must be a member of itself. That violates regularity, so there can be no set of ordinals.

But I hope you will see that the study of non well-founded sets is very obscure and far out of the mainstream of set theory. It's not an equal balance go this way or go that way kind of thing. For all intents and purposes, all sets are well-founded. Practically by definition.

In fact PA is equivalent, as a theory, to ZF minus infinity; that is, ZF with the negation of the axiom of infinity. — fishfry

I would like to discuss this because it is absolutely not self-evident to me that two sets of different rules, i.e. PA versus ZF minus infinity, would be completely equivalent. The rules do not even look like each other. Just look at the axioms. They are simply different. Still, if their equivalence is provable, then I would consider that to be an amazing result.

In fact, in that case, it should be possible to take the axioms of PA, push them through some kind of algebraic transformation process, and then end up with ZF minus infinity. I cannot imagine what this transformation process could look like.

The axiom of infinity allows us to take the "output of the completed induction," — fishfry

That is exactly what I mean by "materializing".

The reason why I used this term, is because this is the term used when you fully calculate and store the output of a view formula in relational databases, instead of keeping it around as a merely virtual construct. So, taking the "output of the completed induction" is called "materializing" in that context.

I just accidentally used a term (materialized view) en provenance from another domain.

In fact, it is not a completely different domain, because relational algebra is a downstream domain from ZF set theory. It completely rests on standard set theory. It is only much closer to practical applications:

Relational algebra, first created by Edgar F. Codd while at IBM, is a family of algebras with a well-founded semantics used for modelling the data stored in relational databases, and defining queries on it. — Wikipedia on relational algebra

Your use of complete is nonstandard and I don't know what you mean. — fishfry

I wasn't aware of the fact that the notation, N = { 0, 1, 2, ... }, is considered complete in set theory (through the axiom of infinity). (It is obviously not considered complete in PA.) So, yes, my use of the term "complete" is not standard in set builder notation in reference to ZF (but not in reference to ZF minus infinity). These things are extremely subtle. It depends on whether the theory in the context of which it is used, has an axiom that can "take the output of the completed induction", i.e. "materialize" it in relational-algebra lingo.

It is also very related to the concept of list comprehension where a similar problem occurs. You can create the list of natural numbers as a virtual construct, but you cannot "materialize" it, because that will cause your system to run out of memory.

Here, the list [0..] represents $\mathbb {N}$ , x^2>3 represents the predicate, and 2*x represents the output expression. List comprehensions give results in a defined order (unlike the members of sets); and list comprehensions may generate the members of a list in order, rather than produce the entirety of the list thus allowing, for example, the previous Haskell definition of the members of an infinite list. — Wikipedia about using virtual constructs that represent the infinite list of natural numbers

So Burali-Forti is a theorem that follows from the axioms of ZF: that the class of ordinals can not be a set. And non well-founded set theory is a thing, but an obscure thing. These two ideas are NOT at some opposite ends of a pendulum or related to one another at all. You are wrong about any important connection or insight here. — fishfry

if Ω is the set of ordinals but Ω is also itself an ordinal, then this situation will result in Ω being a set that contains itself, and therefore, result in a set that is not well-founded.

There are no infinite downward chains of membership. — fishfry

Yes, but that is exactly what would happen if Ω is the set of ordinals but Ω is also itself an ordinal. That is in my impression another reason why Ω cannot be termed a set but must be considered a proper class.

Therefore if the class of ordinals were a set we could take its union to get another ordinal that must be a member of itself. That violates regularity, so there can be no set of ordinals. — fishfry

Yes, that was indeed the connection that I saw.

I would like to discuss this because it is absolutely not self-evident to me that two sets of different rules, i.e. PA versus ZF minus infinity, would be completely equivalent. The rules do not even look like each other. Just look at the axioms. They are simply different. Still, if their equivalence is provable, then I would consider that to be an amazing result. — alcontali

The technical condition is that PA and ZF-infinity (read "ZF minus infinity") are bi-interpretable. Here's one link I found:

https://math.stackexchange.com/questions/315399/how-does-zfc-infinitythere-is-no-infinite-set-compare-with-pa

Also see:

https://mathoverflow.net/questions/551/does-finite-mathematics-need-the-axiom-of-infinity

In fact, in that case, it should be possible to take the axioms of PA, push them through some kind of algebraic transformation process, and then end up with ZF minus infinity. I cannot imagine what this transformation process could look like. — alcontali

Andrés E. Caicedo's checked answer in the Stackexchange thread outlines the procedure.

