Maybe you are right: sets cannot be empty. — Mephist
This is a "set_or_nothing", not a "set" — Mephist
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
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? — fishfry
When I put the same question to Metaphysician Undercover, he admitted that it's not the empty set he objects to, but rather the entirety of set theory. That's a nihilistic position but at least it's a position. You have none that I can see. — 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
I didn't change idea: there is no contradiction in the axiomatic definition of sets given by ZFC, at least for what has been discovered until now. It has not even been proved that ZFC is not contradictory, however; but since nobody has found any contradiction in ZFC after 100 years of using it, I would guess that it is consistent.
By the way, dependent type theory - at least a subset of the version used in coq - has been proved to be consistent (but of course it is not complete - no way to avoid Godel's incompleteness theorem). — Mephist
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
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
For Mephist's part, he read a book on category theory but knows very little actual math — fishfry
For example jgill or anybody else that can be surely qualified as a mathematician. Could you please . . . — Mephist
Your "proof" of inconsistency, as I just said before, is not something that contemporary mathematics would accept as valid. — Mephist
When I challenged you on this point, you admitted that it's not only the empty set, but set theory in its entirety that you object to. — fishfry
Yes, however in my opinion Anders Kock's book ( https://users-math.au.dk/~kock/sdg99.pdf ) is not so difficult to understand. d in my opinion should not be interpreted as a number, but as the base of a vector space made of infinitesimal numbers attached to each of the real numbers of a "base" space — Mephist
Can you show me a proof of consistency of ZFC set theory that doesn't make use of another even more complex and convoluted set theory? — Mephist
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
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
And there you have it from a man (presumably) who cannot process that some things are true by definition within the context of their use.I'm demonstrating to you, that the premises of your logic are false, and you reply, that doesn't matter because for me, and for everyone who uses my system the premises are true, — Metaphysician Undercover
What I object to is the claim of "existence" for objects which have a contradictory description. This is not nihilistic, but a healthy skepticism. The attitude demonstrated by you, that we might assign "existence" arbitrarily is best described as delusional. — Metaphysician Undercover
Of course, the nature of the empty set is essential to understanding what a "set" is, and if a theory has contradictory premises, then I object to the theory in its entirety, it needs to be reformulated — Metaphysician Undercover
It's perfectly possible (and probable) that I wrote something wrong, but I would like to know what's the mistake that I made. — Mephist
By the way, dependent type theory - at least a subset of the version used in coq - has been proved to be consistent (but of course it is not complete - no way to avoid Godel's incompleteness theorem). — Mephist
I don't understand what I am wrong about. — Mephist
I said there is no proof that ZFC is inconsistent (meaning: nobody has never derived a contradiction from ZFC's axioms), but there is even no proof that ZFC is consistent. — Mephist
That's why I prefer type theory to ZFC. — Mephist
Type theory is weaker but is provably consistent. — Mephist
Can you show me what I said wrong? — Mephist
I think the sets that are defined in ZFC are a hierarchical tree-like structure that can be used to model the relation "belongs to" at the same way as the leaves of a tree "belong to" it's root. — Mephist
It lacks symmetry and is too complex. — Mephist
I think in the future it will be substituted by a more elegant and simpler definition. — Mephist
I think it does not correspond to anything in the physical world, so basically yes: it's just an imaginary gadget that obeys the rules of set theory, ad it could be substituted by other similar gadgets that logically equivalent to it. — Mephist
Can you show me a proof of consistency of ZFC set theory that doesn't make use of another even more complex and convoluted set theory? — Mephist
Here's a formal proof in Coq that the Calculus of Constructions is sound: http://www.lix.polytechnique.fr/~barras/publi/coqincoq.pdf — Mephist
I'll not discuss about the empty set any more. Yes, you are right. The empty set exists. You win! — Mephist
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