An additional constraint to the ZFC axioms?

[jordi]

The ZFC axioms are statements combining "atomic formulas" such as "p ∈ A" and "A = B", using AND, OR, imply, NOT, for all and exists.

But (it seems to me, at least) there is the implicit assumption that the "atomic formulas", "p ∈ A" and "A = B", are considered to be propositions, i.e. they are either true or false. But we know there are undecidable statements, such as the continuum hypothesis, in logic and mathematics.

So, it is conceptually thinkable that it is not possible to say if a given atomic formula is true or not. For example, and speaking loosely, the set A could be so large and intricate, that it could become impossible to say if a given p belongs to set A or not.

Would this be a problem for the ZFC axioms? What happens to a given ZFC axiom if an individual part of a formula cannot get a T or F value?

If this is a problem, shouldn't we state that the ZFC axioms require additional constraints, like explicitly stating the law of the excluded middle for atomic formulas?

 

[jim mcnamara]

Tell us what the word axiom means to you. We need to see.

 

[PeterDonis]

undecidable statements

...are not statements that don't have truth values. They are statements whose truth values cannot be logically deduced from the axioms of ZFC. Big difference.

 

[jordi]

In Frederic Schuller lectures:

https://www.physicsforums.com/threa...traint-to-the-zfc-axioms.986356/#post-6318922

it can be seen from page 8, in the axiom of the epsilon-relation, that Schuller is doing what I have said before (better, I am saying what he wrote before). He is stating that the epsilon relationship leads always to propositions (I would say that he should state the same for the equality relationship, but I guess he considers this as obvious).

 

[jordi]

...are not statements that don't have truth values. They are statements whose truth values cannot be logically deduced from the axioms of ZFC. Big difference.

But if you say "only statements that are propositions are considered", and then undecidable statements are not considered, you are using a "dirty trick", since one cannot know, in general, if one is considering a proposition or not.

For example, the statement "this statement is false" is not a proposition, since if it is true, it is false, and and if it is false, it is true.

So, the law of the excluded middle is true ... for propositions. But for statements that are not propositions, the law of the excluded middle is not necessarily true.

What Schuller is doing (and what I say it should be done) is just to explicitly state that the atomic formulas are propositions. And then, ZFC works like a charm, without inconsistencies.

I think it is necessary to make explicit that the atomic formulas are propositions, since we are trying to give the "right properties" of sets (I am taking a platonic view here, assuming sets are objects that exist, and with the axioms, we are just trying to get the "right properties" of those sets). So, what I am assuming is that sets are objects that, apart from satisfying the ZFC axioms, they have two additional properties:

1) It is always possible to check if for two given sets, they are the same set or not
2) It is always possible to check if an object (a set?) is a member of a given set or not

 

[PeterDonis]

In Frederic Schuller lectures

Please give a reference. The link you posted was to my post.

if you say "only statements that are propositions are considered", and then undecidable statements are not considered

Who said undecidable statements are "not considered"? You can express statements like the continuum hypothesis in the language of ZFC. You just can't prove or disprove them from the axioms.

the statement "this statement is false" is not a proposition, since if it is true, it is false, and and if it is false, it is true

Undecidable statements are not like this, so this statement of yours, while true, is irrelevant to this discussion.

the law of the excluded middle is true ... for propositions. But for statements that are not propositions, the law of the excluded middle is not necessarily true

You are confused. The statements you are saying are "not propositions" do not have well-defined truth values at all, so it is meaningless to ask whether any laws of logic like the law of the excluded middle are "true" for them. Undecidable statements do have well-defined truth values; those truth values are simply logically independent of the axioms of ZFC.

It is always possible to check if for two given sets, they are the same set or not

How would you check to see if two infinite sets are the same?

It is always possible to check if an object (a set?) is a member of a given set or not

How would you check to see if a given object was a member of an infinite set?

 

[jordi]

Sorry, the link is inside (look for PDF):

http://mathswithphysics.blogspot.com/2016/07/frederic-schullers-lectures-on-quantum.html

 

[jordi]

Please give a reference. The link you posted was to my post.
Who said undecidable statements are "not considered"? You can express statements like the continuum hypothesis in the language of ZFC. You just can't prove or disprove them from the axioms.
Undecidable statements are not like this, so this statement of yours, while true, is irrelevant to this discussion.
You are confused. The statements you are saying are "not propositions" do not have well-defined truth values at all, so it is meaningless to ask whether any laws of logic like the law of the excluded middle are "true" for them. Undecidable statements do have well-defined truth values; those truth values are simply logically independent of the axioms of ZFC.
How would you check to see if two infinite sets are the same?
How would you check to see if a given object was a member of an infinite set?

