Physics tells us that even bricks are nothing more than probability waves smeared across the universe — fishfry

Aha! So, finally, a resolution to the question I posed about PWs on another thread. The medium through which they travel are brick roads. And they culminate on the shores of the Emerald City!

Thanks! :nerd:

If you don't believe in sets, why go to the trouble of explaining why you don't believe in the empty set? I wonder if that shows that you haven't thought your idea through. Why bother to argue about the lack of elements, when you don't even believe in sets that are chock-full of elements? — fishfry

The problem with the "empty set", which I have described, is a demonstration of the reason why I don't believe in the existence of sets. So I take the trouble of explaining the problem with the empty set to justify why I do not believe in sets. A set is a Platonic Form, and it's nonsense to speak of the Form of nothing. Yet we use "zero", and "nothing" commonly. Therefore I do not believe in this type of Platonic realism because it doesn't give us an appropriate way to represent what "zero" means.

Nobody has claimed sets have "real" existence, whatever that is — fishfry

Sorry fishfry, but you stated quite clearly "if you believe in the existence of any set at all, and you accept the axiom schema of specification, then you must accept the mathematical existence of the empty set."

You're making up nonsense if you are asserting that there is some sort of existence which is not real existence. When I say "real existence", I mean existing, categorically, as distinguished from not existing. What is it, if it's not real existence, an illusion? Real existence is opposed to the illusion of existence. I'd call it deception, claiming that something exists when you know it's not real existence but an illusion. And it's completely nonsensical to claim that there is a type of existence "mathematical existence", which is not real existence. How would that work in set theory? You have a set of existing things, and you have a subset, "mathematical existence". Then you say that "mathematical existence" is somehow outside of the set of existence, because it's not a real existence.

I went through this with aletheist already. Altheist was trying to distinguish different types of existence corresponding to different subjects (fields of study), but refused to recognize that all of these are a member of a single, more general category of "existence" itself. So aletheist refused to recognize ontology as the study of existence in general, what all different sorts of existents have in common, and insisted on placing "mathematical existence" outside of, separate from 'ontological existence". But that is nothing other than an assertion that there is no such thing as ontology. Now you attempt the same move, by saying that mathematical existence is not real existence, you place it outside the field of ontology, which studies "being", "existence", rendering ontology useless by saying that there is a type of existence which cannot be studied by the field of study, ontology, which studies existence in general.

I could easily take you down the rabbit hole of your own words. Is an electron "real?" How about a quark? How about a string? How about a loop? And for that matter, how about a brick? Are there bricks? When we closely examine a brick we see a chemical compound made of molecules, which are made of atoms, which contain protons, neutrons, and electrons, which themselves are nothing more than probability waves smeared across the universe. — fishfry

We can look at these concepts, "electron", "quark", "string", "brick", and see if there are inconsistencies, contradictions, or other forms of fallacious logic, and if not we can say that the thing referenced most likely has some form of existence. So I'm not nihilist, I just believe that contradiction negates the possibility of existence, such that it is impossible that a contradictory thing exists. So things referred to have a probability of existing, until they are proven impossible, then that probability is removed.

Do you believe in the existence of bricks? Physics tells us that even bricks are nothing more than probability waves smeared across the universe. — fishfry

If you define "brick" in this way, as "possibility waves smeared across the universe" it probably doesn't have existence, because physics uses a lot of contradictory mathematics and inconsistent principles, as I am arguing here. But there are other ways that we can define "brick", and use the term, which do not involve logical inconsistencies, and so a "brick" in this sense would have a reasonable probability of existence.

Do you deny science along with math? — fishfry

Science which uses faulty math is obviously faulty, don't you think? The soundness of the conclusions is dependent on the soundness of the premises.

some argue that the real numbers are not truly continuous — aletheist

Who argues that, exactly, besides the Peirceans on this forum? — fishfry

The Peirceans who are not on this forum, for starters; but it goes back at least as far as Aristotle, who recognized that numbers of any kind are intrinsically discrete, rather than continuous. The key word here is "truly"; I have acknowledged that the real numbers are an adequate model of continuity for most mathematical and practical purposes. Nevertheless, conceptually a line is not composed of points, a surface is not composed of lines or points, and a solid is not composed of surfaces or lines or points. Instead, the parts of a line are one-dimensional lines, the parts of a surface are two-dimensional surfaces, and the parts of a solid are three-dimensional solids. Anything of lesser dimensionality is not itself a part (or portion) of that which is truly continuous, but rather a connection (or limit) between its parts.

