Or maybe we can think of vacuity and unity as the bases. Then we have 0 and 1, the binary. But in set theory, with the pairing operation, we can define '1' from '0'..

df: {x y} = z <-> Ak(k e z <-> (k = x v k= y))

dr: {x} = {x x}

df: x = 0 <-> Ay ~xey

df: 1 = {0}

df: n and m are natural numbers -> (n < m <-> n e m)

/

Set theory does provide the structural relations we expect. Even though the objects have "extra-structural" properties (e.g. that 0 is the empty set and 1 is the set {0}), the structural relations are captured (e.g. that 0 < 1).

I found the comments about Cohen's Filter in the article I linked fascinating. Like most math people I knew of his breakthrough results, but was unfamiliar with the actual math. I'd be interested to hear opinions from the set theorists on the forum about this. — jgill

I don't think there are any set theorists here. You're the only mathematician in the house. The rest of us, speaking for myself, are groupies and hangers-on at best. That said, can you say what "this" refers to? Cohen's invention of forcing in general? Or the particular recent result that's floating around the Internet about Martin's Maximum implying (*) or some such? The latter is some serious set-theoretic inside baseball.

I didn't know about structuralism in math! That the number one is an idea, a true idea, seems to me to be the basis of all that follows though, kinda that unity before the plurality. But structuralism in all forms is a really interesting idea! — Gregory

I found the SEP article interesting. It breaks down all the various sub-genres of mathematical structuralism and talks a lot about whether category theory is an example of mathematical structuralism or not, and so forth. Lot of fancy philosophizing :-)

And they point out some of the drawbacks with structuralism. If the natural numbers are not any particular collection of objects, but rather are instances of some "structure," then what exactly is a structure?

The main point is that when we say that 0 is the empty set and 1 is the set containing 0, what we really mean is that these sets represent the natural numbers within set theory. What we don't mean is that these sets actually "are" the natural numbers. Leaving unanswered the question of what the natural numbers really are

My initial guess was that a set is something that contains and not something in its own right. So zero remains a nothingness of anything in that case. Very abstract ideas. Couldn't structure just be that which contains a process and thus, like sets which compose it, it is nothing in itself. This would certainly make mathematics a system of process and divorce it from the notion that anything rests and stays permanent within it

a set is something that contains and not something in its own right — Gregory

That's not the case in set theory.

If you replace numbers with sets, do you keep the same ontology? What of structuralism?

That said, can you say what "this" refers to? Cohen's invention of forcing in general? — fishfry

That would be good. I had heard the expression but had no idea what it was. The article came as a revelation to me. And here I thought the reals consisted of rationals and irrationals. :chin:

You and Tones are far more set theorists than me!

If you replace numbers with sets, do you keep the same ontology? — Gregory

"Replace" might not be a good way of putting. A better way of putting it might be that sets "play the role" of numbers, or something like that.

And what ontology? Different philosophers and/or mathematicians have different conceptualizations of ontology for set theory.

What of structuralism? — Gregory

I addressed that:

Set theory does provide the structural relations we expect. Even though the objects have "extra-structural" properties (e.g. that 0 is the empty set and 1 is the set {0}), the structural relations are captured (e.g. that 0 < 1). — TonesInDeepFreeze

I like the idea of structuralism, and my own personal understanding of set theory is that we may think of the axioms as specifying structural relations rather than worrying whether there is some abstract world of which the axioms are true. In other words, from the axioms, we may consider those abstract relations that the axioms define, rather than worry about whether there is a particular abstract world in which the "objects" in those relations exist.

The question I have though is, even if we don't have to worry about what it means to say the objects exist, haven't we just kicked the can down the road to the question "In what sense do the relations, the structures exist?"?

Buddhism was the original structuralism with their idea of utter dependence. Logic and the world, everything in fact, was dependent but not dependent on something

I'm not a set theorist, but I have some thoughts.

I haven't seen articles before that give a layman's explanation of forcing and of axioms for proving CH. So I appreciate that.

I have my own question. I kinda got the idea behind the explanation of the filter, but I wonder if this is correct:

That filter proves the existence of a certain real number. Then we use other filters to prove the existence of other real numbers. By doing that some uncountable number of ways, we prove that there are sets of real numbers that have cardinality between the cardinality of the set of natural numbers and the cardinality of the power set of the set of natural numbers. Is that correct?

(IMPORTANT. In this post, I take "ZF is consistent" as a background assumption. For example "ZFC is consistent" in this post means "If ZF is consistent then ZFC is consistent".)

'AC' stands for the axiom of choice.
'CH' stands for the continuum hypotheis.
'GCH' stands for the generalized continuum hypothesis.
'ZFC'stands for ZF+AC.

I know nothing about forcing other than this:

(1) Cohen used forcing to prove:

ZF+~AC.

ZFC+~CH is consistent. So, a fortiori, ZFC+~GCH is consistent.

(2) Forcing involves ultrafilters and/or Boolean algebra.


Re (1), Godel had previously proved:

ZFC is consistent. [*]

ZFC+GCH is consistent. So, a fortiori, ZFC+CH is consistent.

Godel did it with the notion of the constructible universe.

Combining Godel and Cohen, we get:

AC is independent from ZF.

CH is independent from ZF.

GCH is independent from ZF.

[*] But what about Godel's second incompleteness theorem that entails "Set theory does not prove its own consistency"? Well, actually the second incompletess doesn't entail that. The second incompleteness theorem does entail, "If set theory is consistent then set theory does not prove set theory is consistent". And, since I put "ZF is consistent" as a blanket assumption for this post, "ZFC is consistent" in this post, stands for "If ZF is consistent then ZFC is consistent". That qualification applies, mutatis mutandis, to both the Cohen theorems and both the Godel theorems above.


FOR REFERENCE:

df. A set of sentences S is consistent iff S does not prove a contradiction.

df. A sentence P is independent from a set of sentences S iff (S does not prove P and S does not prove ~P).

th. A sentence P is independent from a set of sentences S iff (S+P is consistent and S+~P is consistent.

th. If a set of sentences S is consistent, then there is a model in which every sentence in S is true (this is Godel's completeness theorem).

So:

To prove "ZFC is consistent", it suffices to prove there is a model of ZFC. But "ZF is consistent" entails "ZF has a model". So it suffices to prove that "ZF has a model" implies "ZFC has a model".

To prove "ZFC+GCH is consistent", it suffices to prove there is a model of ZFC+GCH. But "ZF is consistent" entails "ZF has a model", which, Godel proved, entails "ZFC has a model". So it suffices to prove that "ZFC has a model" implies "ZFC+GCH has a model".

To prove "ZF+~AC is consistent", it suffices to prove there is a model of ZF+~AC. But "ZF is consistent" entails "ZF has a model". So it suffices to prove that "ZF has a model" implies "ZF+~AC has a model".

To prove "ZF+~CH is consistent", it suffices to prove there is a model of ZF+~CH. But "ZF is consistent" entails "ZF has a model". So it suffices to prove that "ZF has a model" implies "ZF+~CH has a model".

Those are examples of relative consistency. "If theory T is consistent, then theory Y is consistent".

Some years after Godel's results just mentioned, Sierpenski proved that ZF+GCH proves ZFC. So:

Proving "ZF+GCH is consistent", a fortiori, proves "ZFC is consistent".

