Infinite Entities As "concepts"

[HaloNoble6]

Indeed, it seems like "cardinality" is arbitrary.

[LauricAcid]

If points are only units of measurement, then something that has no length, width, or height is being used to measure.

Instead, don't you think the differences between points are the units of measurement, not the points themselves?

[HaloNoble6]

What is your definition of point?

[LauricAcid]

"{ø} = {N sub a}

This is a false statement.

{ø} = N sub a

This is a meaningless statement." [Necessary_Truths]

What does 'N sub a' stand for?

"[...] it's hard to see what it could mean to say that the empty set is equinumerous with a natural number. "

The empty set is a natural number, and every set is equinumerous with itself, therefore the empty set is equinumerous with a natural number.

"I reject the concept of cardinality on the basis that you can't have different measurements (levels) of immeasurability (infinity)."

First, the empty set is not infinite. It's finite. Do you object to speaking of the cardinality of finite sets?

Second, I grant that defintion of infinite cardinality depends on the axiom of choice. But even if you reject the axiom of choice, infinite cardinality would be meaningful in the conditional sense of the axiom being an antecedent in conditional statements.

Third, even without the axiom of choice, and without a definition of cardinality, it is still a theorem that for any set x, there exists a set y, such that a proper subset fo y dominates x (informally speaking, for any set, there's always a larger set).

"The empty set is something. It is the set having no members." [LauricAcid]

"This is a contradiction." [Necessary_Truths]

No, it is not.

First, you have to understand that the word 'the empty set' is a defined term and that it is defined by the only primitive term of set theory (other than identity), which is the 2-place predicate symbol we informally think of as the predicate symbol for the element relation.

The empty set axiom asserts that there exists at least one thing that has no members. (Recall that not even 'set' is an official term of Z set theory). What this thing is is not specified. The axiom of extensionality ensures that there is exactly one such set that has no members. Then we define the empty set to be that set.

This is not contradictory since we can easily show models of the those two axioms.

"A set is only a collection of elements, it is not the collection of elements plus the box you've put them in."

I haven't mentioned any box, nor does set theory.

I recognize that in the everyday sense of 'set', a set is a colllection of elements, so one would think that there is no set that has no elements. But set theory does diverge from this everyday sense. If this bothers you, then you should recognize that the intent of set theory is not to overturn anyone's everyday use of the word 'set' nor to convince anyone of a particular ontology. In set theory (as opposed to class theory), the word 'set' is not part of the official system and is used just to easily convey things about the theory that can be conveyed without using the word 'set' but only at the price of awkardness or lack of informal visualization.

In other words, instead of 'an empty set', we could say 'a memberless object'. Then we agree to countenacne, for the purpose of a theory, a domain in which there is only one memberless object, so in those domains there is 'the memberless object'. That's all there is to it.

"The empty set is nothing, literally."

What do you think the number zero is? Is it a number, which is something, or is it not a thing at all?

/

"Indeed, it seems like "cardinality" is arbitrary." [Felipe]

I suspect that you don't take the notion of cardinality of finite sets to be arbitrary. So it's the notion of cardinality of infinite sets that you think is arbitrary.

Is the notion of equinumerosity arbitrary? Is the notion of well order arbitrary? Is the notion of a choice function arbitrary? If not, then why would infinite cardinality, which is defined on the basis of those concepts, be any more arbitrary?

Edited May 2, 2005 by LauricAcid

[LauricAcid]

"What is your definition of point?"

I started by asking what you think a point is.

Edited May 2, 2005 by LauricAcid

[HaloNoble6]

I did give you a definition, but then you proceeded to call it wrong using your own. So I ask, what is your definition of point?

[HaloNoble6]

The arbitrary aspect of cardinality is that zero is a cardinal number, which means that a set with cardinal number zero is a finite set, yet the term finite means measurable, and "emptiness" is not a measure.

[LauricAcid]

I didn't say that your definition is incorrect, but only pointed to a difficulty with it. I thought that you'd stick to your definition but resolve the difficulties with it.

My defintion of point is: An element in a set. And my definition of 'a space' is 'a set'. Some points have no width, length, or height. And in some contexts, a point is thought of as a location in a space.

So, since, as I understand you, you consider a space without physical entities to contain nothing (or the space is nothing), I wonder what you consider the locations in such a space to be.

Edited May 2, 2005 by LauricAcid

[LauricAcid]

"The arbitrary aspect of cardinality is that zero is a cardinal number, which means that a set with cardinal number zero is a finite set, yet the term finite means measurable, and "emptiness" is not a measure."

Set theory doesn't define 'finite' as a synonym for 'measurable' (and what are your definitions of 'measure' and 'measurable'?) and doesn't impart that emptiness is a measure. So the contradictions you claim don't exist.

And the finititeness of the empty set does not follow from the empty set being a cardinal number. The finiteness of the empty set follows from the empty set being a natural number.