Anything of lesser dimensionality is not itself a part (or portion) of that which is truly continuous, but rather a connection (or limit) between its parts. — aletheist

I think this would end up giving precisely the same mathematical theorems, no? You just restate things in terms of connections and parts.

Like: the metric topology on the interval (0,1) consists of open sets defined by: the empty set (a limit) is in the topology, the whole (0,1) is in the topology, unions of parts and connections/limits are in the topology, countable intersections of parts (be they connections/limits or parts) are in the topology.

The vocabulary of sets lets you phrase all these concepts already. "Parts of a line? They're finite intersections of its interval subsets which have cardinality greater than than 1".

The vocabulary of sets lets you phrase all these concepts already. "Parts of a line? They're finite intersections of its interval subsets which have cardinality greater than than 1". — fdrake

A set is a bottom-up conception, assembling a whole from discrete parts. True continuity is a top-down conception, such that the whole is more fundamental than the parts.

A set is a bottom-up conception, assembling a whole from discrete parts. True continuity is a top-down conception, such that the whole is more fundamental than the parts. — aletheist

Let's say I have some "True continuity" X. Like a line X=(0,1). Let's say I can take "parts" of it in the above manner; arbitrary subintervals. (0,a), (a-1/n,1) are subintervals for any a less than 1 and greater than 1/n. Since they're parts of a true continuity, the true continuity ensures the existence of their intersection; limit the process of intersection over n and get the limit {a}. If I take the union of all such limits, since a was an arbitrary member of (0,1), it produces (0,1). So starting from a true continuity, like a line segment, you can get discrete numbers, then build up the true continuity out of the individual numbers through a union. It's top down and bottom up at the same time.

Do you agree this process is legitimate?

If not, do you reject sets as a concept?

Let's say I have some "True continuity" X. Like a line X=(0,1). — fdrake

Introducing numbers already imposes discreteness. Numbers are for measuring, they cannot constitute a truly continuous line.

So starting from a true continuity, like a line segment, you can get discrete numbers, then build up the true continuity out of the individual numbers through a union. — fdrake

No, again, a line is not composed of points corresponding to numbers. We can only mark them on (not in) a truly continuous line. They then serve as arbitrary and artificial limits/connections between distinct portions/parts.

If not, do you reject sets as a concept? — fdrake

Not at all; again, set theory can be quite useful as the basis for an approximate model of continuity. However, it cannot serve as the basis for true continuity, because it requires discreteness at the outset.

A set is a bottom-up conception, assembling a whole from discrete parts. True continuity is a top-down conception, such that the whole is more fundamental than the parts. — aletheist

Mephist draws on a top-down description of a set; the set as more fundamental than its elements, to explain topology. This is the only way that "empty set" makes sense, if the set is a top-down construction. So a set may be a bottom-up construction, but set theory employs sets as if they are actually top-down, by utilizing the empty set.

The Peirceans who are not on this forum, for starters; — aletheist

Makes sense. I've never met any besides here. Must not hang out among the right philosophers.

I can certainly see Peirce's objection that a true continuum could never be made up of individual points. Did Aristotle reject the notion of an instant of time? Or did Peirce? You could't accept instants if you reject points, I'd imagine.

Did Aristotle reject the notion of an instant of time? Or did Peirce? — fishfry

Not sure about Aristotle, but Peirce indeed explicitly rejected the notion that continuous time is somehow composed of durationless instants. They are artificial creations of thought for marking and measuring time, just like discrete points on a line.

However, it cannot serve as the basis for true continuity, because it requires discreteness at the outset. — aletheist

Do you know of any current mathematical objects that behave more like true continuity in your view?

Do you know of any current mathematical objects that behave more like true continuity in your view? — fdrake

Line figures, surfaces, and solids can be understood in geometry as truly continuous. We use points to model and analyze them, but they are not composed of points.

More fundamentally, my understanding is that category theory is broader than set theory and can serve as a basis for alternative approaches to mathematics that recognize true continuity. The one that currently seems to come closest to Peirce's views is synthetic differential geometry, also known as smooth infinitesimal analysis.

Thanks!