Proving "ZF+~AC is consistent", a fortiori, proves "ZF+~AC+~GCH", which, a fortiori, proves "ZF+~GCH is consistent". So, "ZF+~AC is consistent" proves "ZF+~GCH is consistent".


BACKGROUND:

What is the axiom of choice?

What is the continuum hypotheis?

What is the generalized continuum hypothesis?


df. 0 = the empty set.

df. PS = the set of subsets of S.

df. Y\Z = the set whose members are all and only those members of Y that are not members of Z.

C (the axiom of choice) is the statement:

"For every S, there is a function on the PS\{0} such that for every x in PS\{0}, f(x) e x". We call such an f "a choice function for S".

To visualize the above:

Imagine a nation made up of provinces (and possibly there are infinitely many provinces). From each province we can choose a representative who is a resident of that province.

If S is finite, without the axiom of choice, by a trivial induction on the cardinality of S, we prove there is a choice function for S, so we don't need the axiom of choice to prove there is a choice function for S.

"For every S, there is a function on the PS\{0} such that for every x in PS\{0}, f(x) e x".

The axiom of choice is equivalent with a number of other theorems, especially "Every S has a well ordering" and "Every S is equinumerous with an ordinal".


df. S^x = the set of funtions from x into S.

df. x and y are equinumerous iff there is a bijection between x and y.

df. card(S) = the least ordinal k such that S and k are equinumerous.

df. for ordinals, x < y iff x e y.

df. w = the set of natural numbers

df. R = the set of real numbers

th. card(R) = card(Pw) = card(2^w)

th. There is no surjection from S onto PS. (Cantor's theorem)

th. card(w) < card(R)

CH is the statement:

"There is no S such that card(w) < card(S) < card(R)".

GCH is the statement:

"If X is infinite, then there is no S such that card(X) < card(S) < card(PX).

Cantor failed to prove CH. Hilbert wanted somebody to prove it. Godel proved that we can't disprove GCH, a foritori that we can't disprove CH. Cohen proved that we can't disprove ZFC+~CH, so, a fortiori, we can't dispove ZFC+~GCH.

So some set theorists, who feel that ~CH fits their concept of 'set' have been trying to discover a set theoretic statement that is even more convincingly true to their concept of 'set' and that proves ~CH. Other set theorists, who feel that CH fits their concept of 'set' have been trying to discover a set theoretic statement that is even more convincingly true to their concept of 'set' and that proves CH.

FOR REFERENCE:

Z is basic infinitistic set theory (first order logic with identity, extensionality, schema of separation, pairing, union, power, infinity, regularity). ZF is Z with the axiom schema of replacement added.

The axiom schema of replacement is the statement [I'm simplifying]:

"If R is a function class, then, for every S, there is the T whose members are all and only those y such that there is an x in S such that <x y> in R."

R is a proper class. It is a proper class of ordered pairs. The axiom of replacement is: If R is functional (i.e. if <x y> in R and <x z> in R, then y=z), then for any set S, there exists the set T that is the image of S by R. I.e., if you have a set S, and a functional relation, then the "range" of that relation restricted from the domain S is a set.

That mentions proper a proper class, though Z proves ~Ex x is a proper class. So the actual axiom schema of replacement is a set of axioms, with each axiom mentioning a formula instead of a proper class. The formula "carves out" the proper class.

PS. We don't need definitions of 'cardinality' an 'ordinal' to state CH and GCH. We can state it equivalently:

GCH.

If S is infinite, then is no set X such that all these hold:

S injects into X
There is no bijection between X and S
X injects into 2^S
There is no bijection between S and 2^S

And CH is just a special case of GCH, where S = w.

PPS

Two iconic books that handle forcing in detail are:

'Set Theory' - Jech

'Set Theory: An introduction To Independence Proofs' - Kunen

Jech's book is a great tome. But I might prefer Kunen, because, though I'm not a formalist, I find that a formalist sensibility contributes to good exposition.

However, both those books are at a graduate level. For an introduction to set theory, I always recommend:

Elements Of Set Theory - Enderton, which is a great read

supplemented with

Axiomatic Set Throry - Suppes.

In set theory everything is a set. — fishfry

I didn't know that, but it makes the problems which I've apprehended much more understandable. If everything is a set, in set theory, then infinite regress is unavoidable. A logical circle is sometimes employed, like the one mentioned here to disguise the infinite regress, but such a circle is really a vicious circle.

Sets whose elements are sets whose elements are sets, drilling all the way down to the empty set. — fishfry

I reject "the empty set" for a reason similar to the reason why I rejected a set with no inherent order. it's a fiction which has no purpose other than to hide the shortcomings of the theory. There are very good reasons why "0" ought to represent something in a class distinct from numbers. There are even reasons why "1" ought to be in a distinct class.

No, not at all. First, the elements of a set need not be "the same" in any meaningful way. The only thing they have in common is that they're elements of a given set. — fishfry

This may be the case, but you ought to recognize that being elements of the same set makes them "the same" in a meaningful way. Otherwise, a set would be a meaningless thing. So when you said for instance, that {0,1,2,3,} is a set, there must be a reason why you composed your set of those four elements. That reason constitutes some criteria or criterion which is fulfilled by each member constituting a similarity.

The concept of "set" itself has no definition, as I've pointed out to you in the past. — fishfry

This is a simple feature of common language use. A word may receive its meaning through usage rather than through an explicit definition. That the word has no definition does not mean that it has no meaning, its meaning is demonstrated by its use, as is the case with an ostensive definition. Allowing that a word, within a logical system, has no explicit definition, allows the users of the system an unbounded freedom to manipulate that symbol, (exemplified by TonesInDeepFreeze's claim with "least"), but the downfall is that ambiguity is inevitable. This is an example of the uncertainty which content brings into the formal system, that I mentioned in the other thread.

There is no set of ordinals, this is the famous Burali-Forti paradox. — fishfry

This I would say is a good representation of the philosophical concept of "infinite". Note that the philosophical conception is quite different from the mathematical conception. If every ordinal is a set composed of other ordinals, and there is no limit to the "amount" of ordinals which one may construct, then it ought to be very obvious that we cannot have an ordinal which contains all the ordinals, because we are always allowed to construct a greater ordinal which would contain that one as lesser. So we might just keep getting a greater and greater ordinal, infinitely, and it's impossible to have a greatest ordinal.

I think there is a way around this though, similar to the way that set theory allows for the set of all natural numbers, which is infinite. As you say, "set" has no official definition. And, you might notice that "set" is logically prior to "cardinal number". So all that is required is a different type of set, one which is other than an ordinal number, which could contain all the ordinals. It would require different axioms.

There is no general definition of number. — fishfry

This is not really true now, if we accept set theory. If "set" is logically prior to "number", then "set"
is a defining principle of "number". That is why you and I agreed that each ordinal is itself a set. We have a defining principle, an ordinal is a type of set, and a cardinal is a type of ordinal.

You see you're at best a part-time Platonist yourself. — fishfry

Correction, at my worst I am a part-time Platonist. At my best I am a fulltime Neo-Platonist.

If I put on my Platonist hat, I'll admit that the number 5 existed even before there were humans, before the first fish crawled onto land, before the earth formed, before the universe exploded into existence, if in fact it ever did any such thing. — fishfry

We do not have to go the full fledged Platonic realism route here, to maintain a realism. This is what I tried to explain at one point in another thread. We only need to assume the symbol "5", and what the symbol represents, or means. There is no need to assume that the symbol represents "the number 5", as some type of medium between the symbol, and what the symbol means in each particular instance of use. So when I say that a thing exists, and has a measurement, regardless of whether it has been measured, what I mean is that it has the capacity to be measured, and there is also the possibility that the measurement might be true.

