Maybe it is not necessary to define set membership?

[jordi]

In the past, I have asked in this forum about the concept of set membership, in the context of ZFC.

I guess it is a normal reaction to be a bit surprised by the usual statement in books that the set membership relationship is "undefined".

But I have had this idea: a typical definition of the natural numbers, by von Neumann, is that the number zero is defined as the empty set, the number one is defined as the set consisting of the empty set, and so on.

Since, at least from the Physics point of view, the only sets that we need are the numbers (and it is not problem to define, after the natural numbers, the integers, the rationals, the reals, the complexes, and then other mathematical objects out of those), we could proceed as follows:

We take ZFC at face value, without defining what set membership is. Set membership is just a binary relationship. ZFC axioms just tells us how objects, sets and the set membership relate to each other. We do not need to give a specific model of sets.

But even without defining what set membership is, we can define the number zero as the empty set, one as ... as above. And then, once we have the natural numbers, we define the integers ... and so on. So, we do not need to define what set membership is, in order to define and create the usual objects we use in mathematics. We only need to find the right properties, and define the right axioms (ZFC).

Is this argument correct?

 

[yossell]

The empty set is defined as the set which has no members. So 'x is a member of y' is used to define the empty set. So without a definition of 'x is a member of y' there is no definition of the empty set.

ZFC at face value contains the predicate 'x is a member of y'. And this is a basic predicate of set-theory. In that sense, it is undefined -- but it is a primitive. Yet your post seems to suggest you want to treat it purely formally.

Even if you had the natural numbers, to get the reals, you have to, at some point, take arbitrary subsets of a countable sets -- be it arbitrary Dedekind cuts or whatever. So here the notion of a set of all subsets of an infinite set comes into play, and it's hard to see what would play the role of this if you tried to treat 'x is a member of y' as a purely formal predicate.

There are attempts to try and think of the predicates of set theory as 'defined' structurally -- any relation substituted for 'x is a member of y' that made the axioms of set theory true can count as a membership relation. This approach runs into some difficulties with non-standard models, but it may be what you have in mind. If you were go that way, though, one might just apply this idea directly to Peano Arithmetic, or your theory of the reals or whatever, without needing set theory at all to do physics.

 

[jordi]

I think that most books consider the membership relationship as formal. There is no "constructive definition" of the membership relationship.

For example, the empty set is called that way because the "intuitive membership relationship" suggests so. But if you take the axioms and see the definition of the empty set, in fact, the definition works equally well for any binary relationship.

Or at least, this is my understanding. Mathematical logic books, and set theory books, are scarce, and at least for my taste, they do not "tell the truth" in a straightforward fashion. And this creates some kind of frustration. One learns to "shut up and calculate", à la QM, but the "important questions" remain (apparently, at least under my limited understanding) unclear.

 

[yossell]

There are no constructive definitions of 'and', 'all', '=' either. But this doesn't mean that these are to be thought of as merely formal marks on paper. They are meant to be meaningful.

The empty set is defined as that set which contains no members. I'm not sure where you're going with your second paragraph. Again, it sounds like you want to press a kind of structuralist approach to set theory -- any relation which satisfies the axioms can deserve to be called a membership relation. That approach has some problems, is not (as far as I can see) how set theory is typically presented, but it may work out. But, as I said, you might as well also apply it to number theory and others, thereby avoiding set-theoretic reductions of other mathematical notions altogether. The natural number 1 is just that number which is not the successor of any other -- where successor is any relation which makes the axioms of peano arithmetic come out true.

Funnily enough, I hate the QM 'shut up and calculate' line -- but I don't feel math logic does that at all. There are some questions which it stays neutral on though: but it seems to me a number of math logic and set theory books do have a few sections that discuss and acknowledge the background philosophical and conceptual issues.

 

[SSequence]

But even without defining what set membership is, we can define the number zero as the empty set, one as ... as above. And then, once we have the natural numbers, we define the integers ... and so on. So, we do not need to define what set membership is, in order to define and create the usual objects we use in mathematics.

Even if you had the natural numbers, to get the reals, you have to, at some point, take arbitrary subsets of a countable sets -- be it arbitrary Dedekind cuts or whatever. So here the notion of a set of all subsets of an infinite set comes into play, and it's hard to see what would play the role of this if you tried to treat 'x is a member of y' as a purely formal predicate.