If there is no way to check if two sets are the same, then the ZFC axioms are not always well defined (for example, if there is no way to check if two sets are the same, how is on Earth the axiom of extensionality going to be satisfied?). Instead, with Schuller's new axiom, then yes, everything is always well defined.

The key issue is that "x is a member of A" is not necessarily a proposition, ex-ante. It could be a proposition, or only a statement (for which it is not necessarily the case that it is either true or false). If the statement "x is a member of A" is a statement, but not a proposition, (I believe that) we have a problem.

Schuller solves this straightaway. He postulates that for an object to be called a set, in addition to satisfying the ZFC axioms, a new axiom needs to be satisfied: that for all sets, the statement "x is a member of A" is always a proposition.

Then, the axioms of ZFC are always well defined.

 

[PeterDonis]

page 8, in the axiom of the epsilon-relation

I don't see any such axiom on page 8, or anywhere. The axioms presented are axioms of QM, not ZFC.

 

[jordi]

I don't see any such axiom on page 8, or anywhere. The axioms presented are axioms of QM, not ZFC.

Here:

https://mathswithphysics.blogspot.com/2016/07/lectures-on-geometric-anatomy-of.html

(it seems Schuller has also lectures on QM).

 

[PeterDonis]

if there is no way to check if two sets are the same, how is on Earth the axiom of extensionality going to be satisfied?

The axiom does not say you can always check to see if two sets are the same. It just defines what "the same" means--two sets are the same if they have the same elements. There is no guarantee that you can always prove either that two sets do have the same elements, or that they don't; some statements regarding whether two sets do or do not have the same elements might be undecidable.

It could be a proposition, or only a statement (for which it is not necessarily the case that it is either true or false).

You are confusing two different things: whether a given statement has a well-defined truth value, and whether a given statement's truth value is provable from the axioms of ZFC.

For example, consider the following:

(1) The statement "2 is a member of the set {0, 1, 2}" has a well-defined truth value, and that truth value can be proved to be "True" from the axioms of ZFC.

(2) The statement "2 is not a member of the set {0, 1, 2}" has a well-defined truth value, and that truth value can be proved to be "False" from the axioms of ZFC.

(3) The general statement "x is a member of A" will have a well-defined truth value for any x and A (provided that x and A themselves are well-defined); but that doesn't mean you can always prove its truth value from the axioms of ZFC, for any x and A.

(4) The statement "there are no cardinalities strictly between the cardinality of $\omega$, the set of natural numbers, and the cardinality of the continuum $C$" has a well-defined truth value, but it is known that this truth value cannot be proved to be either True or False from the axioms of ZFC.

(5) The statement "this sentence is false" does not have a well-defined truth value at all.

You might object to statement (4); but consider this more precise reformulation of it:

(4a) The statement "there are no cardinalities strictly between the cardinality of $\omega$, the set of natural numbers, and the cardinality of the continuum $C$" has a well-defined truth value in any semantic model which is consistent with the axioms of ZFC, but it is known that this truth value cannot be proved to be either True or False from the axioms of ZFC for any semantic model which is consistent with the axioms of ZFC.

In other words: there are semantic models consistent with the axioms of ZFC in which the continuum hypothesis is true, and there are other semantic models consistent with the axioms of ZFC in which the continuum hypothesis is false. That is what it means for the continuum hypothesis to be "undecidable" from the axioms of ZFC. If you could prove from the axioms of ZFC that the continuum hypothesis was true, then it would be impossible for any semantic models to exist that were consistent with the axioms of ZFC in which the continuum hypothesis was false. (And similarly if you exchange "true" and "false" in the above sentence.)

In other words: the concept of "truth value" is only meaningful once you have specified a semantic model. If all you have is a set of axioms and theorems proved from them, then nothing has a definite truth value at all. You need to specify something for the axioms to be about before you can even ask the question of what the truth values of statements are. And mathematics by itself cannot tell you what sorts of things a given set of axioms and statements can be about. That has to come from outside mathematics. That is why Bertrand Russell said that mathematics is the subject in which we do not know what we are talking about, nor whether what we are saying is true.