I must say, though, that I am surprised to find you suddenly advocating for mathematical Platonism, after so many posts in which you have denied the existence of mathematical objects. Have you changed your mind without realizing it? — fishfry

If you think that I was advocating for mathematical Platonism, then you misunderstood. I was advocating for realism.

But Meta, really, you are a mathematical Platonist? I had no idea. — fishfry

A mathematical Platonist thinks of ideas as objects. I recognize the reality of ideas, and furthermore I accept the priority of ideas, so I am idealist. But I do not think of ideas as objects, as mathematical Platonists do, I think of them as forms, so I'm more appropriately called Neo-Platonist.

I agree with the points you're raising. I don't know if 5 existed before there were humans to invent math. I truly don't know if the transfinite cardinals were out there waiting to be discovered by Cantor, and formalized by von Neumann. After all, set theory is an exercise in formal logic. We write down axioms and prove things, but the axioms are not "true" in any meaningful sense. Perhaps we're back to the Frege-Hilbert controversy again. — fishfry

This is that vague boundary, the grey area between fact and fiction which we might call "logical possibility". If we adhere to empirical principles, we see that there are individual objects in the world, with spatial separation between them. If we are realist, we say that these objects which are observed as distinct, really are distinct objects, and therefore can be counted as distinct objects. We might see three objects, and name that "3", but "3" is simply what we call that quantity. Being realist we think that there is the same quantity of objects regardless of whether they've been counted and called "3" or not.

But if we give up on the realism, and the empirical principles, there is no need to conclude that what is being seen is actually a quantity of 3. There might be no real boundaries between things, and anything observed might be divisible an infinite number of times. Therefore whatever is observed could be any number of things. This is the world of fiction, which some might call "logical possibility", and you call pure mathematics. Empirical truths, like the fact that distinct objects can be counted as distinct objects, pi as the ratio between circumference and diameter of a circle, and the Pythagorean theorem, we say are discovered. Logical possibilities are dreamt up by the mind, and are in that sense fictions.

I do not mean to argue that dreaming up logical possibilities is a worthless activity. What I think is that this is a primary stage in producing knowledge. We look at the empirical world for example and create a list of possibilities concerning the reality of it. The secondary stage is to eliminate those logical possibilities which are determined to be physically impossible through experimentation and empirical observation. So we proceed by subjecting logical possibilities, and axioms of pure mathematics, to a process of elimination.

That said, can you say what "this" refers to? Cohen's invention of forcing in general?
— fishfry

That would be good. I had heard the expression but had no idea what it was. The article came as a revelation to me. And here I thought the reals consisted of rationals and irrationals. — jgill

Of course the reals consist of rationals and irrationals. That's provable from the axioms. Every model of the reals satisfies the axioms of the reals. FWIW I'm familiar with the work of Natalie Wolchover, the author of the article you linked. She puts the "pop" in pop science; which is to that that she's very good, up to a point; but not past that point. I didn't read the article and can't vouch for anything she might have said.

I can't really describe forcing. Timothy Chow, the author of A Beginner's Guide to Forcing, describes forcing as an "open exposition problem." That is, just as an open problem is a problem nobody knows how to solve, forcing is a subject that nobody knows how to explain to non-specialists.

I strongly recommend his article for anyone interested in the subject; bearing in mind that nobody would be expected to understand much of it, and the more times you read it and the more you read about forcing in general, the better vague understanding you'll get. But there's no known explanation that's any easier than diving in and learning the actual set theory, and it's a notoriously difficult subject.

The basic idea is that we want to know what things are consistent with a given set of axioms, so we try to find models that satisfy the axioms and also satisfy the extra things we're interested in. For example in geometry we can take the Euclidean axioms minus the parallel postulate (PP). We know the PP is consistent because Euclidean geometry is a model of the axioms plus PP. On the other hand in the 1840's, Riemann and others discovered that the axioms plus not-PP also had a model. This means that you can take the other axioms with PP or with the negation of PP, and both resulting systems are consistent. Alternatively, you can say that PP is "independent" of the other axioms; given the axioms, you can neither prove nor disprove PP.

A slightly more sophisticated example if you've seen group theory is that if you take the axioms for a group, you might want to know whether the "Abelian axiom" is consistent and/or provable; namely, is it true that for all x and y in a group, xy = yx.

Well, the integers with addition are an example of a group in which it's true. But the set of invertible 2x2 matrices with multiplication also form a group, and there are examples where commutativity fails. Since there are models of the group axioms with ah]d without commutativity, we would say that the "Abelian axiom" is independent of the group axioms.

Ok. Now with that in mind, down to cases. Cantor called the cardinality of the natural numbers $\aleph_0$. He proved that the cardinality of the real numbers was $2^{\aleph_0}$. And he showed that the next larger cardinal after $\aleph_0$ is $\aleph_1$.

So we have $\aleph_0$ directly followed by $\aleph_1$. And out there among the Alephs is $2^{\aleph_0}$. The question is, might it be the case that $2^{\aleph_0} = \aleph_1$? This question, or rather the claim that equality holds, is the continuum hypothesis (CH).

Cantor was unable to prove CH, and neither was anyone else. In 1940 Kurt Gödel proved that at the very least, CH was consistent with the other axioms of ZF, Zermelo-Fraenkel set theory. He did this by exhibiting a model in which it was true. This model is called Gödel's constructible universe. It's a universe of sets in which all the axioms of ZF are true, and in which CH is true. This showed that at the very least, assuming CH did not introduce any contradiction into ZF that wasn't already there.

That last remark needs explanation. What do I mean that CH doesn't introduce a contradiction that wasn't already there? Recall that Gödel had already proven in 1931 that ZF can't prove itself consistent. So the only way to know if ZF is consistent is to introduce even stronger principles that in effect assume it is. For all we know, set theory is inconsistent.

What Gödel showed, then, is that if ZF is consistent, so is ZF + CH. That is, all these proofs are relative consistency proofs. They don't show that anything is consistent; they only show that IF one system is consistent, then so is that system plus some other stuff.

But what about the negation of CH? Is that consistent with ZF as well? Gödel had shown that there's a model of ZF + CH. Could there be a model of ZF + not-CH? The problem is that nobody had any idea how to cook up alternative models of ZF. This was a real problem.

In 1963 an analyst named Paul Cohen figured it out. By analyst,I mean he was into real analysis -- epsilons and deltas and convergence and such. About as far away from mathematical logic as you can get. He woke up one day and said to himself, "I think I'll take a run at CH." He figured out how to cook up alternative models of ZF. In 1966 he won the Fields medal, the only Fields medal ever given for mathematical logic. I have always assumed that all the other official professional logicians must have been mighty annoyed. Some nonspecialist wakes up one day and solves the greatest unsolved problem in your field.

So ok all of that is preamble. How did he cook up alternative models? He invented a method called forcing. And having come this far, I really can't say much about it; first, because I don't know much about it myself, and second, because as Timothy Chow noted, nobody knows how to explain this to nonspecialists.

The idea basically is analogous to the procedure in abstract algebra where we adjoin roots to fields. That is, suppose that we believe in the rational numbers. We know the rationals satisfy the field axioms: you can add and multiply rationals to get another rational. Multiplication distributes over addition. And every nonzero rational has a multiplicative inverse.