The axiom of infinity allows us to take the "output of the completed induction,"
— fishfry

That is exactly what I mean by "materializing". — alcontali

I'm not convinced. The completed set of natural numbers is posited, or brought into set-theoretic existence, by the axiom of infinity. If you want to call that materialized, well ok I guess ... but ... Is the powerset of $\mathbb N$ materialized by the powerset axioms? If I have two sets $A$ and $B$, is their union materialized by the axiom of union?

There doesn't see to be much depth to your definition of materialization. It just seems to mean, "brought into existence by a given axiom." In which the entire universe of propositions is materialized by the axiom 0 = 1. Is this what you mean?

The reason why I used this term, is because this is the term used when you fully calculate and store the output of a view formula in relational databases, instead of keeping it around as a merely virtual construct. So, taking the "output of the completed induction" is called "materializing" in that context. — alcontali

Oh I see. I understand the example but it's a bit of a stretch as an analogy for the set of natural numbers being given by the axiom of infinity. But if it works for you I do sort of see what you mean.

I just accidentally used a term (materialized view) en provenance from another domain. — alcontali

It's not really analogous to an axiomatic system IMO but sort of works as a vague metaphor.

In fact, it is not a completely different domain, because relational algebra is a downstream domain from ZF set theory. It completely rests on standard set theory. It is only much closer to practical applications:

Relational algebra, first created by Edgar F. Codd while at IBM, is a family of algebras with a well-founded semantics used for modelling the data stored in relational databases, and defining queries on it.
— Wikipedia on relational algebra — alcontali

Yes, perfectly well aware. But that doesn't make your vague analogy any sharper IMO. But this is not an important matter. Everyone is entitled to their own visualizations, intuitions, and conceptual ideas. Whatever works to understand the material.

Your use of complete is nonstandard and I don't know what you mean.
— fishfry

I wasn't aware of the fact that the notation, N = { 0, 1, 2, ... }, is considered complete in set theory (through the axiom of infinity). — alcontali

It surely isn't. There is no such term of art in set theory. Of course the natural numbers with the usual metric (absolute arithmetic difference) is Cauchy-complete. (Tricky. Why?) It's because every Cauchy sequence of natural numbers is eventually constant. But of course this is not what you meant.

What do you mean? There is no such technical term as saying that $\mathbb N$ is "completed" by the axiom of infinity. You just said it was materialized. I'd almost buy the latter, because even though it doesn't mean much, at least it's not a totally nonstandard use of the word complete, which has many other meanings in math but none in this context.

(It is obviously not considered complete in PA.) — alcontali

Oh, complete as in a completed infinity. Yes well the problem here is that "actual and potential infinity" are terms of art in philosophy, not math.

You seem to be trying to make something out of not much. If what you're saying is that you weren't formerly aware that it's the axiom of infinity that bestows set-hood on $\mathbb N$, ok now you know. But you can see from statement of the axiom of infinity that this is exactly what it does. It says that there is a set that contains the empty set and, whenever it contains a set $X$, it also contains the successor $X \cup \{X\}$.

The negation of the axiom of infinity says there's no such set; and since $\mathbb N$ is such a set, the negation of the axiom of infinity outlaws its existence.

So, yes, my use of the term "complete" is not standard in set builder notation in reference to ZF (but not in reference to ZF minus infinity). — alcontali

You're just making stuff up here. I'm trying to figure out why you're meandering down this road. PA is perfectly "complete" in your sense, it contains the conclusions of all its axioms. Or something. Neither PA nor ZF are more or less "complete" than the other. They each contain all and exactly those objects that are permitted by their respective axioms. Yes?

These things are extremely subtle. — alcontali

You're reading in subtleties where there are none. From where I sit you are taking a simple fact and tring to give it great significance. PA has all the natural numbers, and ZF via the axiom of infinity has a completed set of them. But if you took ZF minus powerset, then you wouldn't have full powersets. Does the powerset axiom complete ZF with respect to powersets? There's nothing interesting about this, you are seeing complications and subtleties where truly, I say to you as clearly as I can, there are none.

