First let me put this in context. You said the empty set doesn't exist. — fishfry
One possibility, seemingly proposed by Mephist (if I understood your responses to MU anyway) is that you put in the empty set as a proper class primitive into the theory, so that (1a) is denied but (1) and (2) are still true — fdrake
You get something really similar to that with any mapping t(k):D→Mt(k):D→M, where MM is some manifold in which xx is a point. With the constraint that t(0)=xt(0)=x, the collection of all such maps forms a module ("vector space with elements from a ring"). It's an infinitesimal tangent space attached to the point. It might mutilate intuitions of the real number line, but that doesn't matter, as it seems designed to simplify language and proofs about smooth functions. Whether it's "wrong" or not is just a question of taste and application. — fdrake
As I'm sure you know, if a theory's consistent and has arithmetic, it can't prove its own consistency. You always have to go outside a theory to prove that theory's consistency; so consistency of system X is always just relative consistency to some other system Y. If you want to find a model of some axiom system, you need to construct the model through other rules (even if they're incredibly similar). — fdrake
I'd've thought you'd be quite happy with small categories? :brow: Aren't the models of intuitionistic logic Heyting algebras (from earlier) anyway? — fdrake
They're sets, or categories which are represent-able as sets. So I'm reading this like: "I don't like sets because I don't like the structures that establish ZFC has models. But I like intuitionist dependent type theory because it has a model! (Which is a collection of sets.)" — fdrake
For Mephist's part, he read a book on category theory but knows very little actual math — fishfry
Was that a yes or a no? Stop dancing. You're wrong on the facts, wrong on the math. Why are you trying to placate Metaphysician Undercover's nutty ideas? — fishfry
What do you think a set is, if not anything that obeys the rules of set theory? — fishfry
The fact that he's confused about the empty set, even when shown its existence proof from the axioms of set theory — fishfry
Mephist seems to have no rebuttal to the arguments which demonstrate that the "empty set" is a contradictory concept, and unlike you, seems about ready to face the reality of this. — Metaphysician Undercover
What I object to is the claim of "existence" for objects which have a contradictory description. — Metaphysician Undercover
Does the smiley mean that you don't actually believe what you wrote but that talking to Metaphysician Undercover has caused you to lose your grip? What does the smiley mean? Why did you claim there is no empty set? If you so claim, what do you do with the brief existence proof I just gave? — fishfry
I appreciate your efforts to make sense of this for me. I am not just trolling. — Metaphysician Undercover
To define "extension" as a property which something can both have and have not, at the same time, is just a trick of sophistry, designed to dodge application of the law of non-contradiction — Metaphysician Undercover
I don't see this. I cannot see how you made the contradiction go away. All I see is a trick of sophistry, which hides the contradiction behind the illusion that zero extension is some sort of extension. — Metaphysician Undercover
A segment is defined by giving two points. if the two points are coincident (the same point), then it's the same thing as only one point. If the two points are distinct, the length is the measure of their distance from each-other. I don't see any problem with this definition.So let's look at this example of the segment and the point. You define "point" using "segment". A point is a segment without any length. So the property here is "length". The definition of "length" as you described with "extension" is irrelevant. But from my perspective we need to ensure that "length" is not a sort of property which a thing can both have and not have, at the same time, or else the definition of length would become relevant, as a sophistic trick. The subject, or category is "segment", a point is defined as a type of segment. So we now need a definition of "segment" to make sense of what a point is. Remember that we have just allowed for a segment with no length, so "length" cannot be a defining feature of "segment". How would we proceed to define "segment" now? — Metaphysician Undercover
I submit that this is a similar situation to what we have with "set". If we define "empty set", such that it is a real set which has no extension, then "extension" cannot be a defining feature of "set" without allowing that "extension" is a sort of property which defies the law of non-contradiction.. — Metaphysician Undercover
No I don't see any contradiction here. There is nothing to imply that a point is a line without length. That would be contradictory when #2 says that a line is length — Metaphysician Undercover
Let's look at it this way. The "line" introduces a new property which the preceding "point" has not, "length". The "surface" introduces a property which the preceding "line" has not, "breadth". But the "surface" also maintains the property of the "line", which is "length". Following this pattern, the "line" ought to maintain the property of the "point". But "no parts" is a sort of negation of a property, instead of a proper property. We can say that this negation is the property which the "point" has. So if we understand "no parts" as a negation of all properties, we'd have to understand the property of the point as "no properties", and this would be contradictory. It would be contradictory, to say that the property of a thing is that it has no properties. However, we do not understand "no parts" as "no properties", so the point is defined by what it does not have, and what it does have is left empty or undefined.