Now suppose we want to prove that there is a field that contains the rationals and that also contains the square root of 2. We "adjoin" a meaningless symbol, $\sqrt 2$, to the rationals. We know nothing about this symbol other than that it has the formal property $(\sqrt{2})^2 = 2$.

In order to preserve the field axioms we have to say that all possible additions and multiplications are also in our new "extension field," as it's called. So we have a set of expressions of the form $a + b \sqrt 2$. We can then prove that the resulting system of formal expressions $a + b \sqrt 2$ itself satisfies the field axioms.

This is the best analogy for forcing. We start with a model of set theory, and we carefully add new "thingies," whatever they are, making sure that our new system also satisfies the axioms of set theory. If we're clever, we can arrange things so that CH turns out to be false in our new model. Then we collect our Fields medal. Cohen was clever.

So the idea -- and this is pretty much everything I know about it -- is first, we start with a model of ZF. But wait, since we can't prove within set theory that set theory is consistent, we don't know for sure if there even IS a model of set theory. But no worries. If there is no model, then set theory is inconsistent, and then we can prove ANYTHING, including CH and its negation. So, to get things off the ground, we assume that set theory is consistent and that it has a model.

Then -- and this step comes out of nowhere, pretty much -- since there is a model, there is a countable model. This is the famous, or infamous, Löwenheim–Skolem theorem.

What on earth does it mean to have a countable model of set theory? Doesn't ZF prove that there are uncountable sets? Well yes, it does. But now we have to broaden our understanding of what that means. What does it mean for a set to be countable? It means there is a bijection from that set to the natural numbers.

Suppose we have some set X, and a bijection from X to the naturals. So X is countable. Now suppose we have some model of set theory, and we throw out all the bijections between X and the naturals from the model, making sure we still have a model. Then from "inside the model," X would be uncountable; but from outside the model, from our God-like perspective, we can perfectly well see that X is really countable.

So we learn that uncountability is a "relative" notion. A set may be countable in one model, and uncountable in another. It just depends on which bijections are lying around.

So we assume we have a model of ZF; and we then know that if we do, there must be a countable model; and then we can show that if there's a countable model, there is a countable, transitive model.

Having done that, we take a countable, transitive model of ZF, and carefully add sets to it, making sure that we preserve the axioms of ZF, while cooking up a violation of CH.

Well there you go. I wrote a lot of words and didn't explain a thing about what forcing is. Definitely go read Tim Chow's excellent article.

I reject "the empty set" for a reason similar to the reason why I rejected a set with no inherent order. it's a fiction which has no purpose other than to hide the shortcomings of the theory. There are very good reasons why "0" ought to represent something in a class distinct from numbers. There are even reasons why "1" ought to be in a distinct class. — Metaphysician Undercover

Since no two things are identical in spatiotemporal reality, do you also reject the number 2?

This is the best analogy for forcing. — fishfry

Thank you. That's very clear. I appreciate your commentary.

She puts the "pop" in pop science — fishfry

She popped Cohen's Filter pretty good. :smile:

I'm not a set theorist, but I have some thoughts. — TonesInDeepFreeze

And good ones they are! Thank you for your commentary.

My initial guess was that a set is something that contains and not something in its own right. — Gregory

Sets can contain other sets. In fact a set is "something" in addition to its constituent elements. It's a "something" that allows us to treat the elements as a single whole. If I have the numbers 1, 2, and 3, that's three things. The set {1,2,3} is one thing. It's a very subtle and profound difference. A set is a thing in and of itself. The empty set, the set containing the empty set, and the set containing the set containing the empty set are three distinct sets. Which we could, if we wanted to, take together into a set! Like this:

$\{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}\}$.

So zero remains a nothingness of anything in that case. — Gregory

Zero is not nothing. Zero is a particular point on the real number line. Or the address or location of a point, if you prefer. Zero is a particular thing. It represents the cardinality of the set of purple flying elephants in my left pocket. Zero is something. Nothing is nothing. Which can be read two different ways!

Very abstract ideas. Couldn't structure just be that which contains a process and thus, like sets which compose it, it is nothing in itself. — Gregory

Well, what's a process? If a process is something that can be "contained," it needs explanation.

This would certainly make mathematics a system of process and divorce it from the notion that anything rests and stays permanent within it — Gregory

The natural numbers seem permanent. They don't change from day to day, whether you regard them as sets, as in the von Neumann finite ordinals, or as a process of starting at 0 and taking successors.

I am no expert on these things though.

hank you for your commentary. — jgill

You're very welcome, glad that helped.

You write very well. That must be why I like to engage with you, not that I want to troll you.

Since no two things are identical in spatiotemporal reality, do you also reject the number 2? — Luke

Yes that' the mathematical Platonism I reject. I believe we had a lengthy discussion on this in the other thread, you and I. The number 2 is an unnecessary intermediary between the symbol, and what the symbol represents, or means, in use. Of course you might use the symbol "2" to represent the number 2, but then you are writing fiction.

Sets can contain other sets. In fact a set is "something" in addition to its constituent elements. It's a "something" that allows us to treat the elements as a single whole. If I have the numbers 1, 2, and 3, that's three things. The set {1,2,3} is one thing. It's a very subtle and profound difference. A set is a thing in and of itself. — fishfry

This is what I was asking about earlier, what allows for that unity if not some judgement of criteria, making the elements similar, or the same in some respect., a definition. This is a very important ontological question because we do not even understand what produces the unity observed in an empirical object.

Suppose you arbitrarily name a number of items and designate it as a set. You have created "a thing" here, a set, which is some form of unity. But that unity is completely fictitious. You are just saying that these items compose a unity called "a set", without any justification for that supposed unity. In its simplest from, this is the issue of counting apples and oranges. We can count an apple and orange as two distinct objects, and call them 2 objects. But if we want to make them a set we assume that something unifies them. If we are allowed to arbitrarily designate unity in this way, without any criteria of similarity, then our concept of unity, which some philosophers (Neo-Platonist for example) consider as fundamental loses all its logical strength or significance.

The number 2 is an unnecessary intermediary between the symbol, and what the symbol represents, or means, in use. — Metaphysician Undercover

How can either the number 2 or the numeral "2" represent or mean anything in use if no two things are identical in spatiotemporal reality? Isn't the law of identity the basis of your mathematics?

You write very well. That must be why I like to engage with you, not that I want to troll you. — Metaphysician Undercover

Thank you. I'll get to your second post later, I'm falling behind.

In set theory everything is a set.
— fishfry

I didn't know that, but it makes the problems which I've apprehended much more understandable. — Metaphysician Undercover

Yes. Everything is a set. Or what they call a "pure set," meaning a set whose elements are also sets. There are as I mentioned set theories with urlements, also called atoms, but these are niche theories and not of interest to us at present. So everything is a set that contains other sets.

If everything is a set, in set theory, then infinite regress is unavoidable. — Metaphysician Undercover

No not at all. First, what's wrong with infinite regress? After all the integers go backwards endlessly: ..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ... You can go back as far as you like. I'm fond of using this example in these endlessly tedious online convos about eternal regress in philosophy. Cosmological arguments and so forth. Why can't time be modeled like that? It goes back forever, it goes forward forever, and we're sitting here at the point 2021 in the Gregorian coordinate system.