 

[PeterDonis]

the concept of "truth value" is only meaningful once you have specified a semantic model

In the light of this, let me restate my (5) from the previous post more precisely:

(5a) The statement "this sentence is false" does not have a well-defined truth value at all in any possible semantic model.

In other words, it is impossible to find any semantic model in which that statement can be given a well-defined truth value. There is no possible semantic model in which it is true, and there is no possible semantic model in which it is false.

But that is not the case for, say, the continuum hypothesis.

 

[jordi]

The axiom does not say you can always check to see if two sets are the same. It just defines what "the same" means--two sets are the same if they have the same elements. There is no guarantee that you can always prove either that two sets do have the same elements, or that they don't; some statements regarding whether two sets do or do not have the same elements might be undecidable.

I have an issue with this statement (with the rest of your post, I am OK, it is nothing controversial, I think):

The axiom of extensionality is an iif. And an iif means that both sides have the same truth/false value. I have never seen that an iif can be considered to be valid (as it is, since an axiom is exactly this: stating that that statement is valid) if one, or both, of its "sides" is "undefined".

So, my question to you is: if you think that "There is no guarantee that you can always prove either that two sets do have the same elements, or that they don't", how do you interpret the axiom of extensionality?

 

[PeterDonis]

if one, or both, of its "sides" is "undefined".

You keep on with the same confusion, even though I've already clarified it. The truth value of a statement of the form "x is a member of A" is not "undefined". It's not provable from the axioms of ZFC. Those two things are different things. How many times am I going to have to repeat that?

if you think that "There is no guarantee that you can always prove either that two sets do have the same elements, or that they don't", how do you interpret the axiom of extensionality?

I've already said: as telling you what it means for two sets to be "the same". And, as I've already said, that is not the same thing as asserting that you can always prove whether or not two sets are the same.

Again, how many times am I going to have to repeat the same thing?

 

[jordi]

You keep on with the same confusion, even though I've already clarified it. The truth value of a statement of the form "x is a member of A" is not "undefined". It's not provable from the axioms of ZFC. Those two things are different things. How many times am I going to have to repeat that?
I've already said: as telling you what it means for two sets to be "the same". And, as I've already said, that is not the same thing as asserting that you can always prove whether or not two sets are the same.

Again, how many times am I going to have to repeat the same thing?

You repeat, but you do not prove. You claim, basically, that all statements are either true or false. The only "bad" thing that it can happen is that this truth/falsehood cannot be proven through the considered axioms. But with other axioms, it could be different.

This is what you say. But you have not proved that all statements are either true or false. You claim it. But you do not prove it.

Schuller considers that the statement "x belongs to A" is a statement, not necessarily a proposition. And then, he elevates the statement that "x belongs to A" is a proposition through an additional axiom. I consider this "perfect".

Instead, you claim that "x belongs to A" is a proposition "for sure". Well, I do not see this as obvious. Instead, I follow the argument of Schuller perfectly.

 

[PeterDonis]

Here

What he means by "not a proposition" is what I have been calling "not well-defined". It is not what you have been calling "undecidable". Indeed, on the immediately previous page from the one you referenced (p. 7), he explicitly uses the term "undecidable proposition" (emphasis mine) to describe the thing that his Theorem 1.14 shows must exist.

 

[jordi]

Forget about the undecidability. The key of what I am saying is that with Schuller's "new axiom", it is perfectly clear that "x belongs to A" is a proposition. Without this new axiom, it is not clear at all.

So, by being "x belongs to A" a proposition, the axiom of extensionality is valid "always". Instead, without this "new axiom", it could be that the axiom of extensionality "sometimes" is not valid (parts of it are not either true or false), and this "cannot be".

So, for me, this new axiom is necessary for the whole construction of sets. In fact, it tells us something useful about sets: sets are objects that, apart from satisfying ZFC, it is always true, or false, that an object belongs to them.

It is conceivable that there exist objects B for which the statement "x belongs to B" is neither true nor false.

 

[PeterDonis]

You repeat, but you do not prove. You claim, basically, that all statements are either true or false.

I said that all "well defined" statements have a well-defined truth value in any semantic model.

Your reference says the same thing. See my post #16 just now.

The only "bad" thing that it can happen is that this truth/falsehood cannot be proven through the considered axioms. But with other axioms, it could be different.

That is correct; changing the set of axioms you use can change which statements can or can't have their truth values proved from the axioms. It can also change which statements are or are not well-defined. That is because it changes which semantic models are or are not consistent with the axioms.

Schuller considers that the statement "x belongs to A" is a statement, not necessarily a proposition.

