If yes, can you please try to explain this in a better way? I don't even have much time for this, sorry. I have even to go to the hospital for a couple of days next week. — Mephist
I apologize for interrupting a productive flow of thought. But I was curious what you guys were talking about. Seems pretty esoteric. :chin: — jgill
I apologize for interrupting a productive flow of thought. But I was curious what you guys were talking about. Seems pretty esoteric. — jgill
What do logic and topology have to say about each other?
Specifically; if a logic has a model is there a correspondence between a topological space on the set which models it and how proof works in the logic? — fdrake
Now, the essential change in the point of view that allows you to see the correspondence between topology and logic it this one: consider sets to be more "fundamental" than their elements.
So, if our model are the real numbers, the sets of real numbers are more "fundamental" than the single real numbers. If you think about it, that's what boolean algebra does: boolean algebra speaks about sets and operations between sets (union, intersection, complement): you build sets starting from other sets, without mentioning their elements. — Mephist
For a very nice overview of the history and meaning of all this I recommend the prologue of Mac Lane's Sheaves in Geometry and Logic. One need not understand the details to get the big picture from this very clearly written book. — fishfry
Aristotle demonstrated this premise, that the Form, as a universal type, (what you call "the set") is more fundamental than its elements, leads to an infinite regress and is actually impossible, therefore false. — Metaphysician Undercover
Making the "One" the most fundamental resolves the inherent contradiction of having the empty set as fundamental. The empty set is inherently contradictory because it is something, an object, which at the same time must be nothing. — Metaphysician Undercover
I can summarize. Short answer is that these days you can do logic via category theory; and when you do that, you get intuitionist logic (denial of the law of the excluded middle (LEM) and all that) in a natural way. — fishfry
Combinations, not permutations; i.e., the different proper subsets, and the order of the members does not matter. For a set with n members, its power set has 2^n members.2) Is there an error in thinking of a representation of a powerset as all the permutations of the elements of the original set? — tim wood
This was Cantor's view, which is fairly standard among mathematicians today. However, there is a power set for the real numbers, and a power set for that power set, and so on ad infinitum. That being the case, some argue that the real numbers are not truly continuous, despite comprising what is conventionally called the analytical continuum.4) But if 3, and there is no such point on the line, then (it appears to me) that c = P(N). — tim wood
There are no points in a truly continuous line, period. As a one-dimensional continuum, its parts are all likewise one-dimensional, rather than dimensionless points. We could hypothetically mark points on a line of any multitude--including that of the real numbers and that of their power set--or even beyond all multitude.Is 5 the true statement, that there are points on the line to which no real number can be applied? — tim wood
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 explanations that you can give by words are worth nothing: at the end, the only thing that counts in a mathematical argument are proofs. If what you say cannot be proved, it's not mathematics. I know, neither of us presented any proof of what we said here, but we are on a philosophy forum here, right? — Mephist
"Since the cardinality of the set R of reals is the same as that of the powerset P(N) of the set of natural numbers."
What do logic and topology have to say about each other?
Specifically; if a logic has a model is there a correspondence between a topological space on the set which models it and how proof works in the logic? — fdrake
It's not true that words are worth nothing in mathematics, because the axioms are written in words. My demonstration was a proof, a logical proof that a set cannot be more fundamental than its elements, because that creates an infinite regress. If you are satisfied with an infinite regress you have an epistemological problem. Such mathematics is not supported by sound epistemology. — Metaphysician Undercover
I see mathematical axioms expressed in plain English. — Metaphysician Undercover
From page 4 of the the text referenced:
1) "Since the cardinality of the set R of reals is the same as that of the powerset P(N) of the set of natural numbers."
Please help me out? — tim wood
2) Is there an error in thinking of a representation of a powerset as all the permutations of the elements of the original set? — tim wood
3) if 1 and 2 are correct (and if 2 is correct, then I'm thinking 1 obviously follows), then the question of the cardinality of the continuum, c, becomes the question of the existence of point on the line to which no real number can be applied - for some reason: is this a correct way to think of it? — tim wood
4) But if 3, and there is no such point on the line, then (it appears to me) that c = P(N). — tim wood
5) And it cannot be that simple. which implies there are points on the line that cannot be numbered. — tim wood
6) By "number on the line," I am assuming that to each point on the line is assignable some unique number representable as, say, some numeral in binary form, all of which points/binary numerals represented in the set of permutations of all the zeros and ones. — tim wood
Is 5 the true statement, that there are points on the line to which no real number can be applied? — tim wood
I see that there is a misunderstanding between us on what it means "a logic has a model". — Mephist
consider sets to be more "fundamental" than their elements. — Mephist
There are no points in a truly continuous line, period. — aletheist
— tim wood
Let's start a list of them all.
.1
.01
.11
.001
.011
.101
.111
.0001
.0011
.0101
.0111
.1001
.1011
.1101
...
You get the idea.
This list will eventually take in all the numerals of denumerable length. — tim wood
No, your idea only lists all the bitstrings of FINITE length, of which there are only countably many. For example 1010101010101010... never appears on your list. — fishfry
I found a paper that indicated the the fibers are "L-structures." Not too sure what those are, or what the base set is. I'm not sure I entirely believe it's a discrete topological space. I'm thinking you've probably explained this point to me several times over but I still don't get it. My apologies for giving you a hard time out of frustration at my inability to understand how fiber bundles can be used to model logical structures. — fishfry
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
I'm thinking the number of digits must be countable. And I'm thinking my listing, then, being ordered, is also countable. It's all countable. But clearly that's not correct. — tim wood
But I especially chose a finite set to make it crystal clear — Mephist
Not clear to me. I literally and honestly did not understand what you said in this post. Perhaps it's a lost cause. — fishfry
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 understand than my explanations... — Mephist
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.