Introducing numbers already imposes discreteness. Numbers are for measuring, they cannot constitute a truly continuous line. — aletheist

You are reading more into what @fdrake proposed than there is. He didn't say anything about numbers constituting a line; on the contrary, he was going with your paradigm of continuous line figures - nothing else. And he was trying to show how, with assumptions that seem reasonable even in that paradigm, you still end up with a system that is isomorphic to the set construction. (Surely, we can still talk about the lengths of those line figures? Those are all the numbers that we need to get going.) I am not sure whether it can actually work out that way, but that was the idea, if I understand him correctly.

I am not sure whether it can actually work out that way, but that was the idea, if I understand him correctly — SophistiCat

Regardless, I think the synthetic differential geometry pdf suggested an intuition closer to it. Roughly, augment the real numbers with some set of symbols $D$.

You then stipulate that any function $f$ defined on $D\rightarrow \mathbb{R}\cup{D}$ has some unique constant $b$ such that for all $d \in D$ $f(d)=f(0)+db$. There's also a property that $d^2 = 0$ for all $d \in D$. This looks like placing a family of infinitely small line segments around every point in $\mathbb{R} \cup D$. For functions, it gives a "tangent space" of a function at a point confined to infinitesimal neighbourhoods around it. It "smears out" the real line (and function values) into something that can't be element-wise disassembled in the same way (if you fix a point of $\mathbb{R}$, this corresponds to an infinitesimal neighbourhood around that point in $\mathbb{R}\cup D$)

Let's say I have some "True continuity" X. Like a line X=(0,1). Let's say I can take "parts" of it in the above manner; arbitrary subintervals. (0,a), (a-1/n,1) are subintervals for any a less than 1 and greater than 1/n. Since they're parts of a true continuity, the true continuity ensures the existence of their intersection; limit the process of intersection over n and get the limit {a}. If I take the union of all such limits, since a was an arbitrary member of (0,1), it produces (0,1). So starting from a true continuity, like a line segment, you can get discrete numbers, then build up the true continuity out of the individual numbers through a union. It's top down and bottom up at the same time.

Do you agree this process is legitimate? — fdrake

But you can't start from ANY real number "a". If you define real numbers as limits of rational numbers, "a" should be rational, or should be itself a limit of a sequence of rationals. In a constructivist logic you have to define how "a" is "built".

You then stipulate that any function ff defined on D→R∪DD→R∪D has some unique constant bb such that for all d∈Dd∈D f(d)=f(0)+dbf(d)=f(0)+db. There's also a property that d2=0d2=0 for all d∈Dd∈D. This looks like placing a family of infinitely small line segments around every point in R∪DR∪D. For functions, it gives a "tangent space" of a function at a point confined to infinitesimal neighbourhoods around it. It "smears out" the real line (and function values) into something that can't be element-wise disassembled in the same way (if you fix a point of RR, this corresponds to an infinitesimal neighbourhood around that point in R∪DR∪D) — fdrake

Exactly. This is a sheaf of linear tangent spaces built on the base space of Cauchy sequences of rational numbers. Similar to the sheaf of all vector spaces tangent to a sphere.

Well, you've arrived just where I have a problem, or, rather, where I'm unclear and confused, because I'm not arguing against any well-known fact, but rather I seem to be stuck in some misconception or misperception.

Taking your .10101010..., how long is it? How many zeros and ones? As many as there are counting numbers? Or more? ℵo or ℵ1? — tim wood

ℵo. That's the cardinality of infinite DISCRETE sets.
ℵ1 is the cardinality of powersets of sets whose cardinality is ℵo.

It's not a matter of what exists in reality, it's a matter of what is contradictory in principle. To say " I am going to talk about this object, but this object is not really an object, because there is zero of them", is blatant contradiction. To bring this expression out of contradiction we must amend it. I might say for instance, "I am going to talk about a type of object, of which there are none", or I might say "I am going to talk about a quantity, and this quantity is zero". But if I make the category mistake of conflating these two options to say "I am going to talk about this quantity, zero, as an object itself, and assert that there is none of these objects", then I contradict myself. — Metaphysician Undercover

If you want to prove that ZFC is inconsistent you have to derive "false" using the rules of ZFC's logic. You can't do it using english language, as you are trying to do.
You can't be an art critic without looking at the paintings!

Not sure about Aristotle, but Peirce indeed explicitly rejected the notion that continuous time is somehow composed of durationless instants. They are artificial creations of thought for marking and measuring time, just like discrete points on a line. — aletheist

Thanks. That would make sense. Physics would get more difficult I imagine.

Here is one especially succinct argumentation from Peirce.