Here is how I think about the same thing. I think defining integers, rationals could be developed in a fairly non-controversial way. But now let's think of reals as subsets of $\omega$. The addition and multiplication could be considered as perfectly well-defined (with some kind of "mild" assumptions). But two issues definitely seem to come into play:
(i) The question as to what is exactly meant by a "subset of $\omega$" remains.
(ii) Completeness seems to talk about "non-empty subset of real numbers". I guess the trouble is being precise about what that "really" means.One way seems to be restrict our 'domain of discourse' to some set $A$. In that sense, we could replace:
"an arbitrary real number"
with
"an arbitrary real number in $A$"

"non-empty subset of real numbers"
with
"non-empty subset of real numbers in $A$"My naive understanding [possibly at the risk of too much over-simplification] is that this is the kind of approach that is explored quite extensively in reverse math. But even outside of reverse math, this seems to have been explored.

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Anyway, quite naively [since I don't know much applied math at all], there seem to be two ways to think about "math in physics" I suppose:
(i) One is to see it purely as a symbolic game that yields "actual/real-life" useful results (and not much more)
(ii) The other is to try to think that only "actually correct" math is used in physics.

I am sure you would have read these kind of comments many times before. Regardless, as you may have read (in essays etc.) that number of people have objected to "indispensability argument" saying that "weaker foundations" often suffice. But anyway, I am quite uninformed about the specifics of this and related issues.

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To the extent that I have been able to see is that (somewhat loosely) set theory tries to describe, in a certain way, a "maximal mathematical universe" so to speak. To the extent that it achieves this brings several separate (philosophical) issues, which might be skipped for this thread. But one benefit of it seems to be able to talk about things in a unified way.

But looking at it naively, from a physics viewpoint [unless someone very knowledgeable can point out specific examples], it doesn't seem to me that there is any "direct" reason to worry about why existence or lack of existence of such (mathematical) universe(s) would affect the math used in physics [unless, in some way, it turns out to be absolutely necessary to use these mathematical universe(s)].P.S.
The post is somewhat disjoint because the discussion point in OP seems to be quite broad. Anyway, hopefully it is useful to some extent.

 

[jordi]

Now I am going to say something opposite to what I have said initially: the definition of the natural numbers, as per https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers for example, whereby the zero is the empty set, the one is the set formed by the empty set, and so on, are just an application of the "intuitive" concept of membership, right? Otherwise, if the membership relation were just a generic relationship, we could not defined the natural numbers this way, right?

 

[SSequence]

Well, I don't quite understand what you are saying. But very specific definitions can probably be seen as difference between 'interface' and 'implementation'. The formal language of sets is not multi-sorted (I think this is what it is called). That is, the quantifications are only over a single type of objects, which are sets. That's why the 'implementation' of various objects/concepts (which are more complex) intuitive to us has to be built-in.

 

[yossell]

Now I am going to say something opposite to what I have said initially: the definition of the natural numbers, as per https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers for example, whereby the zero is the empty set, the one is the set formed by the empty set, and so on, are just an application of the "intuitive" concept of membership, right? Otherwise, if the membership relation were just a generic relationship, we could not defined the natural numbers this way, right?

If A and B have different interpretations of 'x is a member of y' then A and B will (usually) disagree over which entity is the empty set.

However, if all you want from a mathematical definition is to characterise a role in a structure -- so that {} is just that entity which contain no "members" (where "member" is just an arbitrary relation which satifies the axioms of set-theory) -- then one would not have to single out some unique and privileged 'empty set.'

It's not clear, in mathematics or its applications, our definitions require anything more than a structural element. If 'your' set of natural numbers are isomorphic to 'my' set of natural numbers, is there any mathematical or applied maths issue that we will disagree with?

Even in set theory, there are arbitrary choices about which set-theoretic objects we identify the numbers with. And, historically, different mathematicians have made different choices. E.g. one chose (Zermelo I think?)
0, {0}, {{0}}... (0 the empty set) as the set-theoretic version of natural numbers; Here, the successor of n is {n}; another (von Neumann?) chose 0, {0}, {0, {0}}...as the set theoretic version of the natural numbers. Here, the successor of n is n u {n}; (this way, each number n has n elements). The axioms of arithmetic are true in either interpretation.

 

[Stephen Tashi]

Set membership is just a binary relationship. ZFC axioms just tells us how objects, sets and the set membership relate to each other. We do not need to give a specific model of sets.

How do you define "binary relationship" without referring to the concept of set? The usual definition for "binary relationship" is that it is a set of ordered pairs. So we encounter both the concept of "set" and the concept of "ordered".

 

[sysprog]

How do you define "binary relationship" without referring to the concept of set? The usual definition for "binary relationship" is that it is a set of ordered pairs. So we encounter both the concept of "set" and the concept of "ordered".

In my view, a binary relation in a 2-valued logic is one of 16 possible input-output relations based on 2 inputs each of which presents one or the other of the 2 possible values, such that the relation outputs one or the other of the 2 possible values, and consequently each such relation may be characterized as a triplet -- the binary relation can be modeled in ways other than set-theoretically -- there are other equivalent constructs.

