I don't like that the empty set is a subset of every set, just as I don't like that X raised to the power zero is one. However, the truth of both is required to preserve the total logic of a system. Denial that X raised to the power zero equals one leads to contradiction, so that's that.

Pay attention now. A set A is NOT a subset of set B if there exists an element x which is in A but not in B.

So if the empty set O is not a subset of set B, then there is an element in O which is not in B.

But this implies there is an element in O which contradicts the definition of empty set, a set which contains no elements.

Conclusion the empty set is a subset of any set ( notice that above B was an arbitrary set).

Q.E.D.

There was a sign at the entrance to Plato's Academy way back when in Athens. It said "Let no one ignorant of mathematics enter through this gate"

Would you be able to enter Plato's Academy if it were still around?

You write, "A set A is NOT a subset of set B if there exists an element x which is in A but not in B.

So if the empty set O is not a subset of set B, then there is an element in O which is not in B."

Maybe I am missing something, but isn't the above like this:

T if G

Therefore, if T then G

I don't recall the name, but isn't that invalid?

And, you quote from my post but then your reply seems unresponsive to and uncorrective of anything in my quote. Do you find an error in what you quoted from me? What is it?

"There was a sign at the entrance to Plato's Academy way back when in Athens. It said "Let no one ignorant of mathematics enter through this gate"

Would you be able to enter Plato's Academy if it were still around?"

"I don't like that the empty set is a subset of every set, just as I don't like that X raised to the power zero is one. However, the truth of both is required to preserve the total logic of a system. Denial that X raised to the power zero equals one leads to contradiction, so that's that."

…that I agree with you that the empty set is a subset of every set?

Do you find an error in my quote, above? If yes, what is the error?

"I don't like that the empty set is a subset of every set, just as I don't like that X raised to the power zero is one. However, the truth of both is required to preserve the total logic of a system. Denial that X raised to the power zero equals one leads to contradiction, so that's that."

…that I agree with you that the empty set is a subset of every set?

Do you find an error in my quote, above? If yes, what is the error?

Sorry about that. I did not get past "I don't like that the empty set..." My apologies.

The main consequence of the existence of the empty set and the fact that is a subset of every set is that existential import has to be disposed of. In order to get from All S is P to Some S is P I need an independent proof of existence. All S is P does not imply Some S is P as can be seen by taking S to be the empty set or the predicate that is not true of any individual.

And that is one of the reasons that Aristotle's syllogistic is defective. Another reason is that n-ary relations for n > 1 cannot be cleanly handled. Hence one cannot show by means of categorical syllogism that all Bears are Mammals implies the tail of a Bear is the tail of a Mammal. No of Aristotle's rules given in Prior Analytics will carry that argument.

Yes, it seems that multiple and nested quantifiers are incapable of being expressed...

Another example: how would express the concept of a mathematical limit without using the predicate calculus? I'm not even sure that Sommer's term logic could do that.

It cannot. . In fact one needs second order logic to prove that every set of real number bounded above has a least upper bound. Sommer'sl logic cannot be used to do mathematics. But then Sommers and Engelbertson did not have that goal. They wanted to beef up syllogistic to handle any verbal argument in a form that is congruent to natural language. Unfortunately first order logic and second order logic (logic if predicates) has a syntax that is quite alien to natural language usage. For example I say in ordinary English that all Bears are Mammals. In first order logic I have to resort to this locution: for all x, if x is a Bear then x is a Mammal. It says the same thing but in a very unnatural manner.

So beefed up term logic has a virtuous goal. To make Aristotle's syllogistic adequate for verbal argument made using plain old natural language. They even beefed it up sufficiently to handle the problem of the tails to wit: if all Bears are Mammal that any tail of a Bear is the tail of a Mammal. I also read somewhere they can even hand three and four term relations in a manner not alien to natural language. What they cannot do however is to derive metamathematical theorems such as the Goedel incompleteness theorem. So Term Logic would not be a good vehicle for doing universal algebra or topology.

It turns out the the needs of mathematics is what drove Frege's development of predicate logic (and also Cantor's work). It was Frege that took the major step away from Aristotle's syllogistic. So there are two logical domains: the logic that is suitable for expressing reasoning in fields like the theory of real and complex variable and the logic that is suitable for rigorous argument using natural language. This would be Jim Dandy for lawyers for example. The natural sciences depend absolutely on the mathematics. Without differential equations one could not do physics at all. But using Term Logic one could write bullet proof contracts and laws.