We are conscious only of the present time, which is an instant, if there be any such thing as an instant. But in the present we are conscious of the flow of time. There is no flow in an instant. Hence, the present is not an instant. — Peirce, c. 1895

It then follows from the first sentence that since the present is not an instant, there is no such thing as an instant at all.

Here is another passage that I found very enlightening when I first came across it in an unpublished manuscript.

Just as it is strictly correct to say that nobody is ever in an exact Position (except instantaneously, and an Instant is a fiction, or ens rationis), but Positions are either vaguely described states of motion of small range, or else (what is the better view), are entia rationis (i.e. fictions recognized to be fictions, and thus no longer fictions) invented for the purposes of closer descriptions of states of motion; so likewise, Thought (I am not talking Psychology, but Logic, or the essence of Semeiotics) cannot, from the nature of it, be at rest, or be anything but inferential process; and propositions are either roughly described states of Thought-motion, or are artificial creations intended to render the description of Thought-motion possible; and Names are creations of a second order serving to render the representation of propositions possible. — Peirce, 1906

Physical reality is a dynamical process of continuous motion, while psychical reality is an inferential process of continuous thought; more generally, continuous semeiosis. Positions and propositions are artificial creations for describing hypothetical instantaneous states of motion and thought/semeiosis, respectively.

If you want to prove that ZFC is inconsistent you have to derive "false" using the rules of ZFC's logic. You can't do it using english language, as you are trying to do.
You can't be an art critic without looking at the paintings! — Mephist

You do recognize that "false" and "true" are assigned to the premises, not by what is determined by the logical system, (which is validity), don't you? Inconsistent, or contradictory premises, is not determined by the logic of the system.

I criticize the axioms according to how they are expressed in English. If the fundamental axioms could not be expressed in English, or other natural languages, they would be meaningless. Terms need to be defined.

If a piece of art has any meaning at all, it would be recognizable by someone other then the artist. So, you don't need to be an artist to be an art critic. The critic assesses the meaning.

You might argue that the axioms have a different meaning in mathematics from what is represented in the English expressions of them, just like the artist might argue that the critic doesn't understand the piece of art. But what type of meaning could this be, if when it is represented in English it is contradictory? Sure, an artist might represent a person as being happy and sad at the very same time, but such illogical nonsense ought to be excluded from a logical discipline such as mathematics.

You do recognize that "false" and "true" are assigned to the premises, not by what is determined by the logical system, (which is validity), don't you? Inconsistent, or contradictory premises, is not determined by the logic of the system. — Metaphysician Undercover

NO. "false" and "true" in first order logic (the logic used in ZFC) are purely SYNTACTICAL expressions. They are determined ONLY by the logic of the system. That's the way it works!

Look at the definition from wikipedia ( https://en.wikipedia.org/wiki/Consistency ).
"""
although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states a theory T is consistent if there is no formula "phi" such that both "phi" and its negation "not phi" are elements of the set of consequences of T.

The set of axioms A is consistent when "phi" and "not phi" belong to "sentences derivable from A" for no formula "phi".
"""

Then, there is a theorem ( https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem )
"""
that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It makes a close link between model theory that deals with what is true in different models, and proof theory that studies what can be formally proven in particular formal systems.
"""

I criticize the axioms according to how they are expressed in English. If the fundamental axioms could not be expressed in English, or other natural languages, they would be meaningless. Terms need to be defined. — Metaphysician Undercover

In a formal logic system TERMS DON'T NEED TO BE DEFINED. That's why it is called "formal" logic.

But what type of meaning could this be, if when it is represented in English it is contradictory? — Metaphysician Undercover

It can be a meaning that has nothing to do with the English meaning of the words. The proof of the theorem shows that a model always exists (if no contradiction is derivable) because it can be built using the strings of symbols of the formal language itself!
Probably that's the part that you strongly disagree with. But if you want to criticize the proof of Godel's completeness theorem, you should at least read it! That's what I meant by "looking at the paintings" before.

We are conscious only of the present time, which is an instant, if there be any such thing as an instant. But in the present we are conscious of the flow of time. There is no flow in an instant. Hence, the present is not an instant.
— Peirce, c. 1895 — aletheist

Ok. But isn't he conflating human experience with reality? He's right that for humans, the present is an experience of flow. But we have no idea what the underlying reality is. Why should human experience be privileged above nature?