No, he doesn't. He says that, if we are using the ZFC axioms, the statement "x is a member of y" is only well-defined if x and y are sets. That is because in the ZFC axioms, sets are the only kinds of entities that exist at all: that is, any semantic model of the ZFC axioms will contain sets, but no other kinds of entities. All Schuller is doing is specifying that. Note carefully his wording (bottom of p. 8 and top of p. 9):

"We remarked, previously, that it is not the task of predicate logic to inquire about the nature of the variables on which the predicates depend. This first axiom clarifies that the variables on which the relation $\in$ depend are sets."

In other words, Schuller has previously defined a system of logic, predicate logic, which deals with properties of predicates regardless of what kinds of entities those predicates apply to. Now he is defining an axiom system for ZFC as a set theory, i.e., the kinds of entities all predicates will apply to in this theory are sets. That's all he is doing. He is not saying that there are statements of the form $x \in y$ which are "statements, but not propositions". He is saying that, as far as ZFC is concerned, the only way to form meaningful statements at all of the form $x \in y$ is if $x$ and $y$ are sets.

Or, to put it another way, you are claiming that Schuller recognizes three categories: "propositions", "statements", and "not well-defined". But that is wrong. He only recognizes two: "propositions" and "not well-defined". There is no third category.

 

[SSequence]

You repeat, but you do not prove. You claim, basically, that all statements are either true or false. The only "bad" thing that it can happen is that this truth/falsehood cannot be proven through the considered axioms. But with other axioms, it could be different.

This is what you say. But you have not proved that all statements are either true or false. You claim it. But you do not prove it.

It is conceivable that there exist objects B for which the statement "x belongs to B" is neither true nor false.

These are reasonable concerns. Originally, intuitionism originated based upon similar concerns. A somewhat related search term is "absolute undecidability".

The logic underlying (traditional) versions of sets though is based upon every single question [that can be asked within it] having a definitive truth/false value.

 

[jordi]

The logic underlying (traditional) versions of sets though is based upon every single question [that can be asked within it] having a definitive truth/false value.

But this is really problematic for me. For example, as said before, the statement "this statement is false" is neither true nor false.

Of course, you could claim that traditional versions of sets exclude this kind of statements. But then, how do you know that the statement "x belongs to A" is for sure either true or false? Couldn't it be that the statement "x belongs to A" is one of those statements that you exclude?

For me, the Schuller decision solves this problem. He allows the possibility, initially, that a statement such as "x belongs to A" is not a proposition. It could be a proposition, or not. That is the most general assumption, and it is clearly valid.

Then, what he does is to explicitly acknowledge that a basic property of sets is that the statement "x belongs to A" is always a proposition. So, he explicitly claims that it cannot be that "x belongs to A" is a non-proposition.

In my opinion, this is a clearly valid and fully informative kind of claim. And it makes that all the ZFC axioms have valid values always.

Maybe set theory books already make this claim, but implicitly: by saying that "membership of" is a relationship, maybe they claim that a relationship always results in a proposition. But I think Schuller decision is "better", since it makes everything explicit and clear.

 

[SSequence]

But this is really problematic for me. For example, as said before, the statement "this statement is false" is neither true nor false.

It isn't clear to me that how this kind of statement can be posed/written in set-theory language.

===========

As far as godel sentence is concerned, as I understand, it is a number-theoretic statement:
$G$: "this statement is not proveable (in the given theory $T$)"
And, assuming $T$ to be sound regarding number-theoretic statements [or just $con(T)$], $G$ will be neither provable nor disproveable in $T$.

Taking example of $PA$, the sentence $G$ for $PA$ is true [because the consistency of $PA$ is proven]. If one wants to be more specific, the sentence $G$ will be true in the intended model for $PA$ (natural numbers). However, it can be false in some other models [see post#12]. But I don't know this topic well, so I don't know what are the "background assumptions" for the non-intended models of $PA$.

For sets this is what I understand (somewhat vaguely). Assuming $con(ZFC)$ to be true, the sentence $G$ for $ZFC$ will be true (in a model where the finite ordinals are the actual natural numbers).

 

[yossell]

I think i see where OP is coming from but, to be honest, I'm having a hard time of making sense of Schuller's discussion.

I do agree with others that the questions decidability is better understood as something to be totally distinguished from the questions of truth-value. We have no way, at present, of knowing whether the Godlback conjecture is true, but that doesn't mean that the conjecture lacks a truth-value; we have no way of telling telling whether an arbitrary statement S of Arithmetic is a member of the set of statements which are true in the standard model of Arithmetic -- but that doesn't mean that the set S is in anyway badly defined.