Oh, Kant rare! Take a closer look at how you are interpreting your process--or, rather, PROCESSES--there. You are package-dealing the process of reaching in and pulling out whatever is there with the process of ACTUALLY COUNTING THE COINS--and that with the additional process of TRYING TO ADD actual coins with non-existent coins.

Start with an empty cup. Now toss six pennies into the cup. How many pennies in the cup? Six.

I have just added six coins to a set of non-existent coins.

I am afraid you are no mathematician, Roger.

Ba'al Chatzaf

It seems to me that ~anywhere~ and ~everywhere~ that someone ~might~ place six coins would be "a set of non-existent coins." The universe, by this view, is through-and-through empty sets, waiting for the ~possible~ addition of six coins, ten thousand coins, three horses, twelve bogus mathematical-theoretical arguments for the existence of empty sets, whatever.

This all smacks of idealism, to me, Ba'al. I don't know if you have heard of, or recall, J. S. Mill's notion that matter is the permanent possibility of sensation. But it sounds a lot like how your view implies the universe is like: the permanent possibility of non-empty sets.

A set is not like a room or a cup. A set is a collection of objects. The room or cup or area on a table or on the ground where the pennies are placed is not a set before the pennies are placed there. It is just a room, cup, etc.

If I showed someone my empty china cabinet and said what do you think of my set of china, he would rightly say: "What set of china?" According to you, I should say, "You must be blind. It's an empty set of china." To which he would rightly say: "Oh, yeah, right next to the empty set of sane ideas that are coming out of your mouth right now."

REB

Is the set of U.S. one thousand dollar bills in your breast pocket empty or not?

Is the set of hands each with seven fingers attached to your arms empty or not?

You see, empty sets exist.

Ba'al Chatzaf

There are no sets in my breast pocket -- not of one thousand dollars bills, nor of anything else. Thare isn't anything in my pocket, unless you count air molecules and the like. But those are not sets. Only if I mentally collect something or other is there a set. Sets do not exist independently of the mind that forms them by mentally collecting this or that as units/members of the set.

Similarly, there are no sets "attached to" my arm either. There are hairs, dead skin cells preparing to flake off (biting my tongue when I say that), etc. But these are not sets, until I or someone mentally collect them into a group.

But for them to be sets, there has to be ~something~ to be mentally isolated as a group. When there isn't anything to be mentally isolated, there can't be a set. A room empty of items doesn't "contain nothing." It ~doesn't~ contain ~anything~. Zero as an "amount" or "number" or "quantity" isn't the presence of nothing. It is the absence of anything. There isn't ~any~ amount. There isn't any qnantity. There isn't any number.

"Empty set" makes no more sense than "itemless collection." To insist that it does is arbitrary nonsense.

I don't like that the empty set is a subset of every set, just as I don't like that X raised to the power zero is one. However, the truth of both is required to preserve the total logic of a system. Denial that X raised to the power zero equals one leads to contradiction, so that's that.

Likewise for the empty set. The subsets of {a1,a2} can be confirmed by checking the unions:

{a1, a2} U {a1, a2} = {a1, a2}

{a1, a2} U {a1} = {a1, a2}

{a1, a2} U {a2} = {a1, a2}

{a1, a2} U { } = {a1, a2}

Therefore, all four ( {a1, a2}, {a1}, {a2} and { } ) are subsets of {a1, a2}.

It isn't pretty, but it appears unavoidable.

I don't see anything wrong with saying, "It's an empty set of china." An empty china cabinet IS an empty set of china.

1. I don't see the two problems as equivalent at all. I have explained elsewhere here on OL and in an essay posted on my web site (www.rogerbissell.com) that any given number taken to the zero power being equal to 1 makes perfect sense if you conceive of it as the number 1 ~not~ subjected to the operation of being multipled by any given number any times. The zero power acts as a command to ~not~ multiply 1 by the given number ~any~ times.

2. A china cabinet is not a set of ~any~ kind. If a china cabinet is empty, it's empty of anything and everything that might fit inside it. By your reasoning, then, an empty china cabinet would also be an empty set of marbles, an empty set of copies of Atlas Shrugged, an empty set of hands with seven fingers, etc.

Just as when you surreptiously introduce division by zero into an algebraic equation and can end up with virtually anything being equal to something else, so can an empty set be an empty set of anything whatever. They're both contradictory nonsense.