Just as it is strictly correct to say that nobody is ever in an exact Position (except instantaneously, and an Instant is a fiction, or ens rationis), but Positions are either vaguely described states of motion of small range, or else (what is the better view), are entia rationis (i.e. fictions recognized to be fictions, and thus no longer fictions) invented for the purposes of closer descriptions of states of motion; so likewise, — Peirce, 1906

Perfectly sensible. Our physics is an approximation or conceptual model to help us describe reality. It's not to be confused with reality. A point I've made many times.

Physical reality is a dynamical process of continuous motion, while psychical reality is an inferential process of continuous thought; more generally, continuous semeiosis. Positions and propositions are artificial creations for describing hypothetical instantaneous states of motion and thought/semeiosis, respectively. — aletheist

Yes of course. Perfectly well agreed. But if all Peirce is saying is that the map is not the territory, that our mathematical and conceptual models are useful fictions to help us manage or conceptualize reality. then of course he's right; but is that all there is? I thought this point was fairly well agreed, even in science. Einstein supersedes Newton supersedes Aristotle. You never get to the end of the process, you just get increasingly better models.

But isn't he conflating human experience with reality? — fishfry

The only reality that we can know is what we learn from experience. We formulate hypotheses to explain our experience (retroduction), work out their necessary consequences and make predictions accordingly (deduction), then test whether those predictions are corroborated or falsified by subsequent experience (induction).

What is reality? Perhaps there isn't any such thing at all. As I have repeatedly insisted, it is but a retroduction, a working hypothesis which we try, our one desperate forlorn hope of knowing anything. — Peirce, 1898

NO. "false" and "true" in first order logic (the logic used in ZFC) are purely SYNTACTICAL expressions. They are determined ONLY by the logic of the system. That's the way it works! — Mephist

The problem though, is that I am talking about judging those premises or axioms which establish those definitions of "true" and "false". If I am to judge them, I must judge them in relation to something else, something outside the system If you are asking me to accept the precepts of the system without judging them, then you are being unreasonable. That's the way it works! We're grown adults, we have free choice to judge these things. It's completely unreasonable for you to say that I must accept the system's axioms in order to judge the system's axioms, when acceptance is dependent on judgement, and acceptance precludes the possibility of fair judgement. A conclusion cannot be incompatible with the premise, so if I accept the axioms, it is literally impossible for me to produce a judgement against them. Therefore you are being completely unreasonable.

In a formal logic system TERMS DON'T NEED TO BE DEFINED. That's why it is called "formal" logic. — Mephist

Logical systems use symbols. A symbol which represents nothing is contradictory nonsense, just like the empty set. So you're just spewing more contradictory nonsense in an effort to justify your earlier contradictory nonsense.

The proof of the theorem shows that a model always exists (if no contradiction is derivable) because it can be built using the strings of symbols of the formal language itself!
Probably that's the part that you strongly disagree with. But if you want to criticize the proof of Godel's completeness theorem, you should at least read it! That's what I meant by "looking at the paintings" before. — Mephist

Yes, that's what I strongly disagree with. Strings of symbols without definitions is nonsense. A symbol which represents nothing is not a symbol. If you are merely talking about a set of rules by which symbols are related to each other, then there is no reason why we can't discuss these rules in plain English. I think that your refusal to discuss this in plain English is evidence that you know that there is deception within the system. So, either you discuss these rules in plain English or I level the accusation that you're attempting to hide deception behind your language.

I don't see how your art analogy works for you. An individual can glance at a painting, and find it ugly without analyzing it, just by apprehending a few prominent features of it. Likewise, we can hear a piece of music, and right away form a dislike for it based on some fundamental aspect of it. Why would you insist that the person must make a thorough analysis, attempting to empathize with the artist's intent, wasting one's time, and even torturing oneself, to justify one's dislike for the piece? When the person can point to a few fundamental, and prominent features, and explain why these features make such an effort unappealing and unwarranted, why not simply accept that, rather than insisting that the person cannot make such a judgement. If fundamental and obvious aspects of the art are unappealing, why insist that the critic must analyze all the finer aspects before making a judgement of dislike?

The only reality that we can know is what we learn from experience. — aletheist

Didn't Plato point out that what we experience is but shadow on a cave? And that the true reality lies outside, unseen and unseeable by us?

But is what we're talking about simply the question of whether what we experience is the same as reality? Did Peirce argue that it is? But how can that be? Science is historically contingent; and the better equipment we have, the better experiments we can do. Our theories of the universe keep changing. The universe, presumably, stays the same.