However, there are those who think that, when it comes to maths, undecidability and truth *should* be connected -- and that's one of the ideas behind intuitionistic logic. This is a logic in which 'P v ¬P' is no longer a theorem.

I have never seen Schuller's axiom 'The expression x is an element of y is a proposition iff x and y are both sets' presented as an axiom of set theory. I can understand why some might think that Schuller's axiom is a step forward. But in fact I think it raises more problems than it solves.

Here are some of difficulties with the axiom.
(1) there's unclarity over object and meta-language; the left hand side refers to an expression: 'x is an element of y'; yet, on the right hand side the x and y are variables of the object language. Though, intuitively, I think I know what's trying to be said, I'm worried that a proper formulation might require set-theory to contain its own meta-language -- which is going to make things confusing.

(2) Set-theory (at least in its modern form) doesn't contain the predicate '...is a proposition'. First order logic was, in part, an attempt to get away from primitive notions like 'proposition' and 'property' and replace them with something less problematic, something less likely to lead to paradox -- something better behaved, something like sets. Reintroducing 'proposition' as part of the primitive vocabulary of set-theory is, arguably, a retrograde step.

(3) We might try and de-mystify 'proposition' by simply saying that a statement counts as a proposition iff it is true or false -- that's all we mean. The simplest way to implement this is say: a statement P is a proposition if and only P or ¬P holds? Put that way, ALL statements of set-theory count as propositions. Because the background logic of set-theory is classical logic, as in classical logic P v ¬P is a theorem. So, if we're going to implement Schuller's axiom in a way that makes it interesting, that allows statements in the theory which are not propositions, it seems we're going to have to move to a non-classical logic -- such as Ssequence's intuitionistic logic.

After all, note this: In the symbolic representation of his axiom '"x is a member of y" is a proposition iff AxAy x is a member of y or ¬ x is a member of y), the rhs is a THEOREM of classical set theory, simply because classical set theory contains classical logic.

That's going to be quite a revision to set-theory you end up with there. I'm not saying it can't be done, but I would like to see the details and I would really like to know how much of classical set-theory remains at the end of it.

(4) If I were to guess at what Schuller wanted to do here, it would be to present set-theory in a two-sorted language, one type of variable x, y, z... ranging over sets, another type of variable X, Y, Z... ranging over non-sets, and formulate the membership predicate so that only sentences of the form 'x is a member of y' were part of the object language. Then 'X is a member of x' and 'x is a member of Z' are not even well-formed.

Indeed, the problem with sentences such as 'This sentence is false' is normally logically dealt with by formalising the logical language in such a way that these sentences are not even expressible within the language. It's not how we do things in a natural language which contains the ability to refer to its own expressions and to talk about these expressions being true or false or lacking truth value. But, as Tarski worried, natural language may not even be a consistent language precisely for these kinds of reasons. Certainly, though perhaps at first sight appealing, the attempt to deal with paradoxical statements by adjusting our logic, moving to non-bivalent logics, using notions of truth, proposition, property and meaning to distinguish the problematic from the non-problematic expressions, turns out to be a harder and more problematic endeavour than it may seem to be at first sight.

 

[jordi]

(2) Set-theory (at least in its modern form) doesn't contain the predicate '...is a proposition'. First order logic was, in part, an attempt to get away from primitive notions like 'proposition' and 'property' and replace them with something less problematic, something less likely to lead to paradox -- something better behaved, something like sets. Reintroducing 'proposition' as part of the primitive vocabulary of set-theory is, arguably, a retrograde step.

(3) We might try and de-mystify 'proposition' by simply saying that a statement counts as a proposition iff it is true or false -- that's all we mean. The simplest way to implement this is say: a statement P is a proposition if and only P or ¬P holds? Put that way, ALL statements of set-theory count as propositions. Because the background logic of set-theory is classical logic, as in classical logic P v ¬P is a theorem. So, if we're going to implement Schuller's axiom in a way that makes it interesting, that allows statements in the theory which are not propositions, it seems we're going to have to move to a non-classical logic -- such as Ssequence's intuitionistic logic.

1. Item (2) above is key, and we need to see how we understand each other. For me, it is clear that both first order logic and set theory are composed of primitive notions called "proposition". In fact, propositions are all there is, in both first order logic and set theory! For example, the rhs of the axiom of extensionality uses the statement "p belongs to A". What is this statement but a proposition, i.e. a statement that can be T or F, and nothing else?