Ba'al's example of the (supposed) empty set of hands with seven fingers (which he says exists!) attached to my arm is a perfect case in point. It is a violation of Occam's razor on steroids...multiplying non-existent entities (empty sets) in total disregard of necessity. By his reasoning, my arm has an infinity of empty sets attached to it. Sorry, I'm gonna have to play the Empiricist card here. No evidence, no logic to support Ba'al's claim.

"2. A china cabinet is not a set of ~any~ kind. If a china cabinet is empty, it's empty of anything and everything that might fit inside it. By your reasoning, then, an empty china cabinet would also be an empty set of marbles, an empty set of copies of Atlas Shrugged, an empty set of hands with seven fingers, etc."

Correct.

Except that it is also empty of things that might NOT fit inside it, so it's more empty than you are allowing.

I drive 300 miles until my truck stops. Can we describe the set of gas in my tank?

Seems your reasoning would disallow even talking about the set of gas in the tank since there is "no set of ~any~ kind" in the tank.

But we can't not talk about the gas in the tank, because when we say, "it's empty," we're talking about: gas in the tank.

What we have here, metaphysically speaking, is a physical container -- something we call a "gas tank" because it was specifically designed to hold gas -- that does have any gas in it, or at least not enough for the practical purpose of fueling your truck.

Your empty gas tank does not contain something called an "empty set" of gas, any more than it contains an empty set of giraffes. "Sets," as Roger has pointed out, are merely mental constructs. As such, they may serve a purpose in logic and mathematics, but that doesn't endow them with any metaphysical status. Similarly, I might decide to organize my books according to the color of their covers, in which I case I might speak of sets of red books, blue books, etc. Or I might organize them into "sets" according to their size, subject matter, etc. But, again, none of my classifications (necessarily) has any metaphysical significance.

I might even leave a space on a shelf for books with polka dot covers, even though I presently don't have any books like this. Then if someone asks me why that space is empty, I might say that I anticipate acquiring some polka dot books in the future. I might even put my answer in positive terms by saying, "That space is for polka dot books." But unless I intentionally created that space for a specific purpose, then it is just an empty space, nothing more. To treat the space as an empty set of polka dots books makes sense only if I had a purpose in creating that space.

Similarly, when we construct empty sets in logic, we need to ask, "An empty set of what?" The what in such cases defines and delimits the empty set and gives it meaning. Without such a predetermination, the notion of an "empty set" would serve no purpose.

"Sets," in short, are simply mental classifications. We can speak about them in various ways, depending on our cognitive purpose, but let's not get carried away by reifying abstractions, especially that abstraction called "nothingness."

The inclusion of the empty set in any set reflects the fact for for material implication -> , False -> p is true, where p is an arbitrary proposition.

One of the biggest mistakes one can make is mixing metaphysics and ontology with formal logic. Formal logic is a machine for grinding out propositions from other propositions with the happy situation than any proposition ground out from a true proposition is itself true.

Here is an analogy. Logic is a machine that cranks wieners from ingredients.. If you put in tasty ingredients you get tasty wieners.

The inclusion of the empty set in any set reflects the fact for for material implication -> , False -> p is true, where p is an arbitrary proposition.

One of the biggest mistakes one can make is mixing metaphysics and ontology with formal logic. Formal logic is a machine for grinding out propositions from other propositions with the happy situation than any proposition ground out from a true proposition is itself true.

Here is an analogy. Logic is a machine that cranks wieners from ingredients.. If you put in tasty ingredients you get tasty wieners.

Ba'al Chatzaf

Were your comments directed to me? If so, I am well aware what an "empty set" is.

The notion, as you illustrated in your post, that any proposition, whether true or false, is implied by a false proposition -- e.g., If 2+2=5. then the earth was destroyed yesterday -- results from the peculiar and arbitrary axioms of propositional logic. Any such proposition is "true" only in a Pickwickian sense, as defined by the axioms of a given system. Whether the axioms of that system make sense is, of course, another matter. But to make sense of such things in "material implication," we first need to clear our minds of what we normally mean by "implication." Strictly speaking, nothing follows necessarily from a false proposition.

Ghs

Later note: I should qualify my last statement. For example, a false proposition implies the truth of its contradictory. But I was thinking of propositions that are logically unrelated, as we find in material implication.

I agree on the need to avoid reifying nothingness. The gauge on my dash successfully avoids such reification, yet it never stops measuring the set of gas in the tank. The algorithms running the gauge constantly track the set of gas in the tank. They don't get trapped in a vicious loop owing to the nonsensicality of measuring non-existent gas, as Roger would have it.