However in set theory there is no infinite regress. That's guaranteed by the axiom of foundation, also known as the axiom of regularity. It says that no set is a member of itself and it also rules out all circular membership chains like $a \in b \in c \in a$ and so forth. In standard set theory all sets are well-founded. That means that if you take its elements, which are themselves sets; and take their elements, which are themselves sets; and drill all the way down; you are guaranteed to hit bottom. There is no possible infinite regress of sets.

For completeness I'll mention that people do study non well-founded sets, but this is yet another niche interest and of no interest to us here. In standard set theory all sets are well-founded. There can never be an infinite regress of sets.

A logical circle is sometimes employed, like the one mentioned here ↪jgill to disguise the infinite regress, but such a circle is really a vicious circle. — Metaphysician Undercover

Oh jeez man, you embarrassed yourself a little here. See now I feel bad pointing out that you embarrassed yourself because you complimented me. LOL.

@jgill was referring to the Riemann sphere, a way of viewing the complex numbers as a sphere. It's based on the simple idea of stereographic projection, a map making technique that allows you to project the points of a sphere onto a plane. There is nothing mystical or logically questionable about this. You should read the links I gave and then frankly you should retract your remark that the Riemann sphere is a "vicious circle." You're just making things up. Damn I feel awful saying that, now that you've said something nice about me.

I reject "the empty set" for a reason similar to the reason why I rejected a set with no inherent order. it's a fiction which has no purpose other than to hide the shortcomings of the theory. — Metaphysician Undercover

I find this deeply inconsistent with other things you've said. Earlier I was making the point that we can have two sets, X and Y, with a bijection between them, and we can say they are "cardinally equivalent," without knowing what that exact cardinal number is. Then later we can define cardinal numbers, and assign one of them to X and Y.

You claimed that the cardinal numbers were already "out there" waiting to be assigned. You used that idea to claim that I was wrong about ordinals being logically prior to cardinals.

So you somehow manage to believe in the existence of cardinal numbers, which include the endless hierarchy of gigantic cardinals given to us by Cantor's theorem: that a set's powerset is always of a strictly larger cardinality than the set. So we have the cardinality of the natural numbers, which is smaller than the powerset of the natural numbers, which is smaller than the powerset of the powerset of the natural numbers, and on and on forever.

You believe in the metaphysical existence of all of these humongously unimaginable cardinals; yet you deny the existence of the empty set on which they're all founded.

That's logically inconsistent.

But never mind that. I don't believe in the existence of the empty set either. Not in reality. If I see a table with nothing on it, there's nothing on it. I do NOT see the empty set sitting on the table. So I agree with you, I don't believe in the empty set.

But I DO believe in the empty set as a formal construction in the game of math. In fact the empty set is the extension of the predicate $x \neq x$. Surely you must agree with that, since you believe in the law of identity.

Can you clarify your remark? If you don't believe in the metaphysical existence of the empty set, I'm in complete agreement. But if you claim to disbelieve in the empty set as a mathematical object, that's like disbelieving in the way the knight moves in chess. You can't disbelieve in it, it's just one of the rules of the game.

And again; if you are so strong on the law of identity, then you must believe that $\emptyset = \{x : x \neq x \}$.

There are very good reasons why "0" ought to represent something in a class distinct from numbers. There are even reasons why "1" ought to be in a distinct class. — Metaphysician Undercover

I don't know what you mean? What do you mean by "class" in this context? Is 2 in its own class? Why not, it's 1 + 1, right? Although in the past you've denied even that, so I hope we're not going down that road again.

No, not at all. First, the elements of a set need not be "the same" in any meaningful way. The only thing they have in common is that they're elements of a given set.
— fishfry

This may be the case, but you ought to recognize that being elements of the same set makes them "the same" in a meaningful way. — Metaphysician Undercover

Jeez Louise man. I say: "The only thing they have in common is that they're elements of a given set." And then you say I "ought to recognize ..." that very thing.

Did you simply not read what I wrote? Do you like to just push my buttons? I say something as clear as day; and you respond by admonishing me that I "ought to recognize" the very thing I've just said. I don't get it. That's why I sometimes think you are trolling me.

Otherwise, a set would be a meaningless thing. So when you said for instance, that {0,1,2,3,} is a set, there must be a reason why you composed your set of those four elements. — Metaphysician Undercover

Ok, {5, my lunch, the Mormon Tabernacle Choir}. What of it?

That reason constitutes some criteria or criterion which is fulfilled by each member constituting a similarity. — Metaphysician Undercover

A very disingenuous point. The elements of a set need have no relation to one another nor belong to any articulable category or class of thought, OTHER THAN being gathered into a set.

This is a simple feature of common language use. A word may receive its meaning through usage rather than through an explicit definition. — Metaphysician Undercover

Ok, you are now agreeing with me on an issue over which you've strenuously disagreed in the past. You have insisted that "set" has an inherent meaning, that a set must have an inherent order, etc. I have told you many times that in set theory, "set" has no definition. Its meaning is inferred from the way it behaves under the axioms.

And now you are making the same point, as if just a few days ago you weren't strenuously disagreeing with this point of view.

But in any event, welcome to my side of the issue. Set has no definition. Its meaning comes exclusively from its behavior as specified by the axioms.

That the word has no definition does not mean that it has no meaning, its meaning is demonstrated by its use, as is the case with an ostensive definition. — Metaphysician Undercover

Completely agreed. And therefore a mathematical set has no inherent order, because that's how sets are used in set theory. Can you see that you've now completely conceded the point?

Allowing that a word, within a logical system, has no explicit definition, allows the users of the system an unbounded freedom to manipulate that symbol, (exemplified by TonesInDeepFreeze's claim with "least"), but the downfall is that ambiguity is inevitable. — Metaphysician Undercover

No question about it. There are philosophers and set theorists who question whether the mathematical conception of set is even coherent. You'll get no disagreement from me on that point. Although just because @Tones forgot to mention that ordinal < means set membership doesn't support your point, it only means @Tones forgot to mention it.

This is an example of the uncertainty which content brings into the formal system, that I mentioned in the other thread. — Metaphysician Undercover

No question. As Hilbert famously pointed out about the axioms of geometry, "One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs."

That's a perfect expression of the formalist position on axiomatic systems.

There is no set of ordinals, this is the famous Burali-Forti paradox.
— fishfry

This I would say is a good representation of the philosophical concept of "infinite". Note that the philosophical conception is quite different from the mathematical conception. If every ordinal is a set composed of other ordinals, and there is no limit to the "amount" of ordinals which one may construct, then it ought to be very obvious that we cannot have an ordinal which contains all the ordinals, because we are always allowed to construct a greater ordinal which would contain that one as lesser. — Metaphysician Undercover

Yes, very good! That's essentially the proof. Any set of ordinals is itself an ordinal. Hence there is no set of all ordinals.

So we might just keep getting a greater and greater ordinal, infinitely, and it's impossible to have a greatest ordinal. — Metaphysician Undercover

Cesare Burali-Forti couldn't have said it better himself.

I think there is a way around this though, similar to the way that set theory allows for the set of all natural numbers, which is infinite. As you say, "set" has no official definition. And, you might notice that "set" is logically prior to "cardinal number". So all that is required is a different type of set, one which is other than an ordinal number, which could contain all the ordinals. It would require different axioms. — Metaphysician Undercover

The collection of all ordinals is a proper class. In standard set theory, ZF or ZFC, there are no official proper classes, so "proper class" is a colloquial expression. There are set theories in which proper classes are formalized. Either way, a proper class is a collection that's too big to be a set. The class of all sets, the class of all ordinals, etc.