2. Item (3) has maybe a guide to the solution of the mental discomfort in this discussion: maybe logicians just state that "the background logic of set-theory is classical logic", and they self-understand, implicity, that any statement that you may have, in either first order logic or set theory, is a proposition. So, we self-restraint ourselves in not allowing statements that are not propositions into the discussion. As a consequence, the statement "p belongs to A" in the axiom of extensionality is a proposition, so the Schuller "new axiom" is unnecessary.

So, maybe it is only an issue of "notation". Maybe logicians and set theorists they are so used to this notation, that they do not make it explicit. But I think it has to be explicit (through the Schuller new axiom or otherwise) that the statement "p belongs to A", which in general could be a proposition or not, IS A PROPOSITION (and it is not a non-proposition).

Then, by being the statement "p belongs to A" a proposition, the axiom of extensionality can always be checked, and the terms will always give T or F, but never "not legit" (as it would happen if "p belongs to A" were a non-proposition).

Does this make any sense? Please, let us try to refine the conversation, because I think we are getting somewhere, but not yet, completely.

 

[jordi]

I think I have a solution:

In the Appendix of Leary's "A friendly introduction to mathematical logic", it is stated:

"Think of a set as a collection of objects. If X is a set, we write a 2 X
to say that the object a is in the collection X. For our purposes, it will be
necessary that any given thing either is an element of a given set or is not
an element of that set. If that sounds obvious to you, suppose we asked
you whether or not 153,297 was in the "set" of big numbers. Depending on
your age and whether or not you are used to handling numbers that large,
your answer might be "yes," "no," or "pretty large, but not all that large."
So you can see that membership in an alleged set might not be all that cut
and dried."

So, from this paragraph, it seems clear that "p belongs to A" is considered to be a proposition, i.e. it can be either true or false, and nothing else.

And the author acknowledges that, in principle, the statement "p belongs to A" does not need to be, necessarily, a proposition. But for his purposes, it is necessary to make any statement "p belongs to A" a proposition.

Then, it seems that my "problem" is just one of notation: mathematicians and logicians assume, mostly implicitly but in cases like Leary explicitly, that the statement "p belongs to A" is a proposition.

Schuller, instead of making that assumption, elevates the fact that the statement "p belongs to A" is a proposition to an axiom.

In the end, it is true that it is an issue of taste. But I prefer Schuller's choice, since it is more explicit.

Do we all agree with this explanation? (apart from the taste).

 

[yossell]

1. Item (2) above is key, and we need to see how we understand each other. For me, it is clear that both first order logic and set theory are composed of primitive notions called "proposition". In fact, propositions are all there is, in both first order logic and set theory! For example, the rhs of the axiom of extensionality uses the statement "p belongs to A". What is this statement but a proposition, i.e. a statement that can be T or F, and nothing else?

But almost any theory -- physical or mathematical -- could be said to be composed of propositions in this sense. So, while I think I see what you mean, this boils down to a fact about the nature of theories.

maybe logicians just state that "the background logic of set-theory is classical logic", and they self-understand, implicity, that any statement that you may have, in either first order logic or set theory, is a proposition.

Yes, but I don't think it's sneaky or hidden or restricted to set theory. It's a general assumption that the theories are formulated in first order or second order logic, and the logic here is classical. You can look at other logics, but the mainstream view is to stick with classical logic.

In this sense 'For every set x and y, x is a member of y or x is not a member of y', 'For every number x, x is prime or x is not prime', 'for every set s, either s is strongly inaccessible or s is not strongly accessible' are all theorems of their respective theories, by classical logical considerations alone.

Now, there are various phenomenon which may make us want to revise classical logic -- for instance, perhaps when we have vague predicates, there are objects which neither satisfy the predicate nor fail to satisfy the predicate; perhaps the set-theoretic and semantic paradoxes show us there are some statements which are such that neither they nor their negation are true. So it's not necessarily unreasonable to abandon classical logic. But non-classical logics have never really become mainstream amongst mathematicians.

Then, by being the statement "p belongs to A" a proposition, the axiom of extensionality can always be checked, and the terms will always give T or F, but never "not legit" (as it would happen if "p belongs to A" were a non-proposition).