It depends on whether the theory in the context of which it is used, has an axiom that can "take the output of the completed induction", i.e. "materialize" it in relational-algebra lingo. — alcontali

Or make a full powerset. Or a union. So is materializing the same as completing? Why go on about this? Without the axiom of pairing, X and Y could be sets but there's no set {X,Y}. Does the pairing axiom complete, or materialize, pairs of sets? Can you see that you're imagining subtlety where there isn't any?

It is also very related to the concept of list comprehension where a similar problem occurs. You can create the list of natural numbers as a virtual construct, but you cannot "materialize" it, because that will cause your system to run out of memory. — alcontali

In Python there's something called a generator. The idea is that I want to iterate through a list but I don't want to pre-create the full list ahead of time. I just supply the algorithm. This I assume is exactly what you mean. I understand the example, I just don't think it's very interesting or meaningful.

Here, the list [0..] represents ℕN , x^2>3 represents the predicate, and 2*x represents the output expression. List comprehensions give results in a defined order (unlike the members of sets); and list comprehensions may generate the members of a list in order, rather than produce the entirety of the list thus allowing, for example, the previous Haskell definition of the members of an infinite list.
— Wikipedia about using virtual constructs that represent the infinite list of natural numbers — alcontali

Sure whatever.

So Burali-Forti is a theorem that follows from the axioms of ZF: that the class of ordinals can not be a set. — alcontali

We've already been through the proof. It's straightforward. First you prove that the union of ordinals is an ordinal, therefore if the class of ordinals is a set so is its union, which violates regularity. Ok done. This is not worth starting a religion over. It doesn't mean anything as significant as what you're trying to read into it.

And non well-founded set theory is a thing, but an obscure thing. These two ideas are NOT at some opposite ends of a pendulum or related to one another at all. You are wrong about any important connection or insight here.
— fishfry

if Ω is the set of ordinals but Ω is also itself an ordinal, then this situation will result in Ω being a set that contains itself, and therefore, result in a set that is not well-founded. — alcontali

Yes, and ...?

There are no infinite downward chains of membership.
— fishfry

Yes, but that is exactly what would happen if Ω is the set of ordinals but Ω is also itself an ordinal. That is in my impression another reason why Ω cannot be termed a set but must be considered a proper class. — alcontali

Yes, that's what regularity prevents. Again, so what? I find myself frustrated. You're trying to convince me that something deep is going on, and there isn't. If your private intuitions help you, all the good. But you haven't written anything of interest; least of all with your materialize and complete terminology. ZF is not complete, you know, at least if it's consistent. You can't say the axiom of infinity completed it using your made-up definition, when it's NOT complete by everyone's standard definition. There's nothing complete about it. It just has more sets than PA because you added another axiom of set existence.

Here's another link that might be of interest, the hereditarily finite sets. This is the class of objects that's described by PA and by ZF-infinity. It's perfectly "complete." It has all the sets and only those sets that it's supposed to as given by the rules of its construction.


To sum up, all I can see is that you're saying that PA is complete with respect to the axioms of PA, and ZF is complete with respect to the axioms of ZF, and ZFC is complete wrt the axioms of ZFC, and so forth. But then ZF is missing some choice functions so it is "incomplete" with respect to the axiom of choice. And ZFC is complete with respect to the axiom of infinity and the axiom of powersets and the axiom of union and the axiom of pairing, but not with respect to the Continuum hypothesis.

There's nothing of interest to this observation; and worse, it's bad terminology because it conflicts with the standard meaning of completeness in an axiomatic system.

This is what I got from your post.

You have admirable patience. Thank goodness run-of-the-mill mathematics avoids all this. :cool:

You have admirable patience. — jgill

Always had an interest in set theory and foundations.

Thank goodness run-of-the-mill mathematics avoids all this. — jgill

Beg to differ. Functional analysis uses the Hahn-Banach theorem, which is equivalent to a weak form of the axiom of choice. There are many areas of "hard" math in which foundational questions arise. The history of analysis from Newton onward is a two hundred year struggle to get the definitions right in order to have a logically coherent theory. And of course there are many deep foundational questions in probability and measure theory related to the axiom of choice and nonmeasurable sets.