There is no general definition of number.
— fishfry

This is not really true now, if we accept set theory. — Metaphysician Undercover

I am disappointed that you didn't accept the historical point I made earlier. Zero, negative numbers, irrational numbers, complex numbers, and transfinite numbers didn't used to be accepted as numbers, and now they are. Likewise p-adic and quaternions, two other types of numbers discovered only in the past couple of centuries. "Number" is a historically contingent concept.

If "set" is logically prior to "number", then "set"
is a defining principle of "number". — Metaphysician Undercover

Not at all. Bricks are the constituents of buildings, but all the different architectural styles aren't inherent in bricks. There are plenty of sets that aren't numbers. Topological spaces aren't numbers. The set of prime numbers isn't a number. Groups aren't numbers. The powerset of the reals isn't a number. Just because numbers are made of sets in the formalism doesn't mean every set is a number.

That is why you and I agreed that each ordinal is itself a set. We have a defining principle, an ordinal is a type of set, and a cardinal is a type of ordinal. — Metaphysician Undercover

So now you agree that ordinals are logically prior to cardinals? I am glad you have internalized this fact to the point where it now seems obvious to you, when only a few days ago you were strenuously disagreeing.

But so what? It's true that in set theory everything is a set, but that doesn't mean everything is a number. I don't follow your logic.

Correction, at my worst I am a part-time Platonist. At my best I am a fulltime Neo-Platonist. — Metaphysician Undercover

I looked that up on Wikipedia and it seemed to be about some kind of mystical emanation from "The One." Lost me, I'm afraid. But I'm shocked that you believe in the vast multitude of gigantic cardinal numbers, while professing disbelief in the empty set.

We do not have to go the full fledged Platonic realism route here, to maintain a realism. This is what I tried to explain at one point in another thread. — Metaphysician Undercover

I don't doubt that you tried to explain this to me and I missed it. Even now I don't think I know the difference between Platonism and realism.

We only need to assume the symbol "5", and what the symbol represents, or means. There is no need to assume that the symbol represents "the number 5", as some type of medium between the symbol, and what the symbol means in each particular instance of use. So when I say that a thing exists, and has a measurement, regardless of whether it has been measured, what I mean is that it has the capacity to be measured, and there is also the possibility that the measurement might be true. — Metaphysician Undercover

I'm afraid you lost me a bit there. The number 5 exists as a formal symbol and concept in set theory. What it is "for real" I am not sure. It's the thing that comes after 4, that's the structuralist idea, I think.

If you think that I was advocating for mathematical Platonism, then you misunderstood. I was advocating for realism. — Metaphysician Undercover

Ok. I admit to being unclear on this. I'm only struck by finding you believing in the pre-existence of the vast array of cardinal numbers, yet disbelieving in the empty sets and set theory in general.

A mathematical Platonist thinks of ideas as objects. I recognize the reality of ideas, and furthermore I accept the priority of ideas, so I am idealist. But I do not think of ideas as objects, as mathematical Platonists do, I think of them as forms, so I'm more appropriately called Neo-Platonist. — Metaphysician Undercover

Ok. Is this a bit structuralist? Natural numbers aren't particular things, but they are the relations among them; that is, 5 is the thing that follows 4, and that's all I need to know about it.

This is that vague boundary, the grey area between fact and fiction which we might call "logical possibility". If we adhere to empirical principles, we see that there are individual objects in the world, with spatial separation between them. If we are realist, we say that these objects which are observed as distinct, really are distinct objects, and therefore can be counted as distinct objects. We might see three objects, and name that "3", but "3" is simply what we call that quantity. Being realist we think that there is the same quantity of objects regardless of whether they've been counted and called "3" or not. — Metaphysician Undercover

Ok. But 3's easy. How about the transfinite cardinals? You believe in them yet disbelieve in set theory? That's a hard row to hoe.

But if we give up on the realism, and the empirical principles, there is no need to conclude that what is being seen is actually a quantity of 3. There might be no real boundaries between things, and anything observed might be divisible an infinite number of times. Therefore whatever is observed could be any number of things. This is the world of fiction, which some might call "logical possibility", and you call pure mathematics. Empirical truths, like the fact that distinct objects can be counted as distinct objects, pi as the ratio between circumference and diameter of a circle, and the Pythagorean theorem, we say are discovered. Logical possibilities are dreamt up by the mind, and are in that sense fictions. — Metaphysician Undercover

You are now willing to agree with me that there may be some virtue to considering math to be an interesting and useful fiction? @Meta I find you agreeing with my point of view in this post.

I do not mean to argue that dreaming up logical possibilities is a worthless activity. — Metaphysician Undercover

You are mellowing! And agreeing with me!! I must be having an effect. I will say that you have achieved some genuine mathematical insight lately.

What I think is that this is a primary stage in producing knowledge. We look at the empirical world for example and create a list of possibilities concerning the reality of it. The secondary stage is to eliminate those logical possibilities which are determined to be physically impossible through experimentation and empirical observation. — Metaphysician Undercover

So you would ban the teaching of Euclidean geometry now that the physicists have accepted general relativity? We disagree on this. Math is the study of that which is logically possible. Math leaves what's real to the physicists. And of course even the physicists no longer have much interest in what's real, but that's a criticism for another time. But math is not bound by what's real. On this we disagree strongly.

So we proceed by subjecting logical possibilities, and axioms of pure mathematics, to a process of elimination. — Metaphysician Undercover

Would you ban Euclidean geometry from the high school curriculum because it turns out not to be strictly true?

Sets can contain other sets. In fact a set is "something" in addition to its constituent elements. It's a "something" that allows us to treat the elements as a single whole. If I have the numbers 1, 2, and 3, that's three things. The set {1,2,3} is one thing. It's a very subtle and profound difference. A set is a thing in and of itself.
— fishfry

This is what I was asking about earlier, what allows for that unity if not some judgement of criteria, making the elements similar, or the same in some respect., a definition. — Metaphysician Undercover

There is no criterion. In fact there are provably more sets than criteria. If by "criterion" you mean a finite-length string of symbols, there are only countably many of those, and uncountably many subsets of natural numbers. So most sets of natural numbers have no unifying criterion whatsoever, They're entirely random.

This is a very important ontological question because we do not even understand what produces the unity observed in an empirical object. — Metaphysician Undercover

I just proved that most sets of natural numbers are entirely random. There is no articulable criterion linking their members other than membership in the given set. There is no formal logical definition of the elements. There is no Turing machine or computer program that cranks out the elements. That's a fact.

Suppose you arbitrarily name a number of items and designate it as a set. — Metaphysician Undercover

Ok.

You have created "a thing" here, a set, which is some form of unity. But that unity is completely fictitious. You are just saying that these items compose a unity called "a set", without any justification for that supposed unity. — Metaphysician Undercover

Tru dat. Or as the kids say, Yes, indubitably so.

After all as I just noted, there are only countably many criteria, formulas, computer programs. But there are uncountably many sets of natural numbers. Most of those sets are entirely random. There is no rhyme or reason to their constituent members. They're just random collections.

Unless you are a constructivist, in which case you deny the existence of random sets. Some go down that path.

In its simplest from, this is the issue of counting apples and oranges. We can count an apple and orange as two distinct objects, and call them 2 objects. But if we want to make them a set we assume that something unifies them. — Metaphysician Undercover

Only their collection into a set.