In the case of the "empty set", the property which it does not have is extension. However, being designated as a "set", it also has whatever property is proper to a "set". If extension is a defining property of a set, we have contradiction because we talking about a thing which is said to have extension (by the type of thing that it is said to be), yet it is also said to have no extension by the value given to that property. — Metaphysician Undercover
All you need to do to turn any non-zero dividing element into a zero dividing element is to multiply it by d. So all elements zero divide. — fdrake
the construction doesn't satisfy the field axioms — fdrake
Moreover, every function from the constructed number line with infinitesimals to itself turns out to be smooth (so, continuous and differentiable). If I've read right anyway. — 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. — aletheist
"Contradiction" has nothing to do with identifying things with physical objects, it relates to how words are defined. So for example, if "set" is defined as something having extension, and "empty set" is defined as a set having no extension, then there is contradiction here. "Empty set" breaks the rules expressed in the definition of "set", and therefore cannot be a set. — Metaphysician Undercover
(https://www.brainyquote.com/quotes/bertrand_russell_402437)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.
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
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
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 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
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
But what type of meaning could this be, if when it is represented in English it is contradictory? — Metaphysician Undercover
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
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
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
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
The problem is that your demonstration, through this technique, produces a misunderstanding of the theory, rather than an understanding. So the criticism is of your technique. You describe topology through reference to set theory, but to understand set theory requires an understanding of extensionality. You demonstrate a misunderstanding of extensionality. The fundamental assumption that a set has extension negates the possibility of an empty set. Therefore your demonstration, which places the set as more fundamental than its elements, implying an empty set, is a demonstration of misunderstanding.
The axiom of extension dictates that a set's identity is established by its elements. Therefore a set without elements can have no identity as "a set", and is therefore not a set. Some set-theorists are wont to obscure this fact by saying that the empty set is unique, when in reality it is distinct from all other sets because it is not a set at all; it has no extension. As I explained, this problem was overcome thousands of years ago by making "One" the fundamental "unique set", as the term "unique" implies. — Metaphysician Undercover
This is a symbol, "0", or "zero". As you seem to be fairly well educated in mathematics, you'll know that it means different things in different contexts. Despite your claim that mathematical languages are very "formal", there is significant ambiguity concerning the definition of "zero". Do you agree that when we refer to "zero" as an existing thing, a number, like in "the number 0", it means a point of division between positive and negative integers? How is this even remotely similar to what "the empty set" means? — Metaphysician Undercover
It's not "the object which represents the empty set" which I am concerned about, it is "the empty set" itself which bothers me. It is a self-contradicting concept. If a set is to be something, an object, then, as an object, it cannot be empty because then it would be nothing. You would have an object, a set, which is at the same time not an object because it's composed of nothing.
So there is a distinction to be made between the definition of the set, "the set of...", and the actual set, or group of those things. If there is none of those defined things, then there is no group, or set of those things, such a defined set is non-existent. There is none of the describe things and therefore no set of those things. There is a defined set, "the set of..." which refers to nothing, no things. It is not an empty set, it is a non-existent set. Only through the category mistake of making the defined set ("the set of..."), into the actual set, can you say that there is this set which is empty. So if we allow that there is this actual set, the set of nothing, then the set becomes something other than the collection of things which forms that set. And we'd have no way to identify any set because the set would not be identified by the things which make it up. — Metaphysician Undercover
So if we have a vector at each point of a sphere that gives the wind direction and velocity at that point, that's a section of a fiber bundle. In set-theoretic terms a section is a right inverse of the projection map. That's how I think about all this. — fishfry
When you and I agree on something, that's really something to be afraid of; better move the hands on the doomsday clock. — Metaphysician Undercover
The problem is that your demonstration, through this technique, produces a misunderstanding of the theory, rather than an understanding. So the criticism is of your technique. You describe topology through reference to set theory, but to understand set theory requires an understanding of extensionality. You demonstrate a misunderstanding of extensionality. The fundamental assumption that a set has extension negates the possibility of an empty set. Therefore your demonstration, which places the set as more fundamental than its elements, implying an empty set, is a demonstration of misunderstanding. — Metaphysician Undercover
The axiom of extension dictates that a set's identity is established by its elements. Therefore a set without elements can have no identity as "a set", and is therefore not a set. — Metaphysician Undercover
Some set-theorists are wont to obscure this fact by saying that the empty set is unique, when in reality it is distinct from all other sets because it is not a set at all; it has no extension. As I explained, this problem was overcome thousands of years ago by making "One" the fundamental "unique set", as the term "unique" implies. — Metaphysician Undercover
Oh I see. Good point. Funny but it never occurred to me to be confused by that. The tangent planes are conceptual thingies attached to each point but they don't "intersect in 3-space" at all. The technical condition is that the total space is the disjoint union of the fibers. I suppose I like this example because it's nice and concrete. For example a vector field is a choice of a single vector from each fiber. So if we have a vector at each point of a sphere that gives the wind direction and velocity at that point, that's a section of a fiber bundle. In set-theoretic terms a section is a right inverse of the projection map. That's how I think about all this. — fishfry
In set theory class many moons ago I proved that "every surjection has a right inverse" is equivalent to the axiom of choice. That makes sense because it says we can always make a simultaneous choice of a tangent vector from each tangent plane. When I found out that a section is what differential geometers call a right inverse, I was enlightened. — fishfry
That's the intuition I'm working with at the moment, special case that it may be. — fishfry
I stand by my remark. The tangent bundle of a sphere is most definitely a fiber bundle. — fishfry
Can you give me the link you want me to look at? There's been so much back and forth and so many links — fishfry