Peek inside the software and examine the values it is holding and you will find that it has a value for the set of gas in the tank: "E." Not, "nonsense." Not, "reification error."

As you said, "[empty sets] may serve a purpose in logic and mathematics."

"The empty set of gas in the tank" isn't pretty rendered in plain english, but it's no problem for logic, no problem for the software running the gauge. It's not nonsense.

"The empty set of gas in the tank" isn't pretty rendered in plain english, but it's no problem for logic, no problem for the software running the gauge. It's not nonsense.

There is no empty set of gas, or a set of any kind, in the tank. The set (or class) is a mental construct and exists only in the mind.

If we wish to speak more correctly, the tank itself would constitute the set (as a Euler circle might represent it on paper), in which case to speak of "the empty set of gas in the tank" would be incorrect, since "set" and "tank" denote precisely the same thing. This would be like saying "There is an empty gas tank in the gas tank."

What we should say here is "The gas tank is empty," not that there is a set within the gas tank that is empty. The "set," in this context, is merely an abstract representation of the tank itself, not something in addition to it and something that could be contained inside it. Thus we could say the following: "The gas tank, whether viewed concretely as a physical object or abstractly as a set, is empty."

Similarly, if we draw a circle to represent widgets but place no symbols signifying widgets inside the circle, then we can say one of two things: "The circle is empty, since it contains no widgets." Or: "The set is empty, since it contains no widgets." But it would be incorrect to say "The circle contains an empty set of widgets." This would be tantamount to saying "The circle contains another circle, and that additional circle contains no widgets."

The set, in this case, simply is the circle viewed from an abstract perspective. We are guilty of a type of conceptual double-counting when we treat the "set" as something in addition to the circle. We commit the same error if we speak of an empty set of gas in the gas tank.

Thus your formulation is not "nonsense," strictly speaking. It is merely false.

Jon: ""The empty set of gas in the tank" isn't pretty rendered in plain english."

George: [Pretties the english]

I see nothing wrong with the prettification.

I repeat, "It's no problem for logic, no problem for the software running the gauge." I can freeze the software and examine its value states. The set of gas it keeps track of has not devolved to nonsense or falsity, it simply has a zero current value.

As you and Roger have it, the software would have to answer: "There is no such set. I cannot answer questions about non-existent sets."

But it doesn't do that. It answers, "I have been keeping tabs on that set, and presently its value is precisely: zero."

But to make sense of such things in "material implication," we first need to clear our minds of what we normally mean by "implication." Strictly speaking, nothing follows necessarily from a false proposition.

I respond: That is correct. The only way to detach the conclusion from the premise is to assert the premise and a false proposition cannot be assert.

Modus ponens applies only when the premise is true.

Jon: ""The empty set of gas in the tank" isn't pretty rendered in plain english."

George: [Pretties the english]

I see nothing wrong with the prettification.

I repeat, "It's no problem for logic, no problem for the software running the gauge." I can freeze the software and examine its value states. The set of gas it keeps track of has not devolved to nonsense or falsity, it simply has a zero current value.

As you and Roger have it, the software would have to answer: "There is no such set. I cannot answer questions about non-existent sets."

But it doesn't do that. It answers, "I have been keeping tabs on that set, and presently its value is precisely: zero."

An abstract class can have a "value" of zero, and there can be "null sets." There is no problem per se in this way of speaking, provided we don't slip into the error of treating such classifications as anything more than convenient mental constructs.

Quite different is the error, say, of referring to an empty cupboard as containing an "empty set" of dishes -- as if the "set" in question is something more than the cupboard itself viewed in abstract terms. There is no such abstract "set" in the cupboard, empty or otherwise. If the cupboard is empty, then so is the corresponding abstract set. Thus to say that the relevant set is empty is merely to say, in different words, that the cupboard itself is empty. To speak this way requires that we view the cupboard in a highly abstract way. But when we lose track of what we are talking about, there is a tendency to reify the concept "set" and treat it as if it is something that exists in addition to the cupboard itself.

This has nothing to do with "prettification." It has to do with speaking accurately. I am currently wearing a shirt. Hence, viewed abstractly, the "set" of shirts I am wearing may be said to contain a single instance. But I am not wearing a shirt plus a "set" consisting of one shirt. And should I remove my shirt, I am not then wearing an empty set of shirts, even though the relevant abstract set may be said to be empty.