An interesting factoid is this. When Cantor discovered transfinite set theory, he was studying the zeros of the trigonometric functions used by Fourier in his studies of heat flow. If you think about it, such a function could have zeros at each of the integers. Or perhaps in addition, at each integer it has zeros at that integer plus 1/n for each n. And maybe it has chains of zeros hanging off some of the 1/n branches. You need a language, or a notation, for keeping track of the way the zeros of trigonometric functions can occur. That's the ordinal numbers.

Transfinite set theory came directly from physical considerations. If you heat up one end of an iron bar under laboratory conditions and carefully observe the heat flow, you will inevitably discover transfinite set theory. That's how it actually happened.

I found a beautiful-looking article called The Trigonometric Background to Georg Cantor's Theory of Sets behind an academic paywall, Grrrrrrrr! Academic paywalls are evil. @jgill you have a university affiliation by any chance?

ps -- Found a great overview of this material written for a general audience.

How did Cantor Discover Set Theory and Topology?. I think I'm going to give this a read.

The technical condition is that PA and ZF-infinity (read "ZF minus infinity") are bi-interpretable. — fishfry

Bi-interpretability looks like an interesting subject, but unfortunately the Wikipedia page does not elaborate PA versus ZF-infinity as an example.

the output of the completed induction ... There is no such term of art in set theory. — fishfry

Well, you did use the term "complete" in the sense of induction-complete. I clearly used it in the same way, and then you suddenly backtrack to claiming that induction-complete would be "no such term of art in set theory".

Completeness of the real numbers
Not to be confused with Completeness (logic).
There are many equivalent forms of completeness, the most prominent being Dedekind completeness and Cauchy completeness (completeness as a metric space). — Completeness of the real numbers

Since our conversation had absolutely nothing to do with completeness of the real numbers, I wonder why you mention Cauchy completeness? It just adds to the confusion.

It's not really analogous to an axiomatic system IMO but sort of works as a vague metaphor. — fishfry

Relational algebra is itself obviously also an axiomatic system. I do not believe that anybody even questions that.

So is materializing the same as completing? — fishfry

It is used as a term for induction-completing (A term you actually introduced by yourself yourself). It has obviously nothing to do with logic completeness, Cauchy completeness, Dedekind completeness, and so on.

Of course the natural numbers with the usual metric (absolute arithmetic difference) is Cauchy-complete. (Tricky. Why?) — fishfry

That is another context in which the term "completeness" is used.

A is perfectly "complete" in your sense, it contains the conclusions of all its axioms. — fishfry

That is about completeness in logic, which is not directly related to the "output of the completed induction". I was not referring to "logically complete".

You can't say the axiom of infinity completed it using your made-up definition, when it's NOT complete by everyone's standard definition. — fishfry

There is not one definition for completeness. It depends on the context. You may have misunderstood what context I was referring to, but that kind of confusion occurs relatively easy with the term "complete".

In fact, I never used the term induction-complete or induction-completed before. I only used it because you used it first. I tend to use the term "materialized" instead of induction-completed. Furthermore, it is probably better not to further overload the term 'complete' with additional meanings.

To sum up, all I can see is that you're saying that PA is complete with respect to the axioms of PA, and ZF is complete with respect to the axioms of ZF, and ZFC is complete wrt the axioms of ZFC, and so forth. — fishfry

No, of course not. That is about logical completeness. I didn't make even one single remark in that context. I don't see why you would understand any of the above in terms of logical completeness.

When a discussion degenerates in "who is smarter than whom", "who knows more than whom", i.e. the typical, ridiculous conversations in the academia, in which they engage because they simply have nothing else to show for, then I tend to back out. That kind of conversations are simply not interesting. In that case, I even prefer -- God forbid -- the slightly less ridiculous conversations in business about whom makes more money than whom, because amounts of money are at least objectively measurable while amounts of knowledge are not.

When a discussion degenerates in "who is smarter than whom", "who knows more than whom", i.e. the typical, ridiculous conversations in the academia, in which they engage because they simply have nothing else to show for, then I tend to back out. — alcontali

Not my intention.