Edited May 2, 2005 by LauricAcid

[HaloNoble6]

I'm not sure that delving into correlating set theory with understanding "space" gets us anywhere. The fact remains that it makes no sense to talk about the size of nothing. Measuring space is in essence measuring the size of an entity that could fit somewhere in space. To measure space as apart from existents is to not measure at all, because only existents are measurable, since only existents are finite.

[HaloNoble6]

I meant to say that the term finite implies measurably, and measurably implies the existence of something being measured.

You say that all natural numbers are finite, how is zero finite? Zero might be an element in the naturals, but it doesn't represent anything but absence.

My definition of finite implies measurably, and measurably implies the existence of something that has extent of some kind. Zero has no extent, it is absence.

[LauricAcid]

I don't insist that you correlate set theory with your views.

But I do wonder what you take to be the ontological status of points in space. For that matter, I wonder what you take to be the ontological status of any mathematical object.

[LauricAcid]

"[,,,] how is zero finite? Zero might be an element in the naturals, but it doesn't represent anything but absence."

By the way you use the term 'finite' and 'zero', I cannot unravel whether zero is finite (since you haven't defined 'measurable' and your definition of 'zero' is not mathematical).

By the way mathematicians use these terms, the finititude of zero follows from zero being a natural number.

[HaloNoble6]

Good, now I understand where you're getting at. I think mathematical objects do not necessarily refer to entities, but that they are concepts of method to explain something about entities. What I mean is, for example, the imaginary number does not represent anything real, but it is a concept used to simplify and compact mathematics in a very elegant way, which inevitably is used to say something about entities and reality.

The body of concepts in mathematics are, in a literal sense, only representations, or concepts of method, not entities. Mathematics can be practiced in way that is completely disjointed from reality, from referring to actually entities. And so, when a mathematician takes the step to represent something in reality with one of his concepts of method, say the concept "variable," that is when mathematics has metaphysical meaning.

I've not thought much of this, but this is very interesting. I'm typing this all on the fly, so forgive me if I stumble a bit.

[LauricAcid]

I think your answer has a lot of good sense in it.

But what is the ontological status of concepts? If we have a concept of a location in space without physical entities, then are we divorced from reality? If not, then what is the ontological status of our concept of a particular location in a space without physical entities? We speak as if the location is "in" the space. Perhaps that is just figurative language. Very well, but then is the location the same thing as the concept of the location? Ah, but you might say, there is no location, only the concept of the location. But what if we back that idea up all the way to natural numbers? Are there natural numbers or only concepts of natural numbers?

Edited May 2, 2005 by LauricAcid

[HaloNoble6]

There are many kinds of concepts at our disposal, and one is a concept that refers to an existent. This is why "empty space" is a concept of method, because it does not directly (but parenthetically) refer to an existent.

I think, in a literally sense, concepts don't exist, but some concepts refer to existents. That is, concept qua concept doesn't exist, they are mental tools we use to refer to reality.

So, if we have an idea of a physical location in space that does not contain physical entities, our idea does not refer to entities, but for the idea to make any sense, we parenthetically, in the back of our minds, know that only entities are measurable and it is the existents of entities that allow us to discuss "empty space," for "empty space" is the absence of the entities that we would otherwise have measured.

To answer your last question, location is not the same thing as the concept of location. Concepts can refer to existents, but they are not the existent.

Same goes for a set like the natural numbers--this set qua set doesn't actually exist, it is a tool that can be used to refer to things that do exist. This doesn't invalidate the concept, it just defines when and how it can be used interchangeably with actually things that exist.

[LauricAcid]

The first four paragraphs of your post make good sense. And I think it needs to be seen how set theory employs what you call concepts of method and that these concepts of method do not entail the ontological absurdities that detractors claim them to entail.

In the present case, the set of natural numbers is an extremely useful concept of method. But to posit (or conceive, as a concept of method, mind you) the existence of the set of natural number is to posit the existence of an infiinite set. And once you've posited the existence of an infinite set, the other (pretty non-controversial) axioms ensure that there are even larger infinite sets. We can't get around that. It's a cold stone truth of logic that if we posit just one infinite set, then (with the other non-controversial axioms) we have to accept that there are sets of even larger infinititude than that of the set of natural numbers, even if only as concepts of method. There shoud be no railing against this. There should be no charges of 'mysticism' or 'false ontology' or any of that.

On the other hand, one is free to reject the axiom of infinity as even a working concept of method. But then it's not clear how one would derive even the basic mathematics of the real numbers, let alone the basic set theory needed to perform the meta-logic needed for first order logic itself. Also, one may reject classical logic and adopt instead an intuitionist logic or some other system. But these have drawbacks too, and are usually much less intuitive and much more complicated just to get off the ground with the formal semantics than classical logic.

"[...] a set like the natural numbers--this set qua set doesn't actually exist, it is a tool that can be used to refer to things that do exist."

1. Aside from the set of natural numbers, which is an infinite set, my question was what do you take to be the ontological status of the numbers themselves. For example, what do you think the number two is? Is it an existing entity or is it a concept of method?

2. If you don't take the set of natural numbers to be an existing set but only a concept of method, then why take any set to be anything other than a concept in method?

Edited May 2, 2005 by LauricAcid