This is the same point I made when discussing the impropriety of speaking of an empty tank as containing an "empty set" of gas. As Roger indicated earlier (though I don't recall the specific context), we can reasonably say that an empty cupboard contains no dishes. But we cannot reasonably say that the cupboard contains an "empty set" of dishes.

As for computers, they will respond as they are designed to respond. What we choose to call a particular operation or function is irrelevant, so far as our topic is concerned.

A set is a "mental container." A meaningful set has criteria for membership. Something that meets the criteria is a member. If nothing meets the criteria, the set is empty.

Example 1: The set C comprised of the intersection of the sets A={1, 2, 3} and B={4, 5, 6}. Since A and B have no common members, C is empty.

Example 2: The set of all coins in my pockets. If my pockets are empty, then the set is empty.

While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity, whose meaning and usefulness are debated by philosophers and logicians. The empty set is not the same thing as nothing; rather, it is a set with nothing inside it and a set is always something. This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. more.

A set is a "mental container." A meaningful set has criteria for membership. Something that meets the criteria is a member. If nothing meets the criteria, the set is empty.

Example 1: The set C comprised of the intersection of the sets A={1, 2, 3} and B={4, 5, 6}. Since A and B have no common members, C is empty.

Example 2: The set of all coins in my pockets. If my pockets are empty, then the set is empty.

While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity, whose meaning and usefulness are debated by philosophers and logicians. The empty set is not the same thing as nothing; rather, it is a set with nothing inside it and a set is always something. This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. more.

Better: Logicians and mathematicians like completeness.

And they shall not have it. The famous Goedel incompleteness theorem shows that any consistent first order logic with the postulates of arithmetic added is incomplete. That is there are closed formulate (no free variables) such that neither they nor their negations are theorems of the system. In a word, provability cannot catch up with truth.

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## BaalChatzaf

Yes. The consistency of First Order Logic which is the basis of all mathematics used in the sciences depends on the existence of the empty set.

Ba'al Chatzaf

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## BaalChatzaf

Pay attention now. A set A is NOT a subset of set B if there exists an element x which is in A but not in B.

So if the empty set O is not a subset of set B, then there is an element in O which is not in B.

But this implies there is an element in O which contradicts the definition of empty set, a set which contains no elements.

Conclusion the empty set is a subset of any set ( notice that above B was an arbitrary set).

Q.E.D.

There was a sign at the entrance to Plato's Academy way back when in Athens. It said "Let no one ignorant of mathematics enter through this gate"

Would you be able to enter Plato's Academy if it were still around?

Ba'al Chatzaf

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## Jon Letendre

You write, "A set A is NOT a subset of set B if there exists an element x which is in A but not in B.

So if the empty set O is not a subset of set B, then there is an element in O which is not in B."

Maybe I am missing something, but isn't the above like this:

T if G

Therefore, if T then G

I don't recall the name, but isn't that invalid?

And, you quote from my post but then your reply seems unresponsive to and uncorrective of anything in my quote. Do you find an error in what you quoted from me? What is it?

"There was a sign at the entrance to Plato's Academy way back when in Athens. It said "Let no one ignorant of mathematics enter through this gate"

Would you be able to enter Plato's Academy if it were still around?"

I don't know. Fuck off, in any case.

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## BaalChatzaf

Can you do this. Substitute O the name of the empty set for A in the definition of not being a subset.

Sure you can. Try real hard, now and it will come to you.

Ba'al Chatzaf

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## Jon Letendre

Can you detect from this I wrote:

"I don't like that the empty set is a subset of every set, just as I don't like that X raised to the power zero is one. However, the truth of both is required to preserve the total logic of a system. Denial that X raised to the power zero equals one leads to contradiction, so that's that."

…that I agree with you that the empty set is a subset of every set?

Do you find an error in my quote, above? If yes, what is the error?

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## BaalChatzaf

Sorry about that. I did not get past "I don't like that the empty set..." My apologies.

Ba'al Chatzaf

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## BaalChatzaf

The main consequence of the existence of the empty set and the fact that is a subset of every set is that existential import has to be disposed of. In order to get from All S is P to Some S is P I need an independent proof of existence. All S is P does not imply Some S is P as can be seen by taking S to be the empty set or the predicate that is not true of any individual.