Beg to differ. Functional analysis uses the Hahn-Banach theorem, which is equivalent to a weak form of the axiom of choice — fishfry

Yes, the Zermelo well-ordering theorem comes into play (I took a year of FA fifty years ago), but is not required if the underlying normed linear space is separable (common). I never had any reason to use Hahn-Banach since I didn't do much in soft analysis, but recently I've been intrigued by functional integration and particularly Feynman's path integral (which is not exactly kosher). I dabbled in a generalization of analytic continued fraction theory, so called by an old friend, Wolf Thron RIP. Nothing there hinging upon transfinite set theory (well, to my knowledge!).

jgill you have a university affiliation by any chance? — fishfry

No more, unfortunately. Retired twenty years ago.

Bi-interpretability looks like an interesting subject, but unfortunately the Wikipedia page does not elaborate PA versus ZF-infinity as an example. — alcontali

Yes, it seems to be more subtle than mere logical equivalence.

Well, you did use the term "complete" in the sense of induction-complete. I clearly used it in the same way, and then you suddenly backtrack to claiming that induction-complete would be "no such term of art in set theory". — alcontali

You are entirely correct. I said, "The axiom of infinity allows us to take the "output of the completed induction," if you think of it that way ..." Guilty as charged. I have apparently been strenuously arguing against my own terminology and blaming it on you. My bad. Thanks for pointing it out.

So is materializing the same as completing?
— fishfry

It is used as a term for induction-completing (A term you actually introduced by yourself yourself). — alcontali

Ok ... materialization means making a set inductively complete. So if we have a set that contains x, it contains the successor of x and the successor of the successor of x, etc., for all possible applications of finitely many occurrences of the successor operation. A set is materialized if it's closed under taking successors.

In this case I believe you're talking about a limit ordinal. A limit ordinal is not the successor of any ordinal; but is rather the "completion," or closure under the successor operation, of its elements. So $0$ is the only finite limit ordinal; and $\omega, 2 \omega, 3 \omega, \dots, \omega^2, \dots$ are all limit ordinals. They're constructed by taking the upward limit of a collection of ordinals.

But why "materialized?" That seems to load the concept with some kind of metaphysical significance. I could live better with "inductively completed," now that I think about it.

In fact, I never used the term induction-complete or induction-completed before. I only used it because you used it first. I tend to use the term "materialized" instead of induction-completed. Furthermore, it is probably better not to further overload the term 'complete' with additional meanings. — alcontali

Guilty as charged. But I don't love the word materialized either. I think the idea of limit ordinals captures the concept.

(1) I don't know in what exact mathematical formulation one says [paraphrasing] "N is a proper class as far as PA is concerned". First order (I'll mean first order throughout) PA is a theory, i.e. a set of sentences closed under deduction. A typical meta-theory for PA is set theory. In set theory we prove that PA has models. And the universes for these models are sets, not proper classes. And one of those universes is N, which is a set.

Indeed, every consistent theory has a model whose universe is a set. Moreover, every consistent theory that has a model with an infinite universe also has a model with a denumerable universe. Moreover, every consistent theory that has a model with an infinite universe also has a model whose universe is N.

Meanwhile in the standard interpretation of PA there is not even mention of sets, proper classes, or universes for models of PA. So, since the standard interpretation of PA doesn't mention it, and, in PA's ordinary meta-theory, universes are sets, not proper classes, I don't know where there would be an actual mathematical formulation of "N is a proper class as far as PA is concerned."

(2) Usual terms for inductive sets are 'inductive set' or 'closed under induction', The induction may pertain to a relation or operation. In the case of ordinals, we may consider the successor operation ( Sx = xu{x} ), and indeed, as mentioned, limit ordinals are closed under the successor operation.

(3) It is not necessary to use the axiom of regularity or unions to show that there is no set that has all the ordinals as members, as follows:

Lemma (easy to prove without using the axiom of regularity): No ordinal is a member of itself.

Now suppose there is a K such that every ordinal is a member of K. Then let L be the subset of K that has only ordinals as members. Then L itself is an ordinal (easy to prove). But then L is a member of itself. Contradiction.

(4) As mentioned, the notion of showing that two theories (even with completely different symbol sets) "say essentially the same thing" is rigorously captured by showing that there is an interpretation of one theory into the other and vice versa. The mathematical formulation of an interpretation of a theory into another is straightforward and not mysterious.

