I came to this forum about 6 months ago because I wanted to present some ideas that in my opinion are original but are not completely formalized (unti...
Yes, that's correct. You can consider a "topos" as a generalized class of all sets. So, the sets are the objects of the category. The final object of ...
Yes, exactly. The intuition of the real number line, in my opinion, is not mutilated but simply different: you have a base space of points that can be...
I would like to hear the opinion of other real mathematicians about what I wrote. For example @"jgill" or anybody else that can be surely qualified as...
I don't understand what I am wrong about. I said there is no proof that ZFC is inconsistent (meaning: nobody has never derived a contradiction from ZF...
OK, I see it's not so easy to finish this discussion about the empty set... :meh: I didn't change idea: there is no contradiction in the axiomatic def...
:sad: I don't know. I have no more ideas how to explain it. Maybe you are right: sets cannot be empty. So you have to define another thing, named "set...
:smile: good to know. Of course you don't have to believe me as a matter of principle. Usually I make a lot of mistakes when I write. Yes, of course. ...
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 n...
OK, I see: This implies that if one reads for example xd1 = 8d1 this not necessarily means x = 8. However if xd = 8d ?d ? x = 8. I don't really unders...
But you cannot multiply by d. You can multiply by (0, d*1), for example, not by d. All non-zero elements are all the elements of the form (x, d*y) whe...
Yes, but d is not a real number. d is a linear operator (like derivatives). The real numbers are of the form (a, d*b). In this case, for example, (0, ...
Why not? Which of the field axioms are not satisfied? Yes, that's true. All subsets of the real line are open, so all functions are continuous (and di...
Hmm... sorry, I didn't even read @"aletheist" posts :gasp: OK, now I read it, but I don't quite agree on all that he writes For example example this p...
Not sure what are R and D in that formula. In Anders Kock's book ( https://users-math.au.dk/~kock/sdg99.pdf ) D is an infinitesimal interval centered ...
OK, let's follow you definition of "set" (that is not the definition used in ZFC set theory, but we are considering an alternative definition because ...
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...
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 mathema...
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 s...
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 ...
I see that I didn't answer on the main topic here, that was about extensionality. The fact that "sets are more fundamental than their elements" is tru...
I was referring to the natural number zero. Natural numbers in set theory are defined as sets: the natural number N is a set that contains N elements....
Yes, that's the same kind of function. The point is that you can have a function whose codomain depends on the argument of the function. In type theor...
Actually, this vector field is a good example of a dependently-typed function. The domain of the function is the surface of the sphere, but what is it...
OK, so I have a question: does the number zero exist? Where's the difference between the number zero and the empty set? In category theory sets are de...
P.S. If you don't like my example because it's made of finite sets, you can "fill the squares" of the total space (it will become a Mobius strip), and...
Yes! :up: Yes! (even if this is not related to the topology of your sets) Moreover, in this case the topology of the total space (the space made of ve...
In my opinion, the misleading part of that example is that the tangent planes seem to have some points in common, since they are immersed in an ambien...
The usual intuition is more like an "airbrush" ( https://en.wikipedia.org/wiki/Fiber_bundle ). The fibers are seen as stick wires coming out from a co...
Well, OK, never mind. However, the book that I gave you the link is very clear and contains proofs and exact definitions. Surely that's easier to unde...
I don't know what are "L-structures", but I think I know what's the source of misunderstanding: the words "discrete" and "continuous" used to refer to...
Yes, but that's not mathematics! The distinction of which concepts are more "fundamental" is very useful to "understand" a theory, but it cannot be ex...
There is a way to translate any mathematical proposition (or axiom) into plain English, but there is no way to translate any English proposition into ...
(continuation: correspondence between a topological space and how proof works in the logic) The rules of logic should be valid for ANY topology and AN...
What I wrote is only an idea, that (in my opinion) is important to understand the "meaning" of a theory, but from the point of view of mathematics all...
I see that there is a misunderstanding between us on what it means "a logic has a model". A logic is a bunch of rules that describe how you can build ...
Comments