As for transfinite math, it rarely if ever comes up in classical analysis. — jgill
As for transfinite math, it rarely if ever comes up in classical analysis. — jgill
Depends on what is meant by 'transfinite math'. 'transfinite' is just another word for 'infinite', and, of course, analysis uses infinite sets. Moreover, there are mathematicians who work (and not in obscurity) with higher cardinals vis-a-vis analysis, though that work might not be prominent in the bread and butter mathematics you have in mind. — TonesInDeepFreeze
How would a difference in size be established between two infinite sets when there is no counting involved?
I'm not sure, but I think bread and butter analysis might touch on the cardinality of the power set of the set of reals (?), but I don't have enough information to dispute that even higher cardinals don't come up much. — TonesInDeepFreeze
I don't see the point in saying that mathematics such as analysis doesn't use infinite sets, when plainly, at the very outset, to even start in the subject, we see that we are using infinite sets. — TonesInDeepFreeze
we could determine S is infinite either by stipulation—e.g., if we are considering the set of all natural numbers, then we thereby know that this set is infinite because there is an infinite amount of them. — Bob Ross
If S1 is a set with size 2 elements ad infinitum and S2 is a set with size 1 of elements ad infinitum, then S1 > S2 (and I don’t need to count them). — Bob Ross
if we are considering the set of all natural numbers, then we thereby know that this set is infinite because there is an infinite amount of them. — Bob Ross
What is the proper interpretation of the cosmological constant Λ? I understand that it corresponds to a vacuum energy density, pervading all reality. Such energy is called dark energy, I gather. Since I'm sketchy on field theory, I don't know how this goes, but somehow this energy density produces a repulsive force beween any two objects in spacetime (within each other's lightcones?). Matter remains cohesive because Λ is very small compared to other forces, so that its effects really only show at an intergalactical scale (megaparsec). — DanCoimbra
ow, somehow this leads to the expansion of the Universe even in the case where the Universe is finite and bounded, which is a possibility considered by cosmologists. In this case, the Universe is increasing in total size, but not increasing *into* anywhere, so it becomes bigger because it has more internal spatial structure. This is what I meant. Why do you think this is incorrect? — DanCoimbra
I just think if mathematical axioms are to be selected, they have to be such that they do not lead to what is contradictory to Existence/Truth (or just semantics in general). — Philosopher19
If a mathematician or a philosopher decides on an axiom or theory that requires belief in the following (or at least logically implies it or leads to it): Nothing can be the set of all things (which logically implies Existence is not the set of all existents), or one infinity is a different bigger than another (or is a different quantity than another), I believe that axiom or theory should be disregarded or at least viewed as contradictory to Existence/Truth (or at least contradictory to the semantic of infinity). — Philosopher19
if each cardinal is STRICTLY larger than the one before it, I suppose they do indeed have different sizes. — Vaskane
[set theory says] Nothing can be the set of all things (which logically implies Existence is not the set of all existents) — Philosopher19
If so, then you understand that a line of an interval of 2 represent twice the length, as the line of an interval of 1. And thus you're perhaps an even lower wisdom score than 8 after I already pointed out several times that there's an error in communication and even held myself accountable for that error, that you're too dumb to understand a line has length/area/size whatever the fuck you wanna call it, after I clearly stated a communication error upon the context ... I mean fuck dude, you're like Marine when he sees red. — Vaskane
It makes a real difference. By saying 'infinity' as a noun and then that there are different sizes of infinity is to picture an object that has different sizes. There is no such object in mathematics.
— TonesInDeepFreeze
I don't think I'm picturing an object. I think I'm just focused on the semantic of Infinity. — Philosopher19
Good faith in posting a critique of mathematics would entail at least knowing something about it.
— TonesInDeepFreeze
I think it is from all that I have seen and heard [...] — Philosopher19
Whether all that I have seen or heard is enough, is another matter. You don't think I have. I think I have. — Philosopher19
There is no object called 'Infinity' in the sense you have been using it.
Here is a way to say what you want to say:
In mathematics, there are sets that are infinite but that have different cardinality from one another.
Better yet:
If x is infinite then there is a y that is infinite and y has greater cardinality than x. — TonesInDeepFreeze
There is no object called 'Infinity' in the sense you have been using it. — TonesInDeepFreeze
There is no x such that for all y, y is a member of x iff y is not a member of y. Proof: — TonesInDeepFreeze
the axiom schema of separation — TonesInDeepFreeze
'There exists a z such that for all y, y is a member of z' contradicts this instance of the axiom schema of separation: For all z, there is a x such that for all y, y is a member of x iff (y is a member of z and yis not a member of y). — TonesInDeepFreeze
We do NOT claim that from "after each natural number there is a next number" and "there is no greatest natural number" that we can infer that there is a set of all the natural numbers. Indeed such an inference IS a non sequitur. And every mathematician and logician knows it is a non sequitur. So, we recognize that to have a set with all the natural numbers we need an AXIOM for that, which is NOT an inference. — TonesInDeepFreeze
Something cannot be both a member of itself and a member of other than itself at the same time. — Philosopher19
I'm not sure what you mean by "So, we recognize that to have a set with all the natural numbers we need an AXIOM for that, which is NOT an inference." — Philosopher19
So what semantic are mathematicians using when they use the world/label "infinite"? — Philosopher19
So what semantic are mathematicians using when they use the world/label "infinite"? — Philosopher19
I have an ability to understand concepts without even knowing of them — Vaskane
Great post, thanks. How do you prove then N is different size to P?Here is a finite definition of an infinite set: "A given set S is infinite iff there exists a bijective function between S and a proper subset of S." Furthermore, such a bijective function can be stated finitely.
Here is an example. Take the set of natural numbers ℕ = { 0, 1, ··· }. Now take a proper subset of ℕ containing only even the numbers, ℙ = { 0 , 2 , ··· }. These two are equinumerous because there is a bijective function f : ℕ → ℙ, given by f(n) = 2n.
The proof that "f" is bijective is finite. So is the proof that ℙ is a proper subset of ℕ. — DanCoimbra
In certain alternative set theories, there are sets that both members of themselves and of other sets. — TonesInDeepFreeze
We go in a circles, as it is with cranks. The crank makes false claims and terrible misunderstandings. Then the crank is corrected and their error is explained. Then the crank ignores all the corrections and just posts the false claims and misunderstanding again as if the corrections and explanations never existed. — TonesInDeepFreeze
Here's actually some advice to all non-mathematicians (from a non-mathematician):This isn't really the place to come to get people to agree with you. I think the math boys really did give you a good amount of feedback that would be hard to get anywhere else. So if you want to run something past us we'll tell you what we think and you can react accordingly. Most of what you say really irks a formally trained mathematician. — Mark Nyquist
How do you prove then N is different size to P? — Corvus
We go in a circles, as it is with cranks. The crank makes false claims and terrible misunderstandings. Then the crank is corrected and their error is explained. Then the crank ignores all the corrections and just posts the false claims and misunderstanding again as if the corrections and explanations never existed.
— TonesInDeepFreeze
Evidently, there's no point in continuing this discussion. — Philosopher19
If you believe your mathematics is free from contradictions or paradoxes — Philosopher19
I see no paradoxes or contradictions or foundational incompleteness in the beliefs that I uphold (mathematical or otherwise). — Philosopher19
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