I want to add my voice to the chorus of people here who have said that it is, at the best of times, misleading to say 'can always be checked'. We can't check whether the number of dinosaurs that ever lived is one really large number or another really large number. But the question: 'is the number of dinosaurs that ever lived even?' has a definite and determinate answer. The principles of set-theory, of number theory aren't themselves involved in the issue of whether the statements that can be formed in the theory can be checked.

But there have been mathematicians who thought that mathematics, unlike statements about dinosaurs, should be understood in terms of what we can check or know. Brouwer, the person behind intuitionistic logic, believed this -- very roughly, he thought statements about maths should be understood as asserting that certain proofs could be constructed. Accordingly, even before Godel, since there are mathematical propositions which we cannot determine, he thought the law of excluded middle couldn't be said to hold -- we do not have a proof of them or a proof of their negation.

I would also add that it's a little misleading to say the terms will always give T or F. Axiom of extensionality is a universally quantified biconditional, not stated in propositional logic.

 

[PeterDonis]

how do you know that the statement "x belongs to A" is for sure either true or false?

You don't. What you do know is that, if x and A are both sets, and they satisfy the axioms of ZFC, then the statement $x \in A$ is either true or false, because ZFC set theory is defined as a theory of sets in which there is an operator $\in$ which is defined to have this property. Outside the context of ZFC, the string of symbols $x \in A$ is meaningless. It's not that it's a statement which might not be either true or false outside the context of ZFC; it's a meaningless string of symbols outside the context of ZFC.

He allows the possibility, initially, that a statement such as "x belongs to A" is not a proposition.

No, he doesn't. As I've already said, when Schuller says the statement $x \in y$ is a proposition, he doesn't mean there is a pre-existing, meaningful statement $x \in y$, which he then shows to be a proposition. He means that there is a string of symbols $x \in y$, which has meaning if and only if $x$ and $y$ are both sets. If they aren't sets, the string of symbols $x \in y$ is meaningless; it's not a statement that just doesn't happen to be a proposition. There is no such thing anywhere in Schuller's presentation as "a statement which isn't a proposition". If you disagree, then please give a specific example.

But in fact, you do agree, since you say:

In fact, propositions are all there is, in both first order logic and set theory!

Yes, exactly. There are only propositions. There are no such things as "statements which are not propositions". So why do you keep claiming that there are?

maybe logicians just state that "the background logic of set-theory is classical logic"

There is no "maybe" about it. Schuller, for example, presents propositional logic first, and then builds ZFC set theory on top of propositional logic. That is how it is done.

Perhaps there are other, different "set theories" that build on something other than propositional logic (though I do not know of any), but not ZFC.

the statement "p belongs to A", which in general could be a proposition or not

You contradict yourself. Above, you said there are only propositions. Now, you claim there are "statements" which could be propositions or not. You can't have it both ways.

the author acknowledges that, in principle, the statement "p belongs to A" does not need to be, necessarily, a proposition

Only because he is using vague ordinary language (his example is "the set of big numbers", which as he describes it is not even well-defined) and not a rigorous axiomatic system. We are not talking about vague ordinary language, We are talking about ZFC, a rigorous axiomatic system, built on propositional logic, another rigorous axiomatic system. If he actually did the work of finding an expression in the language of ZFC for his vague "the set of big numbers", he would find that the number 153,297 either was in it, or wasn't. There would be no room for a wishy-washy "well, it might be in it, or it might not".

 

[PeterDonis]

If I were to guess at what Schuller wanted to do here, it would be to present set-theory in a two-sorted language, one type of variable x, y, z... ranging over sets, another type of variable X, Y, Z... ranging over non-sets

No, that's not what he's doing. This is easily seen by the fact that he never introduces any axioms or theorems of ZFC that contain variables referring to things that aren't sets.

 

[SSequence]

I don't have much to add to what's been said. A few references that may be relevant to the topic.

So it's not necessarily unreasonable to abandon classical logic. But non-classical logics have never really become mainstream amongst mathematicians.

My impression is that intuitionistic logics have become somewhat more mainstream recently? They seem to occur in category theory in someway. This link seems to point out some connections. I don't have any idea about what category theory is [except that it involves "types" in some way, which makes formalizing ordinary mathematical discourse easier]. I don't have any idea why multi-sorted logic wouldn't be achieving a similar goal and why categories would be better than it for this purpose (but anyway this is bit of a tangent).

My very "rough guess" is that the way they occur is in the sense plurality of "mathematical worlds" considered all at once [in the sense of multiplicity of logics]. So, intuitionistic logics seem to occur purely [probably a symbolic calculus] in some of these worlds.