If you are asking me to accept the precepts of the system without judging them, then you are being unreasonable. — Metaphysician Undercover

Were you like this when you learned to play chess? "This is the knight." "But no it's not REALLY a knight. Real knights don't make moves like that, they slay dragons and rescue damsels. I refuse to accept the rules of your game till you tell me what they mean outside of the game."

A conclusion cannot be incompatible with the premise — Metaphysician Undercover

And sometimes a conclusion is indistinguishable from the premise.

:roll:

If you are merely talking about a set of rules by which symbols are related to each other, then there is no reason why we can't discuss these rules in plain English. I think that your refusal to discuss this in plain English is evidence that you know that there is deception within the system. So, either you discuss these rules in plain English or I level the accusation that you're attempting to hide deception behind your language. — Metaphysician Undercover

Yes, I confess that I am trying to hide a deception behind MY language :rofl:
The things that I wrote can be found in any introductory book to mathematical logic, and they are very clear for most of the people that write on this forum. If you really wanted to understand it, just buy a book and read it!

I don't see how your art analogy works for you. An individual can glance at a painting, and find it ugly without analyzing it, just by apprehending a few prominent features of it. — Metaphysician Undercover

Yes, but you are not even trying to take a glance at the painting! You don't have to be an expert to understand how mathematical logic works. And unlike other parts of math, you don't even need to learn some other more fundamental concepts before.

It's completely unreasonable for you to say that I must accept the system's axioms in order to judge the system's axioms, when acceptance is dependent on judgement, and acceptance precludes the possibility of fair judgement. A conclusion cannot be incompatible with the premise, so if I accept the axioms, it is literally impossible for me to produce a judgement against them. Therefore you are being completely unreasonable. — Metaphysician Undercover

You don't have to accept the axioms. You have to prove that assuming those axioms leads to a contradiction. You have to use the rules of logic to produce a sentence of the form "A and not A" (I am not sure if "true" and "false" are terms of first order logic, maybe I made a mistake before saying that you
have to derive "false"). As @fishfry has explained to you many times, this is like in the game chess: you have to show that, starting from a given position, white can checkmate. If it's not possible to produce a sentence of the form "A and not A", it means that the axioms are consistent (not contradictory).

The interpretation of the terms as sets (and then the meaning of the sentences) is a different issue.
You can argue that the terms that ZFC calls "sets" are not exactly correspondent to what we "intuitively" think to be sets, and a lot of people (even mathematicians) have this kind of objections to ZFC. But this is not about the consistency of the theory; this is about it's "meaning".

Were you like this when you learned to play chess? "This is the knight." "But no it's not REALLY a knight. Real knights don't make moves like that, they slay dragons and rescue damsels. I refuse to accept the rules of your game till you tell me what they mean outside of the game." — fishfry

That's not quite right. I learned how the game was played, then decided I didn't want to play it. The fact that it was a game, and the rules referred to nothing "real" probably made me think of it as a waste of time.

You have to prove that assuming those axioms leads to a contradiction. — Mephist

It's not a matter of whether or not the axioms lead to contradiction, it's a matter of whether or not the axioms themselves are contradictory. I demonstrated precisely this, that the axioms are contradictory, with the "empty set". If you've forgotten already, go back and take another look at those posts. Sure, there was some mention that the contradiction might potentially be avoided by introducing exceptions to the rules. But exceptions to the rule just serve to disguise and hide the contradiction in the rules, behind sophisticated complexities. They do not resolve it.

You have to use the rules of logic to produce a sentence of the form "A and not A" (I am not sure if "true" and "false" are terms of first order logic, maybe I made a mistake before saying that you
have to derive "false"). — Mephist

This is where you are wrong Mephist, and I can't seem to get this through your head. You cannot use the rules of the logic without first accepting the axioms. The axioms state the rules. You must agree to play by the rules in order to use the logic. If the axioms themselves are contradictory, then you accept those contradictions, when you proceed to use the logic. Therefore you cannot prove that there is something unacceptable about the axioms, i.e. that they are contradictory, through the use of the logical system, because by accepting the axioms you consent that there is nothing unacceptable about them, i.e. they are not contradictory.