Here's an image of 3 XOR gates being used to implement a crossover that doesn't, other than semiconductorily, violate planarity in a circuit:

$\dots$ and here's an XOR gate functionality implemented by 4 NAND gates:

 

[jordi]

Exactly, a binary relation is a perfectly well defined concept in predicate calculus. Set theory is not necessary for that.

 

[yossell]

How do you define "binary relationship" without referring to the concept of set? The usual definition for "binary relationship" is that it is a set of ordered pairs. So we encounter both the concept of "set" and the concept of "ordered".

This is a good question. However, it's been found that you can get the effect of defining binary relations in systems that go just a little beyond first order logic, but which do not require sets.

More precisely (though from memory), in a system with plural quantification, and which contains the predicate 'x is a part of y' plus the thesis that there are at least denumerably many objects, you can get the effect of ordered pairs and the effect of quantifying over arbitrarily many ordered pairs.

Plural quantification is in some ways similar to quantification over subsets -- the idea is that there are certain terms, such as 'Fred's friends', 'John's children', 'the sets', which do not refer to singular objects (a set of entities) but which refer plurally to *some* things. 'These', 'Those' 'Them' are not singular terms for sets, but plural terms picking out many objects. Plural quantification is like ordinary quantification, but over plurals: There are some X's which are friends of Fred. Note, plural quantification alone won't get you ordered pairs or quantification over ordered pairs.

A motivation for plural quantification comes from the study of sentences such as 'There are some critics who admire only each other' -- a sentence which cannot be formulated in first order logic, yet which doesn't, on the face of it, seem to require a commitment to sets.

The part-whole relationship is, in some ways, like the set-member relationship. But it holds between concrete objects, is more 'physical', and can't be used to generate infinities (as we can by taking {}, {{}}., {{{}}} etc).

Naturally, going beyond standard first order logic, both of these moves can be questioned -- but, in my view, you do seem to be able to get the effect of quantification over ordered pairs without the commitment to set-theory.

The details of the construction can be found in Lewis' Parts of Classes, appendix, though the details are not due to Lewis.

 

[SSequence]

But isn't the relation between first-order-logic and sets complicated? I don't have enough depth of knowledge to write something detailed, but my impression was the following:
----- Some of the basic ideas of predicate logic require very little commitment to a background theory
----- I am reasonably certain that the most powerful theorems etc. would require commitment to a very strong background theory

I am not sure about binary relations but it seems to me that defining them (in full generality) is as complicated as the "collection" over which they are based. So, informally, the membership relation seems to be a predicate [not in very strict set-theoretic sense] with "domain" [quote to emphasize we aren't talking about a set] $S^2$? The symbol $S$ to denote the class of all sets.

So I am not clear how we can define membership relation over some objects without referring to "collection" of all of them. But anyway, I don't know much about the topic (predicate calculus and background assumptions of its theorems ... particularly more advanced ones).

 

[yossell]

But isn't the relation between first-order-logic and sets complicated? I don't have enough depth of knowledge to write something detailed, but my impression was the following:
----- Some of the basic ideas of predicate logic require very little commitment to a background theory
----- I am reasonably certain that the most powerful theorems etc. would require commitment to a very strong background theory

I think that a commitment to first order logic is fairly small. I agree that the *model theory* of first order logic can be very complicated, and the model theory of certain axiom systems of first order logic can require some heavy-duty set-theory.

Second order logic's arguably *is* connected to set theory. For in second order logic we allow ourselves to quantify over all the properties of the first order objects. But the notion of an arbitrary property has been found to be problematic, and mathematicians are happiest when properties are treated as arbitrary sets of the first order domain. So second order quantification can now look like quantification over the powerset of a given domain. And while this is not full blown set-theory, we might worry that there's a bigger commitment to sets here than we might have feared.

But I don't think that first order logic itself makes any commitments to sets.

I am not sure about binary relations but it seems to me that defining them (in full generality) is as complicated as the "collection" over which they are based. So, informally, the membership relation seems to be a predicate [not in very strict set-theoretic sense] with "domain" [quote to emphasize we aren't talking about a set] $S^2$? The symbol $S$ to denote the class of all sets.

It's a little more complicated isn't it -- because in defining the arbitrary binary relations *in full generality* on S we'll need all subsets of SxS -- and that's going to need the powerset of the collection which, when the collection is infinite, is a little more problematic.

So I am not clear how we can define membership relation over some objects without referring to "collection" of all of them.

I think this is a reasonable worry -- if it is the case that the statement 'Every x is F', we need to assume the existence of a set or collection over which the quantify ranges, then even first order logic is committed to sets after all. But my own view is that, even if there were no mathematical obects at all, it would still be true that 'Every person is mortal' and thus that first order quantification is free of a commitment of sets.