If we are allowed to arbitrarily designate unity in this way, without any criteria of similarity, then our concept of unity, which some philosophers (Neo-Platonist for example) consider as fundamental loses all its logical strength or significance. — Metaphysician Undercover

Do you deny the existence of all sets that cannot be cranked out by a Turing machine or at least defined in first-order logic? You can do that if you like. I don't see the use. Consider the following thought experiment. You flip a fair coin a countably infinite number of times. You thereby generate a sequence of 1's and 0's. What invisible magic forces the resulting bitstring to be computable, or describable by an algorithm or formula? Why can't the result be completely random, having no pattern at all? That's by far the most likely outcome.

I mentioned finding a recursive definition of 'prior'. It's simple.

We define the set of symbols prior to a symbol s:

If s is primitive, prior(s) = 0
If s is defined, prior(s) = U{k | Et(t occurs in the definiens for s & k = prior(t))}

Then:

j is prior to s <-> j e prior(s)

How can either the number 2 or the numeral "2" represent or mean anything in use if no two things are identical in spatiotemporal reality? Isn't the law of identity the basis of your mathematics? — Luke

Obviously, "2" refers to two distinct and different things. If there was only one thing we'd have to use "1".

No not at all. First, what's wrong with infinite regress? After all the integers go backwards endlessly: ..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ... You can go back as far as you like. I'm fond of using this example in these endlessly tedious online convos about eternal regress in philosophy. Cosmological arguments and so forth. Why can't time be modeled like that? It goes back forever, it goes forward forever, and we're sitting here at the point 2021 in the Gregorian coordinate system. — fishfry

Again, this is the difference between fiction and fact. We can imagine infinite regress, and imagine time extending forever backward, but it isn't consistent with the empirical evidence. That's the problem with infinite regress, it's logically possible, but proven through inductive (empirical) principles (Aristotle's cosmological argument for example) to be impossible.

jgill was referring to the Riemann sphere, a way of viewing the complex numbers as a sphere. It's based on the simple idea of stereographic projection, a map making technique that allows you to project the points of a sphere onto a plane. There is nothing mystical or logically questionable about this. You should read the links I gave and then frankly you should retract your remark that the Riemann sphere is a "vicious circle." You're just making things up. Damn I feel awful saying that, now that you've said something nice about me. — fishfry

I beg to differ. Didn't we go through this already in the Gabriel's horn thread. It seems like you haven't learned much about the way that I view these issues. You write very well, but your thinking hasn't obtained to that level. Another example of the difference between form and content.

Jeez Louise man. I say: "The only thing they have in common is that they're elements of a given set." And then you say I "ought to recognize ..." that very thing. — fishfry

Are you denying the contradiction in what you wrote? If they are members of the same set, then there is a meaningful similarity between them. Being members of the same set constitutes a meaningful similarity. You said "the elements of a set need not be 'the same' in any meaningful way. The only thing they have in common is that they're elements of a given set." Can't you see the contradiction? If they are said to be members of the same set, then they are the same in some meaningful way. It is contradictory to say that they are members of the same set, and also say that they are not the same in any meaningful way.

Another example of this same sort of contradiction is when people refer to a difference which doesn't make a difference. If you apprehend it as a difference, and speak about it as a difference, then clearly it has made a difference to you. Likewise, if you see two things as elements of the same set, then clearly you have apprehended that they are the same in some meaningful way. To apprehend them as members of the same set, yet deny that they are the same in a meaningful way, is nothing but self-deception. Your supposed set is not a set at all. You are just saying that there is such a set, when there really is no such set. You are just naming elements and saying "those are elements of the same set" when there is no such set, just some named elements. Without defining, or at least naming the set, which they are members of, there is no such set. And, naming the set which they are elements of is a designation of meaningful sameness.

Here is a feature of imaginary things which you ought to learn to recognize. I discussed it briefly with Luke in the other thread. An imaginary thing (and I think you'll agree with me that sets are imaginary things, or "pure abstraction" in your terms) requires a representation, or symbol , to be acknowledged. And, for an imaginary thing, to exist requires being acknowledged. However, the symbol, or representation, is not the imaginary thing. The imaginary thing is something other than the symbols which represent it. So the imaginary thing necessarily has two distinct aspects, the representation, and the thing itself, the former is called form, the latter, content. And this is necessary of all imaginary things.

The important point is that you cannot claim to remove one of these, from the imaginary thing, because both are necessary. So a purely formal system, or pure content of thought, are both impossibilities. And when you say "these things are elements of the same set", you have in a sense named that set, as the set which these things are elements of, thereby creating a meaningful similarity between them. The point being that a meaningful similarity is something which might be created, solely by the mind and that is how the imagination works in the process of creating fictions. But when something is a creation, it must be treated as a creation.

A very disingenuous point. The elements of a set need have no relation to one another nor belong to any articulable category or class of thought, OTHER THAN being gathered into a set. — fishfry

Again, incoherency fishfry. Can't you see that? There is necessarily a reason why you place them in the same set, and this 'reason why' is something other than actually being in the same set. You are not acknowledging that "being gathered into a set" requires a cause, and that cause is something other than being in the same set. So the relation that the things have to one another by being in the same set is not the same as the relation they have to one another by being caused to be in the same set. And things which are in the same set necessarily have relations to each other which are other than being in the same set, because they have relations through the cause, which caused them to be in the same set.

A set is an articulable category, or class of thought! If a set is not a class of thought, then what the heck is it, jeez louise? And don't tell me it might be anything because it is not defined, because even "anything" is a class of thought.

Ok, you are now agreeing with me on an issue over which you've strenuously disagreed in the past. You have insisted that "set" has an inherent meaning, that a set must have an inherent order, etc. I have told you many times that in set theory, "set" has no definition. Its meaning is inferred from the way it behaves under the axioms. — fishfry

It appears like you didn't read what I said. That a word is not defined does not mean that it has no meaning. As I said, it may derive meaning from its use. If the word is used, then it has meaning. So if "set" derives it's meaning from the axioms, then there is meaning which inheres within, according to its use in the axioms.

And now you are making the same point, as if just a few days ago you weren't strenuously disagreeing with this point of view.

But in any event, welcome to my side of the issue. Set has no definition. Its meaning comes exclusively from its behavior as specified by the axioms. — fishfry

What we do not agree on is what "inherent order" means. i really do not see how you get from the premise, that "set" is not defined, but gets its meaning from its use, to the conclusion that a set might have no inherent order. In order for the word "set" to exist, it must have been used. Therefore it is impossible for "set" not to have meaning, and we might say that there is meaning (order, if order is analogous to meaning, as you seem to think), which inheres within. Wouldn't you agree with this, concerning the use of any word? If the word has been used, there is meaning which inheres within, as given by that use. And, for a word to have any existence it must have been used.

Not at all. Bricks are the constituents of buildings, but all the different architectural styles aren't inherent in bricks. There are plenty of sets that aren't numbers. Topological spaces aren't numbers. The set of prime numbers isn't a number. Groups aren't numbers. The powerset of the reals isn't a number. Just because numbers are made of sets in the formalism doesn't mean every set is a number. — fishfry

It appears like you misunderstood. I didn't say every set is a number, to the contrary. I said that if we proceed under the precepts of set theory, every number is a set. Therefore we cannot say that "number" is undefined because "set" is now a defining feature of "number", just like when we say every human beings is an animal, "animal" becomes a defining feature of "human being".