Use of contradictory axioms may lead to absurdities like paradoxes, but a paradox does not prove that any axioms are contradictory, the appearance of paradox could be caused by something else. So when a paradox appears one might go back to the axioms, and determine whether or not there is a contradiction, using principles outside the logical system, the principles in which the system's axioms are based. The axioms are the rules of the logical system and they may just lead to an unsolvable paradox within the system. We do not necessarily know the cause of the paradox though, and sometimes analysis of the paradox cannot lead us to its cause. That's why it's a "paradox". if the paradox is caused by a problem with the axioms, then to solve the problem requires going outside the logical system to determine how the axioms are founded, the relations between them, interpretations of them, etc., and removing inconsistencies.

To take fishfry's example of the chess game, imagine contradictory rules in the game. This could lead to unsolvable problems within the game. From within the game, the problems cannot be solved because the rules leading to the problem are set. However, because the premise of my example is that their are contradictory rules, it appears obvious that the problems are caused by contradictory premises, as contradictory premises can cause problems. So you might think that if there are contradictory rules in any game, the problems they cause would demonstrate clearly, the very contradiction which exists in the rules. But this is not the case, because an equivalent problem might arise as a matter of a difference in interpretation. The "paradox" within the game, by the very nature of a "paradox" doesn't necessarily reveal the source of the problem. And although the rules of the game might include rules of interpretations, those rules of interpretation cannot have rules of interpretation, ad infinitum. Therefore contradiction, and other problems in the rules, or axioms, can only be determined and resolved by reference to principles outside the system composed of those rules, because they might equally be problems of interpretation.

The interpretation of the terms as sets (and then the meaning of the sentences) is a different issue.
You can argue that the terms that ZFC calls "sets" are not exactly correspondent to what we "intuitively" think to be sets, and a lot of people (even mathematicians) have this kind of objections to ZFC. But this is not about the consistency of the theory; this is about it's "meaning". — Mephist

Now perhaps you are starting to grasp what I am arguing. I am arguing the "meaning". Contradiction within the meaning of a theory's axioms is clearly a matter of inconsistency. We might have two distinct senses of "inconsistency" here though, whether the proceedings in a logical system are consistent with the axioms, and whether the axioms themselves are consistent. I am arguing the latter.

So you might hand me the rules of a game, and ask me to play. If I look over the rules and see that there is blatant contradiction in the rules, as in set theory, or even that the rules are open to contrary interpretations, or there are holes, possible situations not covered, I might tell you that I have no desire to play your game. And, I might proceed to show you what I see as blatant contradiction, the empty set. You might accept this and say ok, don't play then. You might also claim that this is only a matter of interpretation, and attempt to work out a satisfactory interpretation with me. But your insistence, that I join the game, even when I see a glaring problem in the rules, and then we try to work out this problem with the rules only after we develop a problem within the play of the game, is completely unreasonable.

That's not quite right. I learned how the game was played, then decided I didn't want to play it. The fact that it was a game, and the rules referred to nothing "real" probably made me think of it as a waste of time — Metaphysician Undercover

You reject all formal systems not based strictly on physical reality? Do you drive on the correct side of the road appropriate to your jurisdiction? Traffic laws are a made-up game too.

Again, nihilism. You reject games, you reject abstraction, you reject science. Fun to argue all day long on an Internet forum but I truly doubt you actually live this way. I bet you obey traffic laws even though they are not laws of nature.

I understand what you mean. But the word "contradiction" in mathematics has the meaning that I said: "A and not A" is not provable for any A.
What you call "contradiction", the impossibility to identify the terms of the language with physical objects, is not considered as a problem in mathematics: it's simply ignored.

Here's a famous quote from Bertrand Russell about mathematics:

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

(https://www.brainyquote.com/quotes/bertrand_russell_402437)

This is at the same time both a limitation and an advantage: it gives you the freedom to invent new concepts, but you lose the relation between mathematical concepts and the physical world!
This is a choice that mathematicians have done at the beginning of 20th century, mainly (I believe) to get rid of the paradoxes arising from the use of infinity and infinitesimals.

However, in my opinion this is the natural development of Aristotle's logic: the formalization of the rules of deduction. The rules of deduction (used in proofs) should not depend in any way on the meaning (or correspondence to real physical objects) of the words.

So, you say that this is all wrong, because you are allowed to create axioms that don't have any correspondence to reality. That's true. But what is the alternative? After all, these mathematical constructions based on "nothing real" happen to be very useful to build models that agree with experiments.
As far as I know, I think it would be possible to reformulate all mathematics without making use of empty sets at all. But would this make any difference?