And that is one of the reasons that Aristotle's syllogistic is defective. Another reason is that n-ary relations for n > 1 cannot be cleanly handled. Hence one cannot show by means of categorical syllogism that all Bears are Mammals implies the tail of a Bear is the tail of a Mammal. No of Aristotle's rules given in Prior Analytics will carry that argument.

Ba'al Chatzaf

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## BaalChatzaf

It cannot. . In fact one needs second order logic to prove that every set of real number bounded above has a least upper bound. Sommer'sl logic cannot be used to do mathematics. But then Sommers and Engelbertson did not have that goal. They wanted to beef up syllogistic to handle any verbal argument in a form that is congruent to natural language. Unfortunately first order logic and second order logic (logic if predicates) has a syntax that is quite alien to natural language usage. For example I say in ordinary English that all Bears are Mammals. In first order logic I have to resort to this locution: for all x, if x is a Bear then x is a Mammal. It says the same thing but in a very unnatural manner.

So beefed up term logic has a virtuous goal. To make Aristotle's syllogistic adequate for verbal argument made using plain old natural language. They even beefed it up sufficiently to handle the problem of the tails to wit: if all Bears are Mammal that any tail of a Bear is the tail of a Mammal. I also read somewhere they can even hand three and four term relations in a manner not alien to natural language. What they cannot do however is to derive metamathematical theorems such as the Goedel incompleteness theorem. So Term Logic would not be a good vehicle for doing universal algebra or topology.

It turns out the the needs of mathematics is what drove Frege's development of predicate logic (and also Cantor's work). It was Frege that took the major step away from Aristotle's syllogistic. So there are two logical domains: the logic that is suitable for expressing reasoning in fields like the theory of real and complex variable and the logic that is suitable for rigorous argument using natural language. This would be Jim Dandy for lawyers for example. The natural sciences depend absolutely on the mathematics. Without differential equations one could not do physics at all. But using Term Logic one could write bullet proof contracts and laws.

Ba'al Chatzaf

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## Roger Bissell

AuthorThere are no sets in my breast pocket -- not of one thousand dollars bills, nor of anything else. Thare isn't anything in my pocket, unless you count air molecules and the like. But those are not sets. Only if I mentally collect something or other is there a set. Sets do not exist independently of the mind that forms them by mentally collecting this or that as units/members of the set.

Similarly, there are no sets "attached to" my arm either. There are hairs, dead skin cells preparing to flake off (biting my tongue when I say that), etc. But these are not sets, until I or someone mentally collect them into a group.

But for them to be sets, there has to be ~something~ to be mentally isolated as a group. When there isn't anything to be mentally isolated, there can't be a set. A room empty of items doesn't "contain nothing." It ~doesn't~ contain ~anything~. Zero as an "amount" or "number" or "quantity" isn't the presence of nothing. It is the absence of anything. There isn't ~any~ amount. There isn't any qnantity. There isn't any number.

"Empty set" makes no more sense than "itemless collection." To insist that it does is arbitrary nonsense.

REB

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## Roger Bissell

Author1. I don't see the two problems as equivalent at all. I have explained elsewhere here on OL and in an essay posted on my web site (www.rogerbissell.com) that any given number taken to the zero power being equal to 1 makes perfect sense if you conceive of it as the number 1 ~not~ subjected to the operation of being multipled by any given number any times. The zero power acts as a command to ~not~ multiply 1 by the given number ~any~ times.

2. A china cabinet is not a set of ~any~ kind. If a china cabinet is empty, it's empty of anything and everything that might fit inside it. By your reasoning, then, an empty china cabinet would also be an empty set of marbles, an empty set of copies of Atlas Shrugged, an empty set of hands with seven fingers, etc.

Just as when you surreptiously introduce division by zero into an algebraic equation and can end up with virtually anything being equal to something else, so can an empty set be an empty set of anything whatever. They're both contradictory nonsense.

Ba'al's example of the (supposed) empty set of hands with seven fingers (which he says exists!) attached to my arm is a perfect case in point. It is a violation of Occam's razor on steroids...multiplying non-existent entities (empty sets) in total disregard of necessity. By his reasoning, my arm has an infinity of empty sets attached to it. Sorry, I'm gonna have to play the Empiricist card here. No evidence, no logic to support Ba'al's claim.

REB

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## Jon Letendre

Roger,

"2. A china cabinet is not a set of ~any~ kind. If a china cabinet is empty, it's empty of anything and everything that might fit inside it. By your reasoning, then, an empty china cabinet would also be an empty set of marbles, an empty set of copies of Atlas Shrugged, an empty set of hands with seven fingers, etc."