This is also of some relevance as a reference: https://plato.stanford.edu/entries/set-theory-constructive.

 

[yossell]

No, that's not what he's doing. This is easily seen by the fact that he never introduces any axioms or theorems of ZFC that contain variables referring to things that aren't sets.

He argues that 'u is a member of u' is not a proposition and then concludes that u is not a set. In those cases where 'u is a member of v' is not a proposition, u and v are not sets. Here is a statement whose variables do not include sets. And he says that his first axiom tells us when u is not a set. Of course, as set theory involves just the propositions, these kinds of statements are not part of axiomatic set theory.

 

[PeterDonis]

He argues that 'u is a member of u' is not a proposition

Yes; what he means is that the string of symbols $u \in u$, where $u$ is defined implicitly by the string of symbols in his Example 2.2, is not well-defined in ZFC--i.e., it's a meaningless string of symbols. He does not mean that the string of symbols $u \in u$ denotes "a statement which is not a proposition".

and then concludes that u is not a set

Yes; what he means is that the symbol $u$, which is defined implicitly by the string of symbols in his Example 2.2, does not denote anything; it's a meaningless symbol. He does not mean that the symbol $u$, defined as in his Example 2.2, denotes "something which is not a set".

IMO, a much better way of conveying what is really meant by Example 2.2 would have been to simply prove the following as a theorem:

$$
\neg \exists u : \forall x : \left( x \notin x \Leftrightarrow x \in u \right)
$$

I agree the terminology in the English version is not very well chosen; I suspect that is an issue of translation from German to English. In German I suspect the terminology he chose would be better suited to convey what I have described.

 

[Stephen Tashi]

Then, it seems that my "problem" is just one of notation: mathematicians and logicians assume, mostly implicitly but in cases like Leary explicitly, that the statement "p belongs to A" is a proposition.

I'll agree - from the presentations of ZFC that I've seen. I don't know if there can be a presentation that is stated only as a formal language. In a formal language, we are concerned with the manipulation of symbols. Given an initial set of expressions that obey a certain syntax, there are rules for deriving other expressions. From this point of view, deriving an expression from others does not prove it because the initial expressions are not assumed to denote things that are "true".

 

[PeterDonis]

the initial expressions are not assumed to denote things that are "true"

Um, what? In the context of a theory like ZFC, the axioms are assumed to be true, and expressions which are tautologies are true by definition, and any expression that can be derived from the axioms and whatever tautologies are required is therefore also true.

Even in propositional logic taken by itself, since a false proposition implies any proposition, the only way to have any distinction at all between propositions is to start with some set of them that are taken as true and work from there. In Schuller's presentation, propositional logic is simply the system in which the only propositions that are taken as true to start with are tautologies (he expresses this as there being no axioms in propositional logic by itself).

 

[SSequence]

I'll agree - from the presentations of ZFC that I've seen. I don't know if there can be a presentation that is stated only as a formal language.

Yes, there is.

Um, what? In the context of a theory like ZFC, the axioms are assumed to be true, and expressions which are tautologies are true by definition, and any expression that can be derived from the axioms and whatever tautologies are required is therefore also true.

I think likely it gets complicated because of models etc. [this was discussed a bit in a recent thread]. But generally, I agree with this. At least, at some level, this would be the intention in the ideal case (e.g. platonism or similar views).

I think perhaps the poster had in mind probably something like how geometries starting with different initial axioms can lead to different conclusions [without anyone of them being "more true" than the other]. I don't know enough about that, but this is definitely a different scenario.

 

[Stephen Tashi]

As to whether ZFC can be presented as a formal language, there seems to be a disagreement.

Yes, there is.

Um, what? In the context of a theory like ZFC, the axioms are assumed to be true, and expressions which are tautologies are true by definition, and any expression that can be derived from the axioms and whatever tautologies are required is therefore also true.

To clarify, I consider the presentation of something as a formal language not to include the assumption that there is a mapping that assigns each (well formed) symbolic expression a truth value - true or false. This does not preclude than an interpretation of the symbolic language might assume the existence of such a mapping.

 

[SSequence]

To clarify, I consider the presentation of something as a formal language not to include the assumption that there is a mapping that assigns each (well formed) symbolic expression a truth value - true or false.

I am a bit confused. If that's the case then it seems one would have to regard a usual formal presentation of PA (and many other theories ...) as not being presented as a formal language.