Numbers are names for quantities. — creativesoul

Correct! Thank you for summarizing all of this bullshit!

I don't know where there would be an actual mathematical formulation of "N is a proper class as far as PA is concerned." — GrandMinnow

If this is in response to something I said, I never claimed such a thing. I use proper class as an informal description of a collection that's not a set. In PA the extension of the unary predicate "n is a natural number" is the collection we call $\mathbb N$. However $\mathbb N$ is not a set in PA. So I say $\mathbb N$ is a proper class. It's the extension of a predicate that isn't a set.

I can see you objecting that PA doesn't talk about sets. But given the objects of PA, namely the natural numbers 0, 1, 2, 3, ... we can certainly define unions, intersections, pairs, powersets, and so forth; and do finite set theory perfectly well. In other words PA is a model of ZF-infinity. In that case we would find that $\mathbb N$ is not a set.

My usage was informal but useful in the sense that when we talk about proper classes like the class of all sets, that's hard to imagine; but $\mathbb N$ is easy to imagine. And if we throw out the axiom of infinity, $\mathbb N$ is a proper class. Unofficially, of course. But $\mathbb N$ is the extension of the predicate "n is a natural number" and it's not a set, so it certainly qualifies. Informally.

Indeed, every consistent theory has a model whose universe is a set. — GrandMinnow

Yes. Have you a demonstration in PA of the consistency of PA? Of course not, since that would violate Gödel's first incompleteness theorem. The only way to get a model of PA is to wave your hands and say the magic words, "Axiom of infinity!" And now you have a model of PA, along with a consistency proof for PA. But without the axiom of infinity you haven't got a model of PA nor a proof of PA's consistency.

I know you know this so I must be missing something. And you know our friends the ultrafinitists, who doubt the consistency of PA. Sounds crazy but that doesn't make them wrong.

I paraphrased what you said:

"via the Peano axioms, in which N is a collection but not a set"

"in PA, the natural numbers are a proper class"

Those are not actual mathematical statements. Not even informally. First order PA (throughout, I will mean first order, which is what people usually mean in modern context unless said otherwise, and as your remark about set theory proving model existence also suggests you're not referring to higher order PA) does not speak of N (not as a formula "in" PA, but only by model interpretation in, say, set theory) nor of a distinction between sets and proper classes. And it would be quite remote, along with needing to work out needed details in execution, to justify those two quoted claims by the fact that PA can be viewed as (ZF-Inf)+~Inf.

In PA the extension of the unary predicate "n is a natural number" is the collection we call NN. — fishfry

No such thing takes place in PA there is no unary predicate 'is a natural number'. Usually the meta-theory for making statements about PA is set theory. And in set theory, the notion of an "extension of a predicate" is given by a model. The universe for the standard model is N, which is a set.

In other words PA is a model of ZF-infinity. — fishfry

PA is a theory, not a model. The correct statement is that PA and (ZF-Inf)+~Inf can be interpreted in each other. But that still doesn't make "N a proper class in PA" a coherent statement.

The only way to get a model of PA is to wave your hands and say the magic words, "Axiom of infinity!" — fishfry

Then one can say that any axiom of any theory is waving hands and saying magic words. But yes, a model of PA is proven to exist from the set theory axioms that include the axiom of infinity. Again, that does make "N a proper class in PA" a coherent statement.

/

Correct statements include these:

If PA is consistent, then PA does not prove that PA has a model.

Set theory proves that N, which is a set, is the universe for a model of PA.

And that does not involve proper classes.

Maybe this will help:

PA is a certain set of sentences. There is no sentence in PA that one would ordinarily regard as saying "N is a proper class". There is a universal quantifier for PA, and with any model (formulated in set theory) for PA, the universal quantifier stands for the universe of the model. And that universe is always a set. In particular, with the standard model, the universe is N, which is a set.

I think maybe what you were driving at originally might be said this way:

The universe for the standard model of PA is N and has as its members all and only the natural numbers, so N itself is not a member of N. And with the standard model, PA itself does not assert the existence of a set that has all the natural numbers as members.

Couching that in terms of proper classes is off.

We can go further, it is not precluded that other models of PA do include all the members of N and N itself as a member of the universe for the model. Most simply, N+ (i.e. Nu{N}) is a denumerable universe for certain models of PA.