Correct.

Except that it is also empty of things that might NOT fit inside it, so it's more empty than you are allowing.

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## Jon Letendre

I drive 300 miles until my truck stops. Can we describe the set of gas in my tank?

Seems your reasoning would disallow even talking about the set of gas in the tank since there is "no set of ~any~ kind" in the tank.

But we can't not talk about the gas in the tank, because when we say, "it's empty," we're talking about: gas in the tank.

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## George H. Smith

What we have here, metaphysically speaking, is a physical container -- something we call a "gas tank" because it was specifically

designedto hold gas -- that does have any gas in it, or at least not enough for the practical purpose of fueling your truck.Your empty gas tank does not contain something called an "empty set" of gas, any more than it contains an empty set of giraffes. "Sets," as Roger has pointed out, are merely mental constructs. As such, they may serve a purpose in logic and mathematics, but that doesn't endow them with any metaphysical status. Similarly, I might decide to organize my books according to the color of their covers, in which I case I might speak of sets of red books, blue books, etc. Or I might organize them into "sets" according to their size, subject matter, etc. But, again, none of my classifications (necessarily) has any metaphysical significance.

I might even leave a space on a shelf for books with polka dot covers, even though I presently don't have any books like this. Then if someone asks me why that space is empty, I might say that I anticipate acquiring some polka dot books in the future. I might even put my answer in positive terms by saying, "That space is

forpolka dot books." But unless I intentionally created that space for a specificpurpose, then it is just an empty space, nothing more. To treat the space as an empty set of polka dots books makes sense only if I had apurposein creating that space.Similarly, when we construct empty sets in logic, we need to ask, "An empty set of

what?" Thewhatin such cases defines and delimits the empty set and gives it meaning. Without such a predetermination, the notion of an "empty set" would serve no purpose."Sets," in short, are simply mental classifications. We can speak about them in various ways, depending on our cognitive purpose, but let's not get carried away by reifying abstractions, especially that abstraction called "nothingness."

Ghs

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## BaalChatzaf

The empty set has no elements in it.

Like the set of 4 corner triangles.

The inclusion of the empty set in any set reflects the fact for for material implication -> , False -> p is true, where p is an arbitrary proposition.

One of the biggest mistakes one can make is mixing metaphysics and ontology with formal logic. Formal logic is a machine for grinding out propositions from other propositions with the happy situation than any proposition ground out from a true proposition is itself true.

Here is an analogy. Logic is a machine that cranks wieners from ingredients.. If you put in tasty ingredients you get tasty wieners.

Ba'al Chatzaf

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## George H. Smith

Were your comments directed to me? If so, I am well aware what an "empty set" is.

The notion, as you illustrated in your post, that any proposition, whether true or false, is implied by a false proposition -- e.g.,

If 2+2=5.-- results from the peculiar and arbitrary axioms of propositional logic. Any such proposition is "true" only in a Pickwickian sense, as defined by the axioms of a given system. Whether the axioms of that system make sense is, of course, another matter. But to make sense of such things in "material implication," we first need to clear our minds of what we normally mean by "implication." Strictly speaking,thenthe earth was destroyed yesterdaynothingfollows necessarily from a false proposition.Ghs

Later note: I should qualify my last statement. For example, a false proposition implies the truth of its contradictory. But I was thinking of propositions that are logically

unrelated, as we find in material implication.## Link to comment

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## Jon Letendre

George,

I agree on the need to avoid reifying nothingness. The gauge on my dash successfully avoids such reification, yet it never stops measuring the set of gas in the tank. The algorithms running the gauge constantly track the set of gas in the tank. They don't get trapped in a vicious loop owing to the nonsensicality of measuring non-existent gas, as Roger would have it.

Peek inside the software and examine the values it is holding and you will find that it has a value for the set of gas in the tank: "E." Not, "nonsense." Not, "reification error."

As you said, "[empty sets] may serve a purpose in logic and mathematics."

"The empty set of gas in the tank" isn't pretty rendered in plain english, but it's no problem for logic, no problem for the software running the gauge. It's not nonsense.