Meta I find you agreeing with my point of view in this post. — fishfry

Didn't it strike you that I was in a very agreeable mood that day? Now I'm back to my old self, pointing out your contradiction in saying that things could be in the same set without having any meaningful relation to each other, other than being in the same set. You just do not seem to understand that things don't just magically get into the same set. There is a reason why they are in the same set.

Maybe at some point we'll discuss the supposed empty set. How do you suppose that nothing could get into a set?

So you would ban the teaching of Euclidean geometry now that the physicists have accepted general relativity? — fishfry

Actually I do not agree with general relativity, so I would ban that first.

Would you ban Euclidean geometry from the high school curriculum because it turns out not to be strictly true? — fishfry

You keep saying things like this, the Pythagorean theorem is not true, now Euclidian geometry in general is not true. I suppose pi is not true for you either? Until you provide some evidence or at least an argument, these are just baseless assertions.

There is no criterion. In fact there are provably more sets than criteria. If by "criterion" you mean a finite-length string of symbols, there are only countably many of those, and uncountably many subsets of natural numbers. So most sets of natural numbers have no unifying criterion whatsoever, They're entirely random. — fishfry

On what basis do you say they are a unity then? You have a random group of natural numbers. Saying that they are a unity does not make them a unity. So saying that they are a "set" does not make them a unity. This is where you need a definition of "set" which would make a set a unity.

I just proved that most sets of natural numbers are entirely random. There is no articulable criterion linking their members other than membership in the given set. There is no formal logical definition of the elements. There is no Turing machine or computer program that cranks out the elements. That's a fact. — fishfry

Then you have no basis to your claim that a set is a unity. And you cannot treat a set as a unified whole. If a set is supposed to be a unified whole, then you cannot claim that "set" is not defined.

A vicious regress is one that must begin at the beginning but starts at the other end. So going from A to B to A when it should go on to C and infinity. An actual infinity is not vicious and a vicious regress is an infinite loop that we don't comprehend (and therefore discard). There is nothing that rules out an infinity in nature because infinities make sense in mathematics and nature acts mathematically. Physically this might not be the whole story, but Aristotelean finitism is truly outdated and a little silly. Logic and mathematics seem perfectly capable of understanding how an infinity would work from an infinite past all the way to infinite space.

jgill was referring to the Riemann sphere, a way of viewing the complex numbers as a sphere . . .— fishfry

I beg to differ — Metaphysician Undercover

I'm missing your point also. What's your gripe about the innocuous Riemann sphere? :chin:

Obviously, "2" refers to two distinct and different things. If there was only one thing we'd have to use "1". — Metaphysician Undercover

"2" can also refer to two distinct but same things, such as "things" of the same type or category. But all categories/classifications are equally as fictitious and man-made as the sets and orders you reject. As you say:

Suppose you arbitrarily name a number of items and designate it as a set. You have created "a thing" here, a set, which is some form of unity. But that unity is completely fictitious. You are just saying that these items compose a unity called "a set", without any justification for that supposed unity. — Metaphysician Undercover

Scientists justified both the inclusion and exclusion of Pluto as a planet at different times. Like Pluto, many individual "things" are borderline cases in their classification. Moreover, nothing guarantees the perpetuity of any category/set, or of what defines ("justifies") the inclusion of its members.

Furthermore, if you base your mathematics on empiricism rather than on "abstraction" or "fiction", then you must also reject fractions, since a half cannot be exactly measured in reality.

If we adhere to empirical principles, we see that there are individual objects in the world, with spatial separation between them. If we are realist, we say that these objects which are observed as distinct, really are distinct objects, and therefore can be counted as distinct objects. We might see three objects, and name that "3", but "3" is simply what we call that quantity. Being realist we think that there is the same quantity of objects regardless of whether they've been counted and called "3" or not.

But if we give up on the realism, and the empirical principles, there is no need to conclude that what is being seen is actually a quantity of 3. There might be no real boundaries between things, and anything observed might be divisible an infinite number of times. Therefore whatever is observed could be any number of things. — Metaphysician Undercover

If there are "no real boundaries between things", then acknowledging that "anything observed might be divisible an infinite number of times" is not to "give up on the realism", but to adhere to it.

I'm missing your point also. What's your gripe about the innocuous Riemann sphere? :chin: — jgill

To make infinite numbers into a circle is to make a vicious circle. It is to say that the beginning is the same as the end. And this is what allows for the faulty view of time which fishfry described.

"2" can also refer to two distinct but same things, such as "things" of the same type or category. — Luke

This is a different sense of "same", not consistent with the law of identity.

But all categories/classifications are equally as fictitious and man-made as the sets and orders you reject. — Luke

When they are based in empirical observation they are not equally fictitious. Remember, fishfry speaks of pure abstraction, and claims that a set might be absolutely random..

Scientists justified both the inclusion and exclusion of Pluto as a planet at different times. Like Pluto, many individual "things" are borderline cases in their classification. Moreover, nothing guarantees the perpetuity of any category/set, or of what defines ("justifies") the inclusion of its members. — Luke

That a person later decides to have been wrong in an earlier judgement, is not relevant.

Furthermore, if you base your mathematics on empiricism rather than on "abstraction" or "fiction", then you must also reject fractions, since a half cannot be exactly measured in reality. — Luke

I do reject fractions, I believe that the principles employed are extremely faulty, allowing that a unit might be divided in any way that one wants. This faultiness I believe, is responsible for the Fourier uncertainty In reality, how a unit can be divided is dependent on the type of unit.

If there are "no real boundaries between things", then acknowledging that "anything observed might be divisible an infinite number of times" is not to "give up on the realism", but to adhere to it. — Luke

That's the case if there are "no real boundaries between things". But I am arguing that empirical evidence demonstrates that there are real boundaries.

"2" can also refer to two distinct but same things, such as "things" of the same type or category. — Luke

This is a different sense of "same", not consistent with the law of identity. — Metaphysician Undercover

So "2" cannot refer to two distinct but same things? You cannot have 2 apples or 2 iPhones, etc?

But all categories/classifications are equally as fictitious and man-made as the sets and orders you reject. — Luke

When they are based in empirical observation they are not equally fictitious. — Metaphysician Undercover

So "2" can refer to two distinct but same things? You can have 2 apples or 2 iPhones, etc?

Scientists justified both the inclusion and exclusion of Pluto as a planet at different times. Like Pluto, many individual "things" are borderline cases in their classification. Moreover, nothing guarantees the perpetuity of any category/set, or of what defines ("justifies") the inclusion of its members. — Luke

That a person later decides to have been wrong in an earlier judgement, is not relevant. — Metaphysician Undercover

The categories we use are either discovered or man-made. If they are discovered, then how can we be "wrong in an earlier judgement" about them; why are there borderline cases in classification; and why does nothing guarantee their perpetuity as categories?

I do reject fractions — Metaphysician Undercover

You need help.

I believe that the principles employed are extremely faulty, allowing that a unit might be divided in any way that one wants. — Metaphysician Undercover

What principles should be employed?

In reality, how a unit can be divided is dependent on the type of unit. — Metaphysician Undercover

How many slices should a cake or a pizza have? Also, doesn't this reintroduce the fractions you rejected?

That's the case if there are "no real boundaries between things". But I am arguing that empirical evidence demonstrates that there are real boundaries. — Metaphysician Undercover

Where's the argument?