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## George H. Smith

There is no empty set of gas, or a set of any kind,

inthe tank. The set (or class) is a mental construct and exists only in themind.If we wish to speak more correctly, the tank itself would constitute the set (as a Euler circle might represent it on paper), in which case to speak of "the empty set of gas

inthe tank" would be incorrect, since "set" and "tank" denote precisely the same thing. This would be like saying "There is an empty gas tank in the gas tank."What we should say here is "The gas tank is empty," not that there is a set

withinthe gas tank that is empty. The "set," in this context, is merely an abstractrepresentationof the tank itself, not something in addition to it and something that could be contained inside it. Thus we could say the following: "The gas tank, whether viewed concretely as a physical objectorabstractly as a set, is empty."Similarly, if we draw a circle to represent widgets but place no symbols signifying widgets inside the circle, then we can say one of two things: "The circle is empty, since it contains no widgets." Or: "The set is empty, since it contains no widgets." But it would be incorrect to say "The circle

containsan emptysetof widgets." This would be tantamount to saying "The circle containsanothercircle, and that additional circle contains no widgets."The set, in this case, simply

isthe circle viewed from an abstract perspective. We are guilty of a type of conceptual double-counting when we treat the "set" as something in addition to the circle. We commit the same error if we speak of an empty set of gasinthe gas tank.Thus your formulation is not "nonsense," strictly speaking. It is merely false.

Ghs

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## Jon Letendre

Jon: ""The empty set of gas in the tank" isn't pretty rendered in plain english."

George: [Pretties the english]

I see nothing wrong with the prettification.

I repeat, "It's no problem for logic, no problem for the software running the gauge." I can freeze the software and examine its value states. The set of gas it keeps track of has not devolved to nonsense or falsity, it simply has a zero current value.

As you and Roger have it, the software would have to answer: "There is no such set. I cannot answer questions about non-existent sets."

But it doesn't do that. It answers, "I have been keeping tabs on that set, and presently its value is precisely: zero."

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## BaalChatzaf

I respond: That is correct. The only way to detach the conclusion from the premise is to assert the premise and a false proposition cannot be assert.

Modus ponens applies only when the premise is true.

Ba'al Chatzaf

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## George H. Smith

An abstract class can have a "value" of zero, and there can be "null sets." There is no problem per se in this way of speaking, provided we don't slip into the error of treating such classifications as anything more than convenient

mentalconstructs.Quite different is the error, say, of referring to an empty cupboard as

containingan "empty set" of dishes -- as if the "set" in question is somethingmorethan the cupboard itself viewed in abstract terms. There is no such abstract "set"inthe cupboard, empty or otherwise. If the cupboard is empty, then so is the corresponding abstract set. Thus to say that the relevant set is empty is merely to say, in different words, that the cupboard itself is empty. To speak this way requires that we view the cupboard in a highly abstract way. But when we lose track of what we are talking about, there is a tendency to reify the concept "set" and treat it as if it is something that exists in addition to the cupboard itself.This has nothing to do with "prettification." It has to do with speaking accurately. I am currently wearing a shirt. Hence, viewed abstractly, the "set" of shirts I am wearing may be said to contain a single instance. But I am not wearing a shirt

plusa "set" consisting of one shirt. And should I remove my shirt, I am not thenwearinganempty setof shirts, even though the relevant abstract set may be said to be empty.This is the same point I made when discussing the impropriety of speaking of an empty tank as

containingan "empty set" of gas. As Roger indicated earlier (though I don't recall the specific context), we can reasonably say that an empty cupboard containsnodishes. But we cannot reasonably say that the cupboardcontainsan "empty set" of dishes.As for computers, they will respond as they are designed to respond. What we choose to

calla particular operation or function is irrelevant, so far as our topic is concerned.Ghs

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## merjet

A set is a "mental container." A meaningful set has criteria for membership. Something that meets the criteria is a member. If nothing meets the criteria, the set is empty.

Example 1: The set C comprised of the intersection of the sets A={1, 2, 3} and B={4, 5, 6}. Since A and B have no common members, C is empty.

Example 2: The set of all coins in my pockets. If my pockets are empty, then the set is empty.

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## BaalChatzaf

As Aristotle said, Nature abhors a vacuum.

Ba'al Chatzaf

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## merjet

Better: Logicians and mathematicians like completeness.

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## BaalChatzaf

And they shall not have it. The famous Goedel incompleteness theorem shows that any consistent first order logic with the postulates of arithmetic added is incomplete. That is there are closed formulate (no free variables) such that neither they nor their negations are theorems of the system. In a word, provability cannot catch up with truth.

Ba'